Abelian subgroup structure of groups of order 16

Abelian normal subgroups of order 8
We can be even more specific regarding the count: for a non-abelian group of order 16, the number of abelian subgroups of order 8 is 1 iff the group has class exactly three (and this subgroup is the centralizer of derived subgroup, where the derived subgroup has order 4), and is 3 iff the group has class exactly two (and these 3 are precisely the subgroups of order 8 containing the center, which is a subgroup of order 4).
 * Existence: By the existence of abelian normal subgroups of small prime power order, there always exists an abelian normal subgroup of order 8 in any group of order 16. To see this, note that if $$n > k(k-1)/2$$, then any group of order $$p^n$$ contains an abelian normal subgroup of order $$p^k$$. Set $$p = 2, k = 3, n = 4$$ and get the desired conclusion.
 * Count: First, note that since index two implies normal, all abelian subgroups of order 8 are normal. Using the congruence condition on number of abelian subgroups of prime index, we get that the number of abelian subgroups is odd, i.e., it is one of the numbers 1,3,5,7,... In fact, we can say more: for a non-abelian group of order 16, the number of abelian normal subgroups of order 8 is either 1 or 3, and for an abelian group of rank $$r$$, the number is $$2^r - 1$$ (so one of the numbers 1,3,7,15).

Below is the information on abelian normal subgroups of order 8:

We now construct a table derived from the above, that lists the total number of abelian normal subgroups of order eight and exponent bounded by some specific number (2, 4, or 8).

Note that the number of abelian normal subgroups of order eight and exponent dividing two need not be odd. However, the number of abelian normal subgroups of order eight and exponent dividing four must be odd, and so must the number of abelian normal subgroups of order eight and exponent dividing eight. See congruence condition on number of abelian subgroups of order eight and exponent dividing four and congruence condition on number of abelian subgroups of prime-cube order.

Abelian normal subgroups of order 4
We note that all groups of order 4 are abelian, so congruence condition on number of subgroups of given prime power order yields that the number of abelian normal subgroups of order 4 is congruent to 1 mod 2, i.e., it is odd.

Below is the information on abelian normal subgroups of order 4:

Abelian normal subgroups of order 2
The abelian normal subgroups of order 2 are precisely the subgroups of order 2 contained inside the socle, which is the first omega subgroup of the center and is an elementary abelian 2-group. If the socle has rank $$s$$, the number of abelian normal subgroups of order 2 inside it is $$2^s - 1$$. Thus, all the counts here are among the numbers 1,3,7,15.

Abelian characteristic subgroups
For groups of order 16, there may or may not exist abelian characteristic subgroups. The situation is discussed below based on the nilpotency class:


 * For an abelian group, the whole group is an abelian characteristic subgroup, and is the unique subgroup that is maximal among abelian characteristic subgroups. There may also exist other nontrivial abelian characteristic subgroups.
 * For a non-abelian group of nilpotency class two, the center is an abelian characteristic subgroup of order four. There are two possibilities: either the group is a group in which every abelian characteristic subgroup is central (in which case the center is the unique maximum among abelian characteristic subgroups) or it is not, in which case there exists an abelian characteristic subgroup of order eight, which is a self-centralizing subgroup and hence a critical subgroup. The former case arises for direct product of Q8 and Z2 (ID: (16,12)) and central product of D8 and Z4 (ID: (16,13)) and the latter case (i.e., the existence of an abelian characteristic subgroup of order 8) occurs for all the remaining groups of nilpotency class two: SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16, and direct product of D8 and Z2.
 * For a group of nilpotency class exactly three, there is a unique subgroup maximal among abelian characteristic subgroups, which is also the unique abelian normal subgroup of order eight. This is the centralizer of derived subgroup and in all cases, it is isomorphic to cyclic group:Z8. It is a self-centralizing subgroup and hence is a critical subgroup.