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](http://www.math.hawaii.edu/~williamdemeo/latticetheory/Palfy-IntervalsInSubgroupLattices-GStA-1993.pdf)).



Can any finite lattice be realized as an intermediate subgroups lattice?
This is an open problem. See


* Wikipedia article: <http://en.m.wikipedia.org/wiki/Finite_lattice_representation_problem>
* [Palfy and Pudlak](http://www.ams.org/mathscinet-getitem?mr=593011)'s result (see open-source description in Palfy's article [Intervals in subgroup lattices of finite groups](http://www.math.hawaii.edu/~williamdemeo/latticetheory/Palfy-IntervalsInSubgroupLattices-GStA-1993.pdf))
* This MO question from 2012: [Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?](https://mathoverflow.net/questions/85724/given-a-lattice-l-with-n-elements-are-there-finite-groups-h-g-such-that-l-c)


