For a compact semisimple Lie group $G$, what is an example of a homogeneous space of $G$ which is not a torus bundle over a [generalized flag manifold](https://en.wikipedia.org/wiki/Generalized_flag_variety) of $G$. Examples for $SU(N)$ would be of most interest.



Homogeneous spaces which are not torus bundle over flag manifolds
Just to summarize the common feature of both Allen's example and my example: there are many Lie subgroups $H$ of $G$ that are contained in no proper parabolic subgroup. Some examples arise from the fact that a proper subgroup of a parabolic may have a normalizer that is not contained in the subgroup; the example $H$ that I intended was the normalizer in $SU(N)$ of a maximal torus $U(1)^{N-1}$. But there are also connected subgroups that are contained in no proper parabolic subgroup, e.g., the intersection of $SU(N)$ with an algebraic subgroup of $\text{GL}\_n(\mathbb{C})$. Another example that I particularly like is the image of the group homomorphism $\rho:SU(2)\to SU(N)$ associated to the $N$-dimensional irreducible representation of $SU(2)$. You can think of $SU(N)/\rho(SU(2))$ as a parameter space for real rational normal curves in $\mathbb{RP}^{N-1}$. 


