Let $G$ be a reductive affine algebraic $\mathbb{C}$-group (not necessarily connected).  Suppose $X$ is an irreducible affine algebraic set over $\mathbb{C}$ where $G$ acts rationally.  Suppose that $H$ is a reductive subgroup of $G$ (again not necessarily connected).  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, and had not emphasized that I was allowing the adjective "reductive" to include disconnected groups (connected component of identity has trivial unipotent radical).  I have editted the problem to reflect my original intentions.