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 $2^r - 1$.

Counts of non-cyclic subgroups

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