  In this note, we prove that $D_8\times C_2^{n-3}$ is the non-elementary
abelian $2$-group of order $2^n$, $n\geq 3$, whose number of subgroups of
possible orders is maximal. This solves a conjecture by Haipeng Qu [7]. A
formula for counting the subgroups of an (almost) extraspecial $2$-group is
also presented.
