List of small groups

The following list in mathematics contains the finite groups of small order up to group isomorphism.

Counts

For n = 1, 2, … the number of nonisomorphic groups of order n is
For labeled groups, see.

Glossary

Each group is named by Small Groups Library as G_oⁱ, where o is the order of the group, and i is the index used to label the group within that order.
Common group names:

Z_n: the cyclic group of order n
Dih_n: the dihedral group of order 2n
D_2n: the dihedral group of order 2n, the same as Dih_n
K₄: the Klein four-group of order 4, isomorphic to and Dih₂
S_n: the symmetric group of degree n, containing the n! permutations of n elements
A_n: the alternating group of degree n, containing the even permutations of n elements, of order 1 for, and order n!/2 otherwise
Dic_n or Q_4n: the dicyclic group of order 4n
Q₈: the quaternion group of order 8, also Dic₂

The notations Z_n and Dih_n have the advantage that point groups in three dimensions C_n and D_n do not have the same notation. There are more isometry groups than these two, of the same abstract group type.
The notation denotes the direct product of the two groups; Gⁿ denotes the direct product of a group with itself n times. G ⋊ H denotes a semidirect product where H acts on G; this may also depend on the choice of action of H on G.
Abelian and simple groups are noted.
The identity element in the cycle graphs is represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16.
In the lists of subgroups, the trivial group and the group itself are not listed. Where there are several isomorphic subgroups, the number of such subgroups is indicated in parentheses.
Angle brackets show the presentation of a group.

List of small abelian groups

The finite abelian groups are either cyclic groups, or direct products thereof; see Abelian group. The numbers of nonisomorphic abelian groups of orders n = 1, 2,... are
For labeled abelian groups, see.

