  We prove the following theorem. Let $G$ be a finite group generated by
unitary reflections in a complex Hermitian space $V=\mathbb{C}^\ell$ and let
$G'$ be any reflection subgroup of $G$. Let $\mathcal{H}(G)$ be the space of
$G$-harmonic polynomials on $V$. There is a degree preserving isomorphism
$\xi:\mathcal{H}(G')\otimes\mathcal{H}(G)^{G'}\overset{\sim}{\longrightarrow}\mathcal{H}$
of graded $\mathcal{N}$-modules, where $\mathcal{N}:=N_{\rm{GL}(V)}(G)\cap
N_{\rm{GL}(V)}(G')$ and $\mathcal{H}^{G'}$ is the space of $G'$-fixed points of
$\mathcal{H}$. This generalises a result of Douglass and Dyer for parabolic
subgroups of real reflection groups.
