  We continue the exploration of various aspects of divisibility of
ultrafilters, adding one more relation to the picture: multiplicative finite
embeddability. We show that it lies between divisibility relations $\mid_M$ and
$\widetilde{\mid}$. The set of its minimal elements proves to be very rich, and
the $\widetilde{\mid}$-hierarchy is used to get a better intuition of this
richness. We find the place of the set of $\widetilde{\mid}$-maximal
ultrafilters among some known families of ultrafilters. Finally, we introduce
new notions of largeness of subsets of $\mathbb{N}$, and compare it to other
such notions, important for infinite combinatorics and