Order Structures on the collection of all the subfunctors of a presheaf

Question

Let $\mathcal{C}$ be a small category and $y: \mathcal{C} \to {\bf Sets}^{\mathcal{C}^{op}}$ be its Yoneda embedding. It is well-known that the family of subobjects of $y(C)$, where $C \in \mathcal{C}$, is a preorder and always contains a "maximum" subobject (i.e. the maximal sieve). It is possible to deduce some other informations? For instance, under which hypotheses the above family forms a complete lattice?