What's the easiest example to show that $Ab(G)$, the set of Abelian subgroups of a group $G$, need not be complete? I heard that $D_4$ was a good example, but $Ab(D_4)=Sub(D_4)$, which is complete.
Edit: I am referring to the partial ordering of $Ab(G)$ and $Sub(G)$ by inclusion. When I say a partially-ordered set $P$ is complete, I mean that every subset $C \subset P$ has a least upper bound, under the ordering.
Edit: When I say $D_4$, I am referring to the dihedral group of order $4$, which is isomorphic to $C_2 \times C_2$, rather than the group of symmetries of a square. This usage is standard in my institution. Therefore $D_4$ is abelian.