Self-duality of the subgroup lattice of finite abelian groups

Question

For each abelian finite group $G$ let $\mathcal L(G)$ be the lattice of all subgroups of $G$.

For which abelian finite groups $G$, is $(\mathcal L(G),\subseteq)$ order-isomorphic to $(\mathcal L(G),\subseteq^{-1})$?

($\subseteq^{-1}$ is the same as $\supseteq$)

Answer 1

Any finite abelian group $G$ is (noncanonically) isomorphic to its dual $\hat{G}=\operatorname{Hom}(G,\mathbb{Q}/\mathbb{Z})$. By duality, the lattice of subgroups of $G$ is anti-isomorphic to the lattice of subgroups of $\hat{G}\cong G$.