  We generalize the theoretical modeling of collective atomic super- and
subradiance to the multilevel case including spontaneous emission from several
excited states towards a common ground state. We show that in a closely packed
ensemble of $N$ atoms with $N-1$ distinct excited states each, one can find a
new class of non-radiating dark states,, which allows for long-term storage of
$N-1$ photonic excitations. Via dipole-dipole coupling only a single atom in
the ground state is sufficient in order to suppress the decay of all $N-1$
other atoms. By means of some generic geometric configurations, like a triangle
of V-type atoms or a chain of atoms with a $J=0 \to J=1$ transition, we study
such subradiance including dipole-dipole interactions and show that even at
finite distances long lifetimes can be observed. While generally hard to
prepare deterministically, we identify various possibilities for a
probabilistic preparation via a phase controlled laser pump and decay.
