Subgroups of $G^n$, where $G$ is a $p$-group

Question

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 $p = 2$.

Answer 1

Let $G=D_8 = \langle a,b \mid a^4=b^2=(ab)^2=1 \rangle$, $G^2 = \langle a_1,b_1\rangle \times \langle a_2,b_2 \rangle$ and $H = \langle a_1a_2,b_1,b_2,a_1^2,a_2^2 \rangle \le G$ with $|H|=32$.

If $H \cong H_1 \times H_2$, with $H_1$ and $H_2$ isomorphic to subgroups of $D_8$, then we must have $|H_1|=8$, $|H_2|=4$ (or vice versa), so $H_2$ is abelian, and hence $|Z(H_1 \times H_2)| =8$. But you can check that $Z(H)=Z(G) = \langle a_1^2,a_2^2 \rangle$ has order $4$.

In fact $H$ is indecomposable.