Congruence condition on number of abelian subgroups of order eight and exponent dividing four

Statement in terms of universal congruence condition
The collection of abelian groups of order $$8$$ and exponent dividing $$4$$ is a fact about::collection of groups satisfying a universal congruence condition for the prime $$2$$.

Hands-on statement
Suppose $$G$$ is a finite p-group where $$p = 2$$. In other words, the order of $$G$$ is a power of $$2$$. Then, if $$G$$ has an abelian subgroup of order $$2^3 = 8$$ and exponent dividing $$4$$, the number of abelian subgroups of $$G$$ of order $$8$$ and exponent dividing $$4$$ is congruent to $$1$$ modulo $$2$$.

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