Order	Id.	G_oⁱ	Group	Non-trivial proper subgroups	Cycle graph	Properties
1	1	G₁¹	Z₁ ≅ S₁ ≅ A₂	–	40px	Trivial. Cyclic. Alternating. Symmetric. Elementary.
2	2	G₂¹	Z₂ ≅ S₂ ≅ D₂	–	40px	Simple. Symmetric. Cyclic. Elementary.
3	3	G₃¹	Z₃ ≅ A₃	–	40px	Simple. Alternating. Cyclic. Elementary.
4	4	G₄¹	Z₄ ≅ Q₄	Z₂	40px	Cyclic.
4	5	G₄²	Z₂² ≅ K₄ ≅ D₄	Z₂	40px	Elementary. Product.
5	6	G₅¹	Z₅	–	40px	Simple. Cyclic. Elementary.
6	8	G₆²	Z₆ ≅ Z₃ × Z₂	Z₃, Z₂	40px	Cyclic. Product.
7	9	G₇¹	Z₇	–	40px	Simple. Cyclic. Elementary.
8	10	G₈¹	Z₈	Z₄, Z₂	40px	Cyclic.
8	11	G₈²	Z₄ × Z₂	Z₂², Z₄, Z₂	40px	Product.
8	14	G₈⁵	Z₂³	Z₂², Z₂	40px	Product. Elementary.
9	15	G₉¹	Z₉	Z₃	40px	Cyclic.
9	16	G₉²	Z₃²	Z₃	40px	Elementary. Product.
10	18	G₁₀²	Z₁₀ ≅ Z₅ × Z₂	Z₅, Z₂	40px	Cyclic. Product.
11	19	G₁₁¹	Z₁₁	–	40px	Simple. Cyclic. Elementary.
12	21	G₁₂²	Z₁₂ ≅ Z₄ × Z₃	Z₆, Z₄, Z₃, Z₂	40px	Cyclic. Product.
12	24	G₁₂⁵	Z₆ × Z₂ ≅ Z₃ × Z₂²	Z₆, Z₃, Z₂, Z₂²	40px	Product.
13	25	G₁₃¹	Z₁₃	–	40px	Simple. Cyclic. Elementary.
14	27	G₁₄²	Z₁₄ ≅ Z₇ × Z₂	Z₇, Z₂	40px	Cyclic. Product.
15	28	G₁₅¹	Z₁₅ ≅ Z₅ × Z₃	Z₅, Z₃	40px	Cyclic. Product.
16	29	G₁₆¹	Z₁₆	Z₈, Z₄, Z₂	40px	Cyclic.
16	30	G₁₆²	Z₄²	Z₂, Z₄, Z₂²,	40px	Product.
16	33	G₁₆⁵	Z₈ × Z₂	Z₂, Z₄, Z₂², Z₈,		Product.
16	38	G₁₆¹⁰	Z₄ × Z₂²	Z₂, Z₄, Z₂², Z₂³,	40px	Product.
16	42	G₁₆¹⁴	Z₂⁴ ≅ K₄²	Z₂, Z₂², Z₂³	40px	Product. Elementary.
17	43	G₁₇¹	Z₁₇	–	40px	Simple. Cyclic. Elementary.
18	45	G₁₈²	Z₁₈ ≅ Z₉ × Z₂	Z₉, Z₆, Z₃, Z₂	40px	Cyclic. Product.
18	48	G₁₈⁵	Z₆ × Z₃ ≅ Z₃² × Z₂	Z₂, Z₃, Z₆, Z₃²		Product.
19	49	G₁₉¹	Z₁₉	–	40px	Simple. Cyclic. Elementary.
20	51	G₂₀²	Z₂₀ ≅ Z₅ × Z₄	Z₁₀, Z₅, Z₄, Z₂	40px	Cyclic. Product.
20	54	G₂₀⁵	Z₁₀ × Z₂ ≅ Z₅ × Z₂²	Z₂, K₄, Z₅, Z₁₀		Product.
21	56	G₂₁²	Z₂₁ ≅ Z₇ × Z₃	Z₇, Z₃	40px	Cyclic. Product.
22	58	G₂₂²	Z₂₂ ≅ Z₁₁ × Z₂	Z₁₁, Z₂	40px	Cyclic. Product.
23	59	G₂₃¹	Z₂₃	–	40px	Simple. Cyclic. Elementary.
24	61	G₂₄²	Z₂₄ ≅ Z₈ × Z₃	Z₁₂, Z₈, Z₆, Z₄, Z₃, Z₂	40px	Cyclic. Product.
24	68	G₂₄⁹	Z₁₂ × Z₂ ≅ Z₆ × Z₄ ≅ Z₄ × Z₃ × Z₂	Z₁₂, Z₆, Z₄, Z₃, Z₂		Product.
24	74	G₂₄¹⁵	Z₆ × Z₂² ≅ Z₃ × Z₂³	Z₆, Z₃, Z₂		Product.
25	75	G₂₅¹	Z₂₅	Z₅		Cyclic.
25	76	G₂₅²	Z₅²	Z₅		Product. Elementary.
26	78	G₂₆²	Z₂₆ ≅ Z₁₃ × Z₂	Z₁₃, Z₂		Cyclic. Product.
27	79	G₂₇¹	Z₂₇	Z₉, Z₃		Cyclic.
27	80	G₂₇²	Z₉ × Z₃	Z₉, Z₃		Product.
27	83	G₂₇⁵	Z₃³	Z₃		Product. Elementary.
28	85	G₂₈²	Z₂₈ ≅ Z₇ × Z₄	Z₁₄, Z₇, Z₄, Z₂		Cyclic. Product.
28	87	G₂₈⁴	Z₁₄ × Z₂ ≅ Z₇ × Z₂²	Z₁₄, Z₇, Z₄, Z₂		Product.
29	88	G₂₉¹	Z₂₉	–		Simple. Cyclic. Elementary.
30	92	G₃₀⁴	Z₃₀ ≅ Z₁₅ × Z₂ ≅ Z₁₀ × Z₃ ≅ Z₆ × Z₅ ≅ Z₅ × Z₃ × Z₂	Z₁₅, Z₁₀, Z₆, Z₅, Z₃, Z₂		Cyclic. Product.
31	93	G₃₁¹	Z₃₁	–		Simple. Cyclic. Elementary.

List of small non-abelian groups

The numbers of non-abelian groups, by order, are counted by. However, many orders have no non-abelian groups. The orders for which a non-abelian group exists are

