Suppose $V$ is a vector space over $\mathbb{C}$ and $G\subset \textrm{GL}(V)$ is a connected linear algebraic group.

Consider the orbit closure $Y = \overline{G.\mathbb{P} L}$, for a subspace $L\subset V$.

Is there a nice description for all the linear spaces contained in $Y$?

I would be really happy with an answer in terms of the representation theory of $G$ and the stabilizer $G_L$.

I know the answer in the case of the Segre variety (I am pretty sure this is classical), and I guess that the answer for other homogeneous varieties might be known. So I'm really interested when the linear space $L$ is at least 2-dimensional.