Let $G_1, G_2$ be two groups with at least one nontrivial proper subgroup each.
Let $S_1, S_2$ be the sets of proper subgroups of, respectively $G_1, G_2$.
Suppose there exists a bijective function $f: S_1 \rightarrow S_2$ such that $\forall A\in S_1, f(A)$ is isomorphic to $A$.
When can I conclude that $G_1, G_2$ are isomorphic?
I think that, if $G_1$ and $G_2$ are finite and abelian we can conclude that they are isomorphic, but I can't prove It. Moreover, I haven't found any counterexample for nonabelian finite groups.