EDIT of EDIT: I guess I should flesh this out in greater detail.
The first thing to note is that the category of groups has a zero object (it's terminal object is also initial). I will write 0 for this.
Say Groups had a subobject classifier $0 \stackrel{true}\longrightarrow \Omega$. Note that the map true must be the unique map out of 0. Also note that the unique map $0 \stackrel{!}\rightarrow A$ is a monomorphism (i.e. injection) for every A. Thus
0----->0 | | ! | | true | | \/ \/ A ---->Ω is a pullback square, where the lower map, $\chi$, is the characteristic map of !. I claim that $\chi$ is a monomorphism. This is because the ker($\chi$) maps to both A and 0 to make the diagram commute, so the inclusion of ker($\chi$) into A factors through 0 by the definition of a pullback. In other words, the kernel is trivial, so $\chi$ is an injection. Thus every group A admits an injection to $\Omega$ which is bad for set theoretic reasons.
