Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there exists a bijection $\phi :\operatorname{Sub}(G\times G) \to \operatorname{Sub}(H\times H)$ such that always $$A\le B~~\leftrightarrow~~\phi(A)\le\phi(B)$$

1) If $G$ and $H$ are infinite abelian groups, are they isomorphic?

2) If $G$ and $H$ are finite non-abelian non-simple groups, are they isomorphic?

**Edit**:

As shane.orourke's answer shows below and also by an example by Schmidt in

```
R. Schmidt. Der Untergruppenverband des direkten Produktes zweier isomorpher Gruppen. J. Algebra 73 (1981), 264–272.
```

The first question and has a negative answer. (Still I'm not sure if Schmidt's example is abelian)

The second question remains unanswered.