**Definition**: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the [normal core][3] of $A$ in $B$.  

**Remark**: The equivalence class of $(A \subset B)$ is the same that the conjugacy class of a [transitive permutation group][1] $G$ with $(A \subset B) \sim (G_1 \subset G)$.  

**Gap computation:**  
Up to equivalence, there are $40225$ inclusions of finite groups $(A \subset B)$ of index $[B : A] \le 30$.    
Among them, $28798$ have a [distributive lattice][2] $\mathcal{L}(A \subset B)$ of intermediate subgroups.  
So the ratio with such a distributive lattice is more than $70 \%$.    

Let $p_n$ be such a ratio for index $\le n$.   

**Questions**: Is $(p_n)$ convergent when $n \to \infty$?  
If yes, let $\alpha = lim(p_n)$. Then $\alpha = 0$ or  $0< \alpha < 1/2$ or $\alpha \ge 1/2$?  
If $\alpha = 0$, what's the asymptotic analysis of $(p_n)$?  


Here is a graph for $p_n$ with $1 \le n \le 30$:   
![enter image description here][3]
 


  [1]: http://www.gap-system.org/Datalib/trans.html
  [2]: http://en.wikipedia.org/wiki/Distributive_lattice
  [3]: https://i.sstatic.net/wgSAH.png