Given any object$X$ in any category$C$, the subobjects of $X$ form a poset (though in general it may be a large one, i.e. a partially ordered proper class; cf. well-powered category). This is called, naturally enough, the poset of subobjects of $X$, or the subobject poset of $X$.

Sometimes it is the poset of regular subobjects that really matters (although these are the same in any (pre)topos).

Properties

If $C$ is finitely complete, then the subobjects form a meet-semilattice, so we may speak of the semilattice of subobjects.

In any coherent category (such as a pretopos), the subobjects form a distributive lattice, so we may speak of the lattice of subobjects.

The reader can probably think of other variations on this theme.

Related concepts

If one opts for the alternative^{1} definition that subobjects are monomorphisms into the object (not isomorphism classes thereof), then one gets a preorder of subobjects instead. In any case, the poset of subobjects$Sub(X)$ in our sense is the posetal reflection of the preorder $Mono(X)$ of subobjects in the alternative sense, and of course the reflection quotient map $Mono(X) \to Sub(X)$ is an equivalence.