  We provide a general recursive method for constructing transfer systems on
finite lattices. Using this we calculate the number of homotopically distinct
$N_\infty$ operads for dihedral groups $D_{p^n}$, $p > 2$ prime, and cyclic
groups $C_{qp^n}$, $p \neq q$ prime. We then further display some of the
beautiful combinatorics obtained by restricting to certain homotopically
meaningful $N_\infty$ operads for these groups.
