Let $G$ be a reductive affine algebraic $\mathbb{C}$-group. Suppose $X$ is an irreducible affine algebraic set over $\mathbb{C}$ where $G$ acts rationally. Suppose that $H$ is a reductive subgroup of $G$. Let $x\in X$.

If the orbit $G\cdot x$ is closed in $X$ (in the ball topology), is the sub-orbit $H\cdot x$ also closed ?

NOTE: Originally, I left off the assumption that $H$ is a reductive subgroup. I have editted the problem to reflect my original intention.