There are a lot of possibilities. Here are a few examples (of homogeneous spaces):
- symmetric spaces (e.g. $S^n=SO(n+1)/SO(n)$, projective spaces $P^n(\mathbb C)=SU(n+1)/S(U(n)\times U(1))$, $P^n(\mathbb H)=Sp(n+1)/Sp(n)\times Sp(n)$, Grassmannian spaces $SO(p+q)/SO(p)\times SO(q), SU(p+q)/S(U(p)\times U(q)), Sp(p+q)/Sp(p)\times Sp(q)$, hyperbolic spaces),
- different realizations of the spheres, e.g. $SU(n+1)/SU(n)=S^{2n+1}$, $Sp(n+1)/Sp(n)=S^{4n+3}$, $Spin(9)/Spin(7)$ (be careful with the embedding), etc.
- Lie groups: $G/e$ with $G$ any Lie group.
- $G/\Gamma$ with $\Gamma$ a discrete cocompact subgroup of $G$ (e.g. nilmanifolds if $G$ is nilpotent).
- Flag manifolds, Stiefel manifolds, Aloff-Wallach spaces, Ledger-Obata spaces, Generalized Wallach spaces.
- Isotropy irreducible spaces (e.g. $SO(\dim K)/K$ with $K$ a compact simple Lie group and the embedding is given by the adjoint representation.
Did I forget someone obvious?