**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** (the computation at index $32$ is due to [Gordon Royle][4])  
Up to equivalence, there are $2841561$ inclusions of finite groups $(A \subset B)$ of index $[B : A] \le 32$.    
Among them, $373533$ have a [distributive lattice][5] $\mathcal{L}(A \subset B)$ of intermediate subgroups.  
So the ratio with such a distributive lattice is less than $15 \%$.   

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


Here is a graph for $p_n$ with $1 \le n \le 32$:   
![enter image description here][6]  

Here is the table giving at $T$  the number of transitive permutation group $G$ (up to conjugacy) of degree $n$ and at  $D$ the number with the lattice $\mathcal{L}(G_1 \subset G)$ distributive (with degree $32$ due to [Gordon][4]):

$\scriptsize{ \begin{array}{c|c}
 n  &1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18 \newline 
                             \hline
T &1&1&2&5&5&16&7&50&34&45&8&301&9&63&104&1954&10&983  \newline
                             \hline
D &1&1&2&4&5&15&7&39&32&44&8&249&9&62&104&1055&10&894  
\end{array} }  $   
 
$\scriptsize{ \begin{array}{c|c}
 n  & 19&20&21&22&23&24&25&26&27&28&29&30&31&32 \newline 
                             \hline
T &8&1117&164&59&7&25000&211&96&2392&1854&8&5712&11&2801324  \newline
                             \hline
D &8&923&163&58&7&15627&208&95&2151&1541&8&5461&11&344731  
\end{array} }$  
 


  [1]: https://i.sstatic.net/wgSAH.png
  [2]: http://www.gap-system.org/Datalib/trans.html
  [3]: http://www.gap-system.org/
  [4]: http://mathoverflow.net/users/1492/gordon-royle
  [5]: http://en.wikipedia.org/wiki/Distributive_lattice
  [6]: https://i.sstatic.net/xXJjO.png