Let $G$ be a group (for now discrete). A subgroup $H$ of $G$ is called a commensurated subgroup of $G$, if $H\cap xHx^{-1}$ is a finite index subgroup of $H$ for all $x\in G$. These subgroups are also called Hecke subgroups or almost normal subgroups. My question is:

Is there any non-elementary, closed, discrete and commensurated subgroup $H$ of a non-discrete locally compact group $G$?

Elementary examples are finite subgroups and normal subgroups.

Edit: Regarding Jean Raimbault's comment, we consider all closed discrete nearly normal subgroups of $G$ as elementary examples too. (nearly normal means it is commensurable to some normal subgroup).

I just noticed that another class of very elementary examples can be considered as follows: Consider $G=\Gamma\times \Delta$, where $\Gamma$ is discrete and $\Delta$ is an arbitrary non-discrete locally compact group and $H$ is a commensurated subgroup of $\Gamma$.