Order	Id.	G_oⁱ	Group	Non-trivial proper subgroups	Cycle graph	Properties
6	7	G₆¹	D₆ ≅ S₃ ≅ Z₃ ⋊ Z₂	Z₃, Z₂	40px	Dihedral group, Dih₃, the smallest non-abelian group, symmetric group, smallest Frobenius group.
8	12	G₈³	D₈	Z₄, Z₂², Z₂	40px	Dihedral group, Dih₄. Extraspecial group. Nilpotent.
8	13	G₈⁴	Q₈	Z₄, Z₂	40px	Quaternion group, Hamiltonian group. The smallest group G demonstrating that for a normal subgroup H the quotient group G/''H need not be isomorphic to a subgroup of G''. Extraspecial group. Dic₂, Binary dihedral group <2,2,2>. Nilpotent.
10	17	G₁₀¹	D₁₀	Z₅, Z₂	40px	Dihedral group, Dih₅, Frobenius group.
12	20	G₁₂¹	Q₁₂ ≅ Z₃ ⋊ Z₄	Z₂, Z₃, Z₄, Z₆	40px	Dicyclic group Dic₃, Binary dihedral group, <3,2,2>.
12	22	G₁₂³	A₄ ≅ K₄ ⋊ Z₃ ≅ ⋊ Z₃	Z₂², Z₃, Z₂	40px	Alternating group. No subgroups of order 6, although 6 divides its order. Smallest Frobenius group that is not a dihedral group. Chiral tetrahedral symmetry.
12	23	G₁₂⁴	D₁₂ ≅ D₆ × Z₂	Z₆, D₆, Z₂², Z₃, Z₂	40px	Dihedral group, Dih₆, product.
14	26	G₁₄¹	D₁₄	Z₇, Z₂	40px	Dihedral group, Dih₇, Frobenius group.
16	31	G₁₆³	K₄ ⋊ Z₄	Z₂³, Z₄ × Z₂, Z₄, Z₂², Z₂	40px	Has the same number of elements of every order as the Pauli group. Nilpotent.
16	32	G₁₆⁴	Z₄ ⋊ Z₄	Z₄ × Z₂, Z₄, Z₂², Z₂	40px	The squares of elements do not form a subgroup. Has the same number of elements of every order as Q₈ × Z₂. Nilpotent.
16	34	G₁₆⁶	Z₈ ⋊ Z₂	Z₈, Z₄ × Z₂, Z₄, Z₂², Z₂		Sometimes called the modular group of order 16, though this is misleading as abelian groups and Q₈ × Z₂ are also modular. Nilpotent.
16	35	G₁₆⁷	D₁₆	Z₈, D₈, Z₂², Z₄, Z₂	40px	Dihedral group, Dih₈. Nilpotent.
16	36	G₁₆⁸	QD₁₆	Z₈, Q₈, D₈, Z₄, Z₂², Z₂	40px	The order 16 quasidihedral group. Nilpotent.
16	37	G₁₆⁹	Q₁₆	Z₈, Q₈, Z₄, Z₂	40px	Generalized quaternion group, Dicyclic group Dic₄, binary dihedral group, <4,2,2>. Nilpotent.
16	39	G₁₆¹¹	D₈ × Z₂	D₈,, Z₂³, Z₂², Z₄, Z₂	40px	Product. Nilpotent.
16	40	G₁₆¹²	Q₈ × Z₂	Q₈, Z₄ × Z₂, Z₄, Z₂², Z₂	40px	Hamiltonian group, product. Nilpotent.
16	41	G₁₆¹³	⋊ Z₂	Q₈, D₈, Z₄ × Z₂, Z₄, Z₂², Z₂	40px	The Pauli group generated by the Pauli matrices. Nilpotent.
18	44	G₁₈¹	D₁₈	Z₉, D₆, Z₃, Z₂		Dihedral group, Dih₉, Frobenius group.
18	46	G₁₈³	Z₃ ⋊ Z₆ ≅ D₆ × Z₃ ≅ S₃ × Z₃	Z₃², D₆, Z₆, Z₃, Z₂		Product.
18	47	G₁₈⁴	⋊ Z₂	Z₃², D₆, Z₃, Z₂		Frobenius group.
20	50	G₂₀¹	Q₂₀	Z₁₀, Z₅, Z₄, Z₂		Dicyclic group Dic₅, Binary dihedral group, <5,2,2>.
20	52	G₂₀³	Z₅ ⋊ Z₄	D₁₀, Z₅, Z₄, Z₂		Frobenius group.
20	53	G₂₀⁴	D₂₀ ≅ D₁₀ × Z₂	Z₁₀, D₁₀, Z₅, Z₂², Z₂		Dihedral group, Dih₁₀, product.
21	55	G₂₁¹	Z₇ ⋊ Z₃	Z₇, Z₃		Smallest non-abelian group of odd order. Frobenius group.
22	57	G₂₂¹	D₂₂	Z₁₁, Z₂		Dihedral group Dih₁₁, Frobenius group.
24	60	G₂₄¹	Z₃ ⋊ Z₈	Z₁₂, Z₈, Z₆, Z₄, Z₃, Z₂		Central extension of S₃.
24	62	G₂₄³	SL ≅ Q₈ ⋊ Z₃	Q₈, Z₆, Z₄, Z₃, Z₂		Binary tetrahedral group, 2T = <3,3,2>.
24	63	G₂₄⁴	Q₂₄ ≅ Z₃ ⋊ Q₈	Z₁₂, Q₁₂, Q₈, Z₆, Z₄, Z₃, Z₂		Dicyclic group Dic₆, Binary dihedral, <6,2,2>.
24	64	G₂₄⁵	D₆ × Z₄ ≅ S₃ × Z₄	Z₁₂, D₁₂, Q₁₂, Z₄ × Z₂, Z₆, D₆, Z₄, Z₂², Z₃, Z₂		Product.
24	65	G₂₄⁶	D₂₄	Z₁₂, D₁₂, D₈, Z₆, D₆, Z₄, Z₂², Z₃, Z₂		Dihedral group, Dih₁₂.
24	66	G₂₄⁷	Q₁₂ × Z₂ ≅ Z₂ ×	Z₆ × Z₂, Q₁₂, Z₄ × Z₂, Z₆, Z₄, Z₂², Z₃, Z₂		Product.
24	67	G₂₄⁸	⋊ Z₂ ≅ Z₃ ⋊ D₈	Z₆ × Z₂, D₁₂, Q₁₂, D₈, Z₆, D₆, Z₄, Z₂², Z₃, Z₂		Double cover of dihedral group.
24	69	G₂₄¹⁰	D₈ × Z₃	Z₁₂, Z₆ × Z₂, D₈, Z₆, Z₄, Z₂², Z₃, Z₂		Product. Nilpotent.
24	70	G₂₄¹¹	Q₈ × Z₃	Z₁₂, Q₈, Z₆, Z₄, Z₃, Z₂		Product. Nilpotent.
24	71	G₂₄¹²	S₄	A₄, D₈, D₆, Z₄, Z₂², Z₃, Z₂		Symmetric group. Has no normal Sylow subgroups. Chiral octahedral symmetry, Achiral tetrahedral symmetry.
24	72	G₂₄¹³	A₄ × Z₂	A₄, Z₂³, Z₆, Z₂², Z₃, Z₂		Product. Pyritohedral symmetry.
24	73	G₂₄¹⁴	D₁₂ × Z₂	Z₆ × Z₂, D₁₂, Z₂³, Z₆, D₆, Z₂², Z₃, Z₂		Product.
26	77	G₂₆¹	D₂₆	Z₁₃, Z₂		Dihedral group, Dih₁₃, Frobenius group.
27	81	G₂₇³	Z₃² ⋊ Z₃	Z₃², Z₃		All non-trivial elements have order 3. Extraspecial group. Nilpotent.
27	82	G₂₇⁴	Z₉ ⋊ Z₃	Z₉, Z₃², Z₃		Extraspecial group. Nilpotent.
28	84	G₂₈¹	Z₇ ⋊ Z₄	Z₁₄, Z₇, Z₄, Z₂		Dicyclic group Dic₇, Binary dihedral group, <7,2,2>.
28	86	G₂₈³	D₂₈ ≅ D₁₄ × Z₂	Z₁₄, D₁₄, Z₇, Z₂², Z₂		Dihedral group, Dih₁₄, product.
30	89	G₃₀¹	D₆ × Z₅	Z₁₅, Z₁₀, D₆, Z₅, Z₃, Z₂		Product.
30	90	G₃₀²	D₁₀ × Z₃	Z₁₅, D₁₀, Z₆, Z₅, Z₃, Z₂		Product.
30	91	G₃₀³	D₃₀	Z₁₅, D₁₀, D₆, Z₅, Z₃, Z₂		Dihedral group, Dih₁₅, Frobenius group.

