When is the subobject functor conservative?

Question

If $\mathcal{C}$ is a category with pullbacks, then the functor $Sub(-):\mathcal{C}^{op} \rightarrow \mathbf{Pos}$ takes an object to its poset of subobjects, and a morphism $f:X \rightarrow Y$ in $\mathcal{C}$ induces a morphism $f^{-1}: Sub(Y) \rightarrow Sub(X)$. When does this this functor reflect isomorphisms? For example, if $\mathcal{C}$ is the category of sets this is the case.