  We consider maximum packings of edge-disjoint $4$-cliques in the complete
graph $K_n$. When $n \equiv 1$ or $4 \pmod{12}$, these are simply block
designs. In other congruence classes, there are necessarily uncovered edges; we
examine the possible `leave' graphs induced by those edges. We give particular
emphasis to the case $n \equiv 0$ or $3 \pmod{12}$, when the leave is
$2$-regular. Colbourn and Ling settled the case of Hamiltonian leaves in this
case. We extend their construction and use several additional direct and
recursive constructions to realize a variety of $2$-regular leaves. For various
subsets $S \subseteq \{3,4,5,\dots\}$, we establish explicit lower bounds on
$n$ to guarantee the existence of maximum packings with any possible leave
whose cycle lengths belong to $S$.
