Can any finite 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).
 A: The more recent paper I've found about this problem is by Michael Aschbacher in 2013:
Overgroup lattices in finite groups of Lie type containing
a parabolic
His introduction is a short survey of the last advanced, he recall Palfy–Pudlak theorem and question, and focus on the following John Shareshian's conjecture. His paper is a first step for a proof of this conjecture:  
Let $B_n$ be the subgroups lattice of the cyclic group $C_m$ with $m=p_1p_2 \dots p_n$ square free and $p_i$ prime number.
Let $\mathcal{L}$  be a finite lattice and $\mathcal{L}'$ the poset $\mathcal{L}-\{l,g  \}$ with $l$ and $g$ the least and greatest elements of $\mathcal{L}$.   
Shareshian's Conjecture: If $\mathcal{L}'$ is a disconnected graph with connected components $(B'_{n_i})_{i=1, \dots , k}$ and  $n_i \ge 3$, then $\mathcal{L}$ is not an intermediate subgroups lattice.    
The smallest lattice $\mathcal{L}$ coming from this conjecture is the following:   
 
with $k=2$, $n_1=n_2=3$
A: This is an open problem. See


*

*Wikipedia article: http://en.m.wikipedia.org/wiki/Finite_lattice_representation_problem

*Palfy and Pudlak's result (see open-source description in Palfy's article Intervals in subgroup lattices of finite groups)

*This MO question from 2012: Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?