Classifying groups of small order

Small groups of prime power order pⁿ are given as follows:

Order p: The only group is cyclic.
Order p²: There are just two groups, both abelian.
Order p³: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p² by a cyclic group of order p. The other is the quaternion group for and Heisenberg group#Heisenberg group modulo an [odd prime p|the Heisenberg group modulo p] for.
Order p⁴: The classification is complicated, and gets much harder as the exponent of p increases.

Most groups of small order have a Sylow p subgroup P with a normal p-complement N for some prime p dividing the order, so can be classified in terms of the possible primes p, p-groups P, groups N, and actions of P on N. In some sense this reduces the classification of these groups to the classification of p-groups. Some of the small groups that do not have a normal p-complement include:

Order 24: The symmetric group S₄
Order 48: The binary octahedral group and the product
Order 60: The alternating group A₅.

The smallest order for which it is not known how many nonisomorphic groups there are is 2048 = 2¹¹.

Small Groups Library

The GAP computer algebra system contains a package called the "Small Groups library," which provides access to descriptions of small order groups. The groups are listed up to isomorphism. At present, the library contains the following groups:

those of order at most 2000 except for order 1024 ;
those of cubefree order at most 50000 ;
those of squarefree order;
those of order pⁿ for n at most 6 and p prime;
those of order p⁷ for p = 3, 5, 7, 11 ;
those of order pqⁿ where qⁿ divides 2⁸, 3⁶, 5⁵ or 7⁴ and p is an arbitrary prime which differs from q;
those whose orders factorise into at most 3 primes.

It contains explicit descriptions of the available groups in computer readable format.
The smallest order for which the Small Groups library does not have information is 1024.