It is known that, given a complex vector space $V$ of dimension $\mathit{dim}(V)=n+1$, all irreducible representations for the group $\mathit{GL}(V)$ are parametrized by Young tableaux $\lambda$. For example for the Young tableaux $\lambda=(d)$ we have that $$S_{\lambda}V=\mathit{Sym}^d(V)$$ and for $\lambda=(1,\dots,1)$ that $$S_{\lambda}V=\bigwedge^k(V)$$ In particular every irreducible $\textit{GL}(V)-$module $S_{\lambda}V$ can be seen as a subvector space of a suitable tensor power $V^{\otimes N_{\lambda}}$.

My question now is the following: is it true that all homogeneous varieties arising as the orbit of a highest weight vector in $S_{\lambda}V$ are only either a Veronese, a Grassmannian or a flag variety?