Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice.

Let $s(n):= max\{|\mathcal{L}(G)| \text{ for } |G|=n \}$.

There is an OEIS page for the sequence $s(n)$: 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

Now let $[H,G]$ be an interval of finite groups and $|[H,G]|$ its cardinal.

Let $t(n):= max\{|[H,G]| \text{ for } |G:H|=n \}$.

I didn't find an OEIS page for the sequence $t(n)$.

This is computable for the indices $<32$, using GAP or MAGMA (I've no access to, currently)

**Question**: Is there an integer $n$ such that $t(n)>s(n)$?

*Remark*: If we observe that $\forall n \ t(n)=s(n)$, then a proof of this fact would answer this post.

Else, I'm interested in the smallest counter-example.

*Remark*: Alexander Hulpke has checked that $t(n)=s(n)$, $\forall n \le 47$ (see comment below).