$newcommandla{langle}newcommandra{rangle}$Let $G=mathbb{Z}/p^{i_1}timescdotstimesmathbb{Z}/p^{i_r}$ with $i_1leqldotsleq i_r$ be a finite abelian $p$-group. Then there can be many choices of generators ${x_1,ldots,x_r}$ such that the order of $x_j$ is $p^{i_j}$ and $G=la x_1ratimescdotstimes la x_rra$.

Let $H$ be a subgroup of $G$. Then $H$ is of the same form with less or equal number of factors.

Does there exist a choice of generators ${x_1,ldots,x_r}$ of $G$ as above such that $H$ is a product of subgroups of $la x_jra$?

If it is not true, is there an easy counterexample?