# Abelian subgroup structure of groups of order 2^n

This article gives specific information, namely, abelian subgroup structure, about a family of groups, namely: groups of order 2^n.
View abelian subgroup structure of group families | View other specific information about groups of order 2^n

## Abelian subgroup structure by order of groups $n$ $2^n$ Number of groups of order $2^n$ Information on groups Information on abelian subgroup structure
0 1 1 trivial group --
1 2 1 cyclic group:Z2 subgroup structure of cyclic group:Z2
2 4 2 groups of order 4 subgroup structure of groups of order 4
3 8 5 groups of order 8 subgroup structure of groups of order 8
4 16 14 groups of order 16 abelian subgroup structure of groups of order 16
5 32 51 groups of order 32 abelian subgroup structure of groups of order 32
6 64 267 groups of order 64 abelian subgroup structure of groups of order 64
7 128 2328 groups of order 128 abelian subgroup structure of groups of order 128
8 256 56092 groups of order 256 abelian subgroup structure of groups of order 256
9 512 10494213 groups of order 512 abelian subgroup structure of groups of order 512

## Summary on existence, congruence, and replacement

### Summary on existence

Note that if there exists an abelian normal subgroup of a particular order, there exist abelian normal subgroups of all smaller orders, because any normal subgroup contains normal subgroups of all orders dividing its order. $n$ $2^n$ Largest $k$ for which every group of order $2^n$ contains an abelian subgroup of order $2^k$ Corresponding value $2^k$ Largest $k$ for which every group of order $2^n$ contains an abelian normal subgroup of order $2^k$ Corresponding value $2^k$
0 1 0 1 0 1
1 2 1 2 1 2
2 4 2 4 2 4
3 8 2 4 2 4
4 16 3 8 3 8
5 32 3 8 3 8
6 64 4 16 4 16
7 128 4 16 4 16

### Summary on congruence condition

Below are values of $n$ and $k$, the $n$ values in the rows and the $k$ values in the columns. A "Yes" indicates that in any group of order $2^n$, the number of abelian subgroups of order $2^k$ is either 0 or odd. A "No" indicates that there exists a group of order $2^n$ where the number of abelian subgroups of order $2^k$ is a nonzero even number. $n$ $2^n$ $k = 0, 2^k = 1$ $k = 1, 2^k = 2$ $k = 2, 2^k = 4$ $k = 3, 2^k = 8$ $k = 4, 2^k = 16$ $k = 5, 2^k = 32$ $k = 6, 2^k = 64$
0 1 Yes
1 2 Yes Yes
2 4 Yes Yes Yes
3 8 Yes Yes Yes Yes
4 16 Yes Yes Yes Yes Yes
5 32 Yes Yes Yes Yes Yes Yes
6 64 Yes Yes Yes Yes Yes Yes Yes
7 128 Yes Yes Yes Yes Yes Yes (?) Yes Yes
8 256 Yes Yes Yes Yes Yes Yes (?) No Yes Yes