To add to Robert Bryant's answer, the coadjoint orbits of a Lie group always have a natural symplectic structure, the (Lie-)Kirillov-Kostant(-Souriau) form. These are the symplectic leaves of the natrual Lie-Poisson structure on $\mathfrak{g}^\ast$. For $x \in \mathfrak{g}^\ast$, we can identify the coadjoint orbit $\mathcal{O}_x$ with $G/H_x$, where $H_x$ is the stabilizer of $x$ under the coadjoint action.  For $x$ such that $H_x$ is Cartan (which should happen generically), this endows $G/H$ with a symplectic structure. However, this depends on the choice of $x$. For example, even in the absolute simplest case of $G = SU(2)$, the coadjoint orbits are spheres of different radii, so the orbits $\mathcal{O}_x$ and $\mathcal{O}_y$ are not symplectomorphic unless $y$ is conjugate to $x$.