Let $G$ be a group and $\mathfrak{L}(G)$ be set of all subgroups of $G$. Clearly, $\mathfrak{L} (G)$ is a lattice.

If we know that $\mathfrak{L} (G)$ is symmetric then what can be said about the group $G$ ?

Any reference and observation would be appriciated. I also wonder whether is there any such nonsolvable groups ?

By saying "symmetric", I mean lattice is isomorphic to its reverse lattice.

**Example:** Elemantary abelian $p$ groups, the groups that all Sylow subgroups of prime orders are such examples.

**Note** I had asked this question there.