# posets – When are quotients of the Boolean lattice dissective?

Let $$B_n$$ be the Boolean lattice and $$G$$ a subgroup of $$S_n$$ acting on $$B_n$$. Let $$P_G=B_n/G$$ denote the quotient poset.

Question 1: When is $$P_G$$ a lattice? When is it distributive in that case?

An example where it is not a lattice is for $$n=5$$ and the subgroup $$G=langle(1,2,3,4,5)rangle$$. But in this case it is still dissective, where dissective means that the MacNeille completion of the poset is a distributive lattice, see https://nreadin.math.ncsu.edu/papers/dissective.pdf .

Question 2: When is the poset $$P_G$$ dissective? When is it dissective but not a lattice?

I am not sure whether one can expect a general answer, but restrictions on G (cyclic, abelian…) or specific example of $$P_G$$ being dissective and not a lattice are also welcome