I am looking for non-trivial examples for the following situation:

• $G$ is a locally compact group
• $H\subset G$ a closed subgroup
• Both are unimodular and non-discrete
• The quotient space $G/H$ is compact, but $G$ is not compact

Trivial cases would be $G=G_1\times G_2$, $H=H_1\times H_2$ with $H_1=G_1$ and $H_2$ being discrete in $G_2$, or $G_2$ being compact. The same goes for semi-direct products instead of direct products.

Examples with $G$ being topologically simple would be nice.

I am looking for non-trivial examples of the following:

• $G$ is a locally compact group
• $H\subset G$ a closed subgroup
• Both are unimodular and non-discrete
• The quotient space $G/H$ is compact, but $G$ is not compact

Trivial cases would be $G=G_1\times G_2$, $H=H_1\times H_2$ with $H_1=G_1$ and $H_2$ being discrete in $G_2$, or $G_2$ being compact. The same goes for semi-direct products instead of direct products.

Examples with $G$ being topologically simple would be nice.