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Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.

Question: Can any finite lattice be realized as an intermediate subgroups lattice?

Remark: It's true for all the finite distributive lattices (see theorem 2.1 here).

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This is an open problem. See

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  • $\begingroup$ Thank you! I observe that Palfy believes it's not true (see the sentence before theorem 1.7 p 484), so because his paper is more than 20 years old, perhaps we know a counter-example today. $\endgroup$
    – Sebastien Palcoux
    Feb 8, 2015 at 21:41
  • $\begingroup$ The section 4 of Palfy's paper is dedicated to height 2 lattices. Is it still open in this case? Else, are they some progress since 20 years? $\endgroup$
    – Sebastien Palcoux
    Feb 8, 2015 at 22:11
  • $\begingroup$ $M_{19}$ is not correct, the list of small values $n$ for which there is not a known representation for $M_{n}$ was, in Palfy's paper (p487): $n=16, 23, 35, 36, 37, 40...$. $\endgroup$
    – Sebastien Palcoux
    Feb 8, 2015 at 23:59
  • $\begingroup$ I've unaccepted your answer because I've found recent advanced (2013) of the problem (see my answer). Are you agree with this decision? $\endgroup$
    – Sebastien Palcoux
    Feb 9, 2015 at 14:54
  • $\begingroup$ @SébastienPalcoux Your answer is very interesting! As for whether to accept my answer or your own answer you might go by this: Questions that turn out to be less-famous open problems, but still well-known to experts, are answered by giving a reference to somewhere that the openness of the problem is discussed. -- Noah Snyder, meta.mathoverflow.net/a/368/4600 $\endgroup$
    – Bjørn Kjos-Hanssen
    Feb 9, 2015 at 16:14
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The more recent paper I've found about this problem is by Michael Aschbacher in 2013:
Overgroup lattices in finite groups of Lie type containing a parabolic

His introduction is a short survey of the last advanced, he recall Palfy–Pudlak theorem and question, and focus on the following John Shareshian's conjecture. His paper is a first step for a proof of this conjecture:

Let $B_n$ be the subgroups lattice of the cyclic group $C_m$ with $m=p_1p_2 \dots p_n$ square free and $p_i$ prime number.
Let $\mathcal{L}$ be a finite lattice and $\mathcal{L}'$ the poset $\mathcal{L}-\{l,g \}$ with $l$ and $g$ the least and greatest elements of $\mathcal{L}$.

Shareshian's Conjecture: If $\mathcal{L}'$ is a disconnected graph with connected components $(B'_{n_i})_{i=1, \dots , k}$ and $n_i \ge 3$, then $\mathcal{L}$ is not an intermediate subgroups lattice.

The smallest lattice $\mathcal{L}$ coming from this conjecture is the following:

enter image description here

with $k=2$, $n_1=n_2=3$

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    $\begingroup$ This picture is not quite right. The top and bottom elements from each Boolean algebra should be removed. (That is, the two coatoms and the two atoms in the picture should not be there.) To be clear, I do not know whether the pictured lattice is isomorphic with an interval in the subgroup lattice of a finite group. $\endgroup$
    – John Shareshian
    Feb 15, 2015 at 2:31
  • $\begingroup$ One more small correction. I believe that the conjecture you attribute to me is a special case of two more general conjectures, one due to Michael Aschbacher and the other due to me. $\endgroup$
    – John Shareshian
    Feb 15, 2015 at 2:39
  • $\begingroup$ @JohnShareshian: Thank you, I will fix the picture. Michael Aschbacher attributes this conjecture to you in his paper above p72. Do you prefer I call it Aschbacher-Shareshian's conjecture? $\endgroup$
    – Sebastien Palcoux
    Feb 15, 2015 at 12:26
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    $\begingroup$ Thanks for pointing that out. I won't argue with Michael. $\endgroup$
    – John Shareshian
    Feb 15, 2015 at 16:22

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