  We investigate the subradiance properties of $n\geq 2$ multilevel fermionic
atoms loaded into the lowest motional level of a single trap (e.g.~a single
optical lattice site or an optical tweezer). As pointed out in our previous
work [arXiv:1907.05541], perfectly dark subradiant states emerge from the
interplay between fermionic statistics and dipolar interactions. While in
[arXiv:1907.05541] we focused on the $n=2$ case, here we provide an in-depth
analysis of the single-site dark states for generic filling $n$, and show a
tight connection between generic dark states and total angular momentum
eigenstates. We show how the latter can also be used to understand the full
eigenstate structure of the single-site problem, which we analyze numerically.
Apart from this, we discuss two possible schemes to coherently prepare dark
states using either a Raman transition or an external magnetic field to lift
the Zeeman degeneracy. Although the analysis focuses on the single-site
problem, we show that multi-site dark states can be trivially constructed in
any geometry out of product states of single-site dark states. Finally, we
discuss some possible implementations with alkaline-earth(-like) atoms such as
$^{171}$Yb or $^{87}$Sr loaded into optical lattices, where they could be used
for potential applications in quantum metrology and quantum information.
