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.