Property of isomorphic subgroups in finite groups

Question

I have the following question:

Does there exist a finite group $G$ and two subgroups $U,H\leq G$, s.t. the following properties are satisfied:

a) $H\cong U$.

b) There is no subgroup $L$, s.t. $U\lneq L \lneq G$

c) There exists a subgroup $K$, s.t. $H\lneq K \lneq G$.

I already know, that this implies that $H$ is not conjugated to $U$.

Thanks for help.

Answer 1

Let $G$ be a finite non-abelian simple group. Then in $G \times G$, the diagonal subgroup

$$D = \{(g,g): g \in G\}$$

is maximal (see this question) and $D \cong G$, but $G \times 1$ is not maximal.