Let's say we have a homogeneous space $H\setminus G$.

Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure?

I am particularly interested in a situation when $H\setminus G$ is maximally-noncompact, i.e. $H$ is a maximally compact subgroup of $G$.

I hope, my question does not sound too broad. Maybe, this question has a trivial answer, but from a background of a theoretical physicist, it is not obvious.