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 their Small Groups library as G"o""i", where "o" is the order of the group, and "i" is the index of the group within that order.
Common group names:
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; "G""n" 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. (For groups of order , the simple groups are precisely the cyclic groups Z"n", for prime "n".) The equality sign ("=") denotes isomorphism.
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.
List of small abelian groups.
The finite abelian groups are either cyclic groups, or direct products thereof; see abelian groups. The numbers of nonisomorphic abelian groups of orders "n" = 1, 2, ... are
For labeled Abelian groups, see .
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
Classifying groups of small order.
Small groups of prime power order "p""n" are given as follows:
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:
The smallest order for which it is "not" known how many nonisomorphic groups there are is 2048 = 211.
Small groups library.
The group theoretical computer algebra system GAP contains 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:
It contains explicit descriptions of the available groups in computer readable format.
The smallest order for which the SmallGroups library does not have information is 1024.