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](https://math.stackexchange.com/questions/1311040/the-groups-with-symmetric-subgroups-lattice).



The groups with symmetric subgroups lattice
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.  

For example, a locally finite group with dual is metabelian and lattice-isomorphic to an abelian group and actually self-dual (Corollary 8.2.5), and so any finite group with dual is supersolvable, and every non-normal Sylow has prime order (Corollary 8.2.6). Thus a non-solvable example must be infinite. Indeed, [Tarski monsters](http://en.wikipedia.org/wiki/Tarski_monster_group) are non-solvable examples.


