# Subgroups of order 8 in groups of order 16

This article contains summary information about all subgroups of order 8 inside groups of order 16. See also groups of order 16 | groups of order 8 | subgroup structure of groups of order 16 | supergroups of groups of order 8

This article describes the occurence of groups of order 8 as subgroups inside groups of order 16.

## Numerical information on counts of subgroups

### Counts of all subgroups

General assertion Implication for the counts in this case
congruence condition on number of subgroups of given prime power order The number of subgroups of order 8 is odd.
The number of normal subgroups of order 8 is odd.
The number of p-core-automorphism-invariant subgroups (which in this case means the number of subgroups invariant under automorphisms in the 2-core of the automorphism group) of order 8 is odd.
index two implies normal, or more generally, see equivalence of definitions of maximal subgroup of group of prime power order The number of subgroups of order 8 equals the number of normal subgroups of order 8.
formula for number of maximal subgroups of group of prime power order: In a finite $p$-group with minimum size of generating set $r$, the number of maximal subgroups is $(p^r - 1)/(p - 1)$. The number of subgroups of order 8 in a group of order 16 is 1, 3, 7, or 15, these values occurring when the minimum size of generating set is respectively 1, 2, 3, and 4. For a non-abelian group of order 16, the only possibilities are 3 and 7, corresponding to minimum size of generating set being 2 or 3.

### Counts of abelian subgroups

General assertion Implication for the counts in this case
existence of abelian normal subgroups of small prime power order: A group of order $p^n$ contains at least one abelian normal subgroup of order $p^r$ if $n > r(r-1)/2$. Any group of order 16 contains an abelian normal subgroup of order 8.
index two implies normal number of abelian subgroups of order 8 = number of abelian normal subgroups of order 8.
congruence condition on number of abelian subgroups of prime-cube order The number of abelian subgroups of order 8 is odd.
The number of abelian normal subgroups of order 8 is odd.
The number of abelian 2-core-automorphism-invariant subgroups of order 8 is odd.
congruence condition on number of abelian subgroups of prime index. Even more spceifically, the following: in a non-abelian group of order $p^n$, the number of abelian subgroups of index $p$ is 0, 1, or $p + 1$. In an abelian group, it is of the form $(p^r - 1)/(p - 1)$ where $r$ is the minimum size of generating set. In a non-abelian group of order 16, the number of abelian subgroups of order 8 is 1 or 3.
In an abelian group of rank $r$, the number is $2^r - 1$.