This post is a *relative version* of General bound for the number of subgroups of a finite group

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, 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

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 for a discussion on the estimate of this sum.

We have slightly better with $\sum_{k>1, k|n}^n {n-1\choose k-1}$.