What groups have a second maximal subgroup below exactly four maximal subgroups?

Question

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.)

Answer 1

(Updated to reflect John Shareshian's excellent answer and suggestions.)

Yes, there are infinitely many examples.

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.

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.

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. MR779379 DOI:10.1515/crll.1985.356.140 GDZ:GDZPPN002202190

Heineken, H. "A remark on subgroup lattices of finite soluble groups." Rendiconti del Seminario Matematico della Università di Padova, 77 (1987), 135-147 MR904616 NUMDAM: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. MR1474164 DOI:10.1006/jabr.1997.7069

Answer 2

I think that with some work you can characterize the examples where $G$ is solvable as follows. (I did not check carefully, so you can take this with a grain of salt.)

If $H$ has trivial core in $G$ and $[H,G]$ is $M_4$, then one of the following holds:

1) $G$ is the semidirect product $H(V+V)$, where $V$ is an irreducible $F_3[H]$-module such that the only elements of $GL(V)$ commuting with $H$ are the scalar transformations $1$ and $-1$.

(This case occurs when every maximal subgroup containing H has nontrivial core in $G$. Your example $C_2:C_3xC_3$ is of this type. This condition should be sufficient and you can construct tons of examples of this type without too much trouble.)

2) $G$ is the semidirect product $MV$, where $V$ is an irreducible $F_3[M]$-module. There is a $1$-dimensional subspace $W$ of $V$ such that $H=N_M(W)=C_M(W)$, and $C_V(H)=W$.
In particular, no element of $M$ has $W$ as an eigenspace with eigenvalue $-1$.

(This case occurs when some maximal $M$ containing H has trivial core in $G$. Your example $S_3$ is of this type. I don't know if there are lots of examples with $H$ maximal in $M$)

I can recommend that if you are interested in understanding how to find intervals of type M_n, you look at the work of Andrea Lucchini, including his joint paper with Robert Baddeley in Journal of Algebra. There is also recent work of Michael Aschbacher on the interval representation problem in general that is most certainly worth examining.