Let $G$ be a finite $p$-group and let $n > 0$. Let $G^n$ be the direct product of $n$ copies of $G$.
Are all subgroups of $G^n$ isomorphic to $H_1 \times \dotsm \times H_n$ for some subgroups $H_1, \dots, H_n$ of $G$?
Comments. The question is being isomorphic to a direct product of subgroups, and not being equal to a direct product of subgroups. A negative answer to a similar question for $G = S_3$ and $n = 2$ was given here. This question and its answer, which relies on Goursat's lemma might also be relevant.
P.S. I am especially interested in the case $p = 2$.