Given two isomorphic subgroups of a finite abelian group with the same number of minimal generators, are the quotients isomorphic?

Question

Let $G$ be a finite abelian group.

We know that the least size of a minimal generating set of a proper nontrivial subgroup $H \leqslant G$ might be the same as $G$, for example ($\Bbb Z_6 \cong \Bbb Z_3 \times \Bbb Z_2 $ has a subgroup isomorphic to $\Bbb Z_2$, and both $\Bbb Z_6$ and $\Bbb Z_2$ can be generated by $1$ element). Also, in general, given two isomorphic subgroups $H_1$ and $H_2$ of $G$, it is not necessarily true that $G/H_1$ and $G/H_2$ are isomorphic (Isomorphic quotient groups).

However, if the least size of a minimal generating set of each $H_i$ is the same as that of $G$, and $H_1 \cong H_2$, does this implies that $G/H_1 \cong G/H_2$?

Answer 1

There are two subgroups isomorphic to $C_2 \times C_4$ in $C_4 \times C_8$ with non-isomorphic quotient groups.