I would like to know examples of $G/K$ where $G$ is a locally compact group and $K$ is a compact subgroup of $G$.


I know about Riemannian symmetric spaces of Euclidean, compact and non-compact type. They can be realized as $G/K$. I am wondering what are other prominent spaces (discrete and continuous) which can be written as $G/K$, where $G$ can be unimodular or non-unimodular. Perhaps homogeneous tree is an example outside the set of Riemannian symmetric spaces. 



Examples of locally compact group quotiented by compact subgroup, but not Riemannian symmetric space 
There are lots of Riemannian homogeneous spaces that are not symmetric. For example, consider the chain $\mathrm{SO}(2)\subset \mathrm{SO}(3)\subset \mathrm{SO}(4)$. The quotient $\mathrm{SO}(4)/\mathrm{SO}(2)$ is not a symmetric space.


