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

The combinatorics of $N_\infty$ operads for $C_{qp^n}$ and $D_{p^n}$