**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][1] of $A$ in $B$.  

**Remark**: The equivalence class of $(A \subset B)$ is the same that the conjugacy class of a [transitive permutation group][2] $G$ with $(A \subset B) \sim (G_1 \subset G)$.  

**[GAP][3] computation:**  
Up to equivalence, there are $40237$ inclusions of finite groups $(A \subset B)$ of index $[B : A] \le 31$.    
Among them, $28802$ have a [distributive lattice][4] $\mathcal{L}(A \subset B)$ of intermediate subgroups.  
So the ratio with such a distributive lattice is more than $70 \%$, i.e. such inclusions are the majority.   

We would like to know if this majority is a "small index" phenomenon, or if it's true in general:    

Let $p_n$ be such a ratio for index $\le n$, let $\alpha_n = \liminf_{r>n} (p_r)$ and  $\alpha = \lim_{n \to \infty} (\alpha_n) $.

**Question**:  $\alpha = 0$ or  $0< \alpha < 1/2$ or $\alpha \ge 1/2$?  
  (Optional: If $\alpha = 0$, what's its asymptotic analysis of $(\alpha_n)$?)  


Here is a graph for $p_n$ with $1 \le n \le 31$:   
![enter image description here][5]
 


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