I am looking for a finite group $G$ with the following property: there is a (core-free) subgroup $H < G$ such that the interval $\{ K : H < K < G\}$ in Sub[$G$] contains exactly four maximal subgroups. (In other words, $[H, G] \cong M\_4$.) 


I have used GAP to search for such groups and, to my surprise, I could find only three: $S\_3$, $C\_3 \times C\_3$, and $(C\_3 \times C\_3) : C\_2$. So far, all other examples reduce to these after modding out by a normal subgroup (so they are not examples if we require $H$ be core-free). 


I've searched through most of the groups of order less than 960. Though, I can't promise my GAP code is free of bugs that may be causing me to miss something.


**Question:**
Does anyone know of other finite groups, besides $S\_3$, $C\_3 \times C\_3$, and $(C\_3 \times C\_3) : C\_2$, with an upper interval isomorphic to $M\_4$? (If not, I would welcome any ideas that could help explain why this should be a rare phenomenon.)



What groups have a second maximal subgroup below exactly four maximal subgroups?
(Updated to reflect John Shareshian's excellent answer and suggestions.)



> 
> **Yes**, there are infinitely many examples.
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First off, some negative results: If $[G:H] ≤ 31$, then $G/Core(G,H)$ acting on H is one of your three examples. If H is contained in a core-free maximal subgroup M with $[G:M] ≤ 50$ (subject to a few caveats), then in fact G is your first example. These "$M$" correspond to John Shareshian's type (2) examples.


However, I think your search of small groups must have had an error:  

 The group $G =$ `SmallGroup(648,725)` has Sylow $2$-subgroup $H$ that is core-free, and the interval $G/H$ is the bounded version of a $4$-element anti-chain.  

 This corresponds to the next smallest $H$ (after $H≅2$) in John Shareshian's type (1) examples.


As a general comment, no interval is so rare that it only occurs finitely many times:



> 
> Given any interval $[G/H]$ there is an anti-isomorphic interval in a wreath product of the permutation group $(G,H)$ with a non-abelian simple group.
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Applying the construction an even number of times produces an infinite sequence of examples with a given (core-free) interval. In particular, there is an example with $|G|=279936000000$ and $|H|=360$, with $H$ core-free and the interval $G/H$ the bounded version of a $4$-element anti-chain, $M\_4$. This takes the "seed" $(G,H)$ to be the regular representation of the non-abelian group of order $6$, and the non-abelian simple group of order $60$.


This is very similar to [F. Ladisch's answer to your previous question](https://mathoverflow.net/questions/62495/which-finite-nonabelian-groups-have-long-chains-of-subgroups-as-intervals-in-thei/62546#62546).


Here's what I've found in the literature (both of which are referenced in Roland Schmidt's book; I did not find much else):


Kurzweil (1985) puts your second and third example into context ($G/N=H$ acting on an isotypic semisimple module $N$, so that the interval $[G/H]$ is a projective space). Your first example is just "small", I think. More importantly, it gives the method of replicating examples using wreath products as the second example on page $148$.


Heineken (1987) pins down the structure of solvable $G$ with second maximal $H$ such that the interval $[G/H]$ is not a (bounded) antichain, $M\_n$. He has some results that say $n−1$ is usually a prime power.


John Shareshian points out that Baddeley–Lucchini (1997) is dedicated to the question of which $M\_n$ can occur as intervals in the subgroup lattice of a finite group. In particular, this shows that the $n−1$ being a prime power result is definitely restricted to solvable groups: Feit showed both $n=7$ and $n=11$ occur. The paper is definitely focused on the non-solvable case.  

 The Math Review is also really good.



> 
> Kurzweil, Hans.
>  "Endliche Gruppen mit vielen Untergruppen."
>  J. Reine Angew. Math. 356 (1985), 140–160.
>  MR[779379](http://www.ams.org/mathscinet-getitem?mr=779379)
>  DOI:[10.1515/crll.1985.356.140](http://dx.doi.org/10.1515/crll.1985.356.140)
>  GDZ:[GDZPPN002202190](http://resolver.sub.uni-goettingen.de/purl?GDZPPN002202190)
> 
> 
> Heineken, H.
>  "A remark on subgroup lattices of finite soluble groups."
>  Rendiconti del Seminario Matematico della Università di Padova, 77 (1987), 135-147
>  MR[904616](http://www.ams.org/mathscinet-getitem?mr=904616)
>  NUMDAM:[RSMUP\_1987\_\_77\_\_135\_0](http://www.numdam.org/item?id=RSMUP_1987__77__135_0)
> 
> 
> Baddeley, Robert; Lucchini, Andrea
>  "On representing finite lattices as intervals in subgroup lattices of finite groups."
>  J. Algebra 196 (1997), no. 1, 1–100.
>  MR[1474164](http://www.ams.org/mathscinet-getitem?mr=1474164)
>  DOI:[10.1006/jabr.1997.7069](http://dx.doi.org/10.1006/jabr.1997.7069)
> 
> 
> 


