Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.

Question: Can any finite complete lattice be realized as an intermediate subgroups lattice?

Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.

Question: Can any finite lattice be realized as an intermediate subgroups lattice?

Remark: It's true for all the finite distributive lattices (see theorem 2.1 here).