# The groups with symmetric subgroups lattice

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.

• @YCor: Thank you. I edited the question. Jun 4 '15 at 20:12

You should have a look at Roland Schmidt's book Subgroup lattices of groups, especially Chapter 8, "Dualities of subgroup lattices". In this chapter, groups $G$ are studied such that there is another group $\overline{G}$ (a "dual"), such that the subgroup lattice of $G$ is isomorphic to the reverse of the subgroup lattice of of $\overline{G}$. A group with the property in your question is "self-dual" in Schmidt's terminology.