Can this lattice be realized as an intermediate subgroups lattice?

Question

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 $G$.

Let the lattice $\mathcal{L}$ as follows:

Question: Can $\mathcal{L}$ be realized as an intermediate subgroups lattice?

Remark: I've checked by GAP that there is no example for $[G:H]<32$.

Answer 1

Yes. The first example was given by Aschbacher in the following 2008 paper (example 8.5).

Aschbacher, Michael. On intervals in subgroup lattices of finite groups. J. Amer. Math. Soc. 21 (2008), no. 3, 809–830. link

Before this paper, Watatani (1994) had proven that with two possible exceptions, any finite lattice with $\leq 6$ elements could be realized as an intermediate subfactor lattice. In the paper above Aschbacher finds examples for these two lattices.