The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$.  
The lattice  $B_{3}$ is the following:   
   
![enter image description here][1] 

*Question*: What are the rank $3$ boolean intervals of the form $[H,G]$, with $G$ a simple group?   

*Remark*: For $\vert G \vert \leq 4000000$, we have found (by GAP):

 - $A_8$ (of order $20160$) with a subgroup of index $315$,
 - $PSU(3,5)$ (of order $126000$) with a subgroup of index $6000$, 
 - $PSp(6,2)$ (of order $1451520$) with a subgroup of index $2835$,
 - $PSU(4,3)$ (of order $3265920$) with a subgroup of index $25515$.    

Can we have a classification in general?

  [1]: https://i.sstatic.net/RTPpu.png
  [2]: http://homepages.ulb.ac.be/~tconnor/atlaslat/