Let $(H\_1 \subset G\_1)$ and $(H\_2 \subset G\_2)$ be [core-free](http://groupprops.subwiki.org/wiki/Core-free_subgroup) **maximal inclusions** of finite groups. 


Their product, the inclusion $(H\_1 \times H\_2 \subset G\_1 \times G\_2)$, admits four obvious intermediate subgroups : $H\_1 \times H\_2$, $G\_1 \times G\_2$, $H\_1 \times G\_2$ and $G\_1 \times H\_2$. 



> 
> **Question** : If $(H\_1 \times H\_2 \subset G\_1 \times G\_2)$ admits a non-obvious
>  intermediate subgroup, what are the properties of $(H\_1
>  \subset G\_1)$ and $(H\_2 \subset G\_2)$ ?
> 
> 
> 


Here are examples of "prospective" properties, ordered from the weakest to the strongest : 


* $[G\_1 : H\_1]=[G\_2 : H\_2]$ ?
* $(H\_1 \subset G\_1) \sim (H\_2 \subset G\_2)$ ?
* $(H\_1 \subset G\_1) \sim (H\_2 \subset G\_2) \sim (\{e\} \subset \mathbb{Z}\_p)$ ?



Products of maximal inclusions of finite groups with a non-obvious intermediate
If I understand correctly, each $H\_i$ is maximal in $G\_i$ and core-free. In particular, unless $G\_i$ has prime order $p$, $H\_i$ is not normal in $G\_i$.


Let $X$ be a subgroup of $G\_1 \times G\_2$. For $i=1,2$ let $X^i$ be the projection of $X$ onto $G\_i$ and let $X\_i$ be the intersection of $X$ with $G\_i$. Then $X\_i$ is a normal subgroup of $X^i$, and $X^1/X\_1 \cong X^2/X\_2$. If $X$ contains $H\_1 \times H\_2$ as in your situation, then each $X\_i$ contains $H\_i$. As $H\_i$ is maximal in $G\_i$, $X\_i \in \{H\_i,G\_i\}$. If I have understood your assumptions correctly, unless $G\_i$ has prime order $p$, it follows that $X\_i=X^i$ (in which case also $X\_{3-i}=X^{3-i}$ and $X=X^1 \times X^2$). So, unless each $G\_i$ has prime order $p$, the four groups you list are the only intermediate subgroups.


