The groups with symmetric subgroups lattice

Question

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

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

Any reference and observation would be appriciated.

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

Answer 1

This is covered in Chapter 8 of Schmidt's book "Subgroup lattices of groups"; everything in this answer is there, but I summarize it here for convenience.

Say a group $G$ has a dual $\hat{G}$ if their subgroup lattices are anti-isomorphic. Your question is about when $G$ is dual to itself.

Baer (1939) proved that every group w/ dual is torsion, i.e. all its elements have finite order.

For locally finite groups (which means every finitely generated subgroup is finite), there is a complete classification of groups with duals. (This is Theorem 8.2.2 in Schmidt, due to Suzuki and Zacher):

If $G$ is a locally finite group, and $\delta$ is a lattice anti-isomorphism to $L(\hat{G})$, then:

1. $G = \prod_{\lambda} G_{\lambda}$ where the $G_{\lambda}$ are coprime to one another
2. $\hat{G} = \prod_{\lambda} \delta(\hat{G}_\lambda)$, where $\hat{G}_\lambda := \langle G_{\mu} : \mu \neq \lambda \rangle$ and the $\delta(\hat{G}_\lambda)$ are coprime to one another
3. For every $\lambda$, there exists (not nec. distinct) primes $p,q$ such that one of the following holds:

• $G_\lambda$ is a cyclic $p$-group, $\delta(\hat{G}_\lambda)$ is a cyclic $q$-group, or
• $G_\lambda$ and $\delta(\hat{G}_\lambda)$ are both either elementary abelian of order $p^n$ or a semi-direct product of an elementary abelian group of order $p^{n-1}$ by a group of order $q$, or
• $G_\lambda$ and $\delta(\hat{G}_\lambda)$ are finite non-Hamiltonian $p$-groups with modular subgroup lattices

Maybe from these more can be said when $G$ is self-dual ($G = \hat{G}$)...