  We elucidate the correspondence between a particular class of superconformal
field theories in six dimensions and homomorphisms from discrete subgroups of
$SU(2)$ into $E_8$, as predicted from string dualities. We show how this match
works for homomorphisms from the binary icosahedral group $SL(2,5)$ into $E_8$,
correcting previous errors in both the mathematics and physics literature. We
use this correspondence to list the homomorphisms from binary dihedral groups,
the binary tetrahedral group, and the binary octahedral group into $E_8$--a
novel mathematical result. The partial ordering specified by renormalization
group flows suggests an ordering on these homomorphisms similar to the known
ordering of nilpotent orbits of a simple Lie algebra dictated by the Hasse
diagram.
