Let $K$ be an imaginary quadratic field, $\mathcal{O}_K$ be its ring of integers and $\mathcal{O}(n)$ be an order of $K$ of conductor $n\in\mathbb{N}$. It is known that there is a surjection between class groups
$Cl(\mathcal{O}(n))\twoheadrightarrow Cl(\mathcal{O}_K)$.
My question is the following: is it possible (in some cases) to find a splitting of this surjection, or, more weakly, is it possible for $Cl(\mathcal{O}_K)$ to be (isomorphic to) a subgroup of $Cl(\mathcal{O}(n))$?