An [Eulerian][1] subgroup lattice is boolean (see [here][2]), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-boolean Eulerian lattice is the following:
[![enter image description here][3]][3]  
It is the face lattice of the square polytope (see [here][4]), let's call this lattice $P_4$.  

By some tools used in [this paper][5] and the computation of Gordon Royle cited [here][6], we can prove that if $P_4=[H,G]$ as a lattice, then $|G:H| \ge 135$ (and so $|G| \ge 270$, because $H \neq 1$). Moreover, we checked (GAP) that $|G| \ge 512$, and for $G$ simple, $|G| \ge |{\rm PSL}(2,191)|= 3483840$.   



The existence of a lattice which is not the lattice of an interval of finite groups is an open problem.   
In the following paper (p72): [Overgroup lattices in finite groups of Lie type containing a parabolic][7]
Michael Aschbacher conjectures (after John Shareshian) that a lattice $L$ such that the poset $\overline{L}:=L \setminus \{\hat{0},\hat{1} \}$ is disconnected with connected components the posets $\overline{B}_{n_1}, \dots , \overline{B}_{n_r}$, with $B_{n_i}$ a boolean lattice of rank $n_i \ge3$ (and $r \ge 2$), is not the lattice of an interval of finite groups. But if all the $n_i$ are equal and odd, then $L$ is a non-boolean Eulerian lattice ([here][8] the smallest example).  

All these evidences lead to wonder:  
**Question**: Is there a non-boolean Eulerian interval of finite groups?

From an eventual positive answer for the relative version of K.S. Brown's problem and dual (i.e.  $\sum_{K \in [H,G]} \mu(K,G)|G:K|$ and $\sum_{K \in [H,G]} \mu(H,K)|K:H|$ are nonzero) and the property [here][9], we can deduce the following extension of (dual) Ore's theorem on boolean intervals:

*Conjecture*: Let $[H,G]$ be an Eulerian interval of finite groups. Then:  

-  $\exists V$ irr. $\mathbb{C}$-rep. of $G$ such that $G_{(V^H)} = H$ (see [here][10]).
-  $\exists g \in G$ such that $\langle Hg \rangle = G$ (see [here][11]).


This conjecture could help the investigation.


  [1]: https://en.wikipedia.org/wiki/Eulerian_poset
  [2]: https://mathoverflow.net/a/281048/34538
  [3]: https://i.sstatic.net/9x7S6.png
  [4]: https://en.wikipedia.org/wiki/Abstract_polytope#/media/File:A_Square_and_its_Hasse_Diagram.PNG
  [5]: https://arxiv.org/pdf/1610.07253.pdf
  [6]: https://mathoverflow.net/q/252997/34538
  [7]: http://ac.els-cdn.com/S0021869313000847/1-s2.0-S0021869313000847-main.pdf?_tid=2f8219d8-b03e-11e4-8faa-00000aab0f02&acdnat=1423474366_aa3f3e97d7423ea941a91ef949e3125c
  [8]: https://mathoverflow.net/a/196074/34538
  [9]: https://mathoverflow.net/q/280917/34538
  [10]: https://mathoverflow.net/a/207608/34538
  [11]: https://mathoverflow.net/a/195331/34538