This post is a *relative version* of http://mathoverflow.net/q/132675/34538

Let $[H,G]$ be a interval of finite groups with $|G:H| = n$.

*Question:* What is a good upper-bound of $|[H,G]|$, as a function of $n$?  

If $H=\{ e\}$, then the best possible upper-bound is essentially $n^{(\frac{1}{4}+o(1)) \log_2(n)}$ ([here][1], Corollary 1.6).  
Should we expect the same in general?

There is an OEIS page for the maximal cardinal of a subgroup lattice for a group of order $n$: [A018216][2]    
1, 2, 2, 5, 2, 6, 2, 16, 6, 8, 2, 16, 2, 10, 4, 67, 2, 28, 2, 22, 10, 14, 2, 54, 8, 16, 28, 28, 2, 28, 2, 374, 4, 20, 4, 78, 2, 22, 16, 76, 2, 36, 2, 40, 12, 26, 2, 236, 10, 64, 4, 46, 2, 212, 14, 98, 22, 32, 2, 80, 2, 34, 36, 2825, 4, 52, 2, 58, 4, 52, 2, 272 

I didn't find an OEIS page for the maximal cardinal of an interval of finite groups, at index $n$.  
This should be computable for the indices $<32$, using GAP or MAGMA.  

*Remark*: Because an element $K \in [H,G]$ admits a unique partition by cosets $Hg$, we have:    
$$|[H,G]| < \sum_{k \mid n} {n \choose k}$$
See [this post][3] for a discussion on the estimate of this sum.  
We have slightly better with $\sum_{k>1,  k|n}^n  {n-1\choose k-1}$.


  [1]: http://www.ams.org/tran/1996-348-09/S0002-9947-96-01665-0/S0002-9947-96-01665-0.pdf
  [2]: http://oeis.org/A018216
  [3]: https://math.stackexchange.com/q/1808549/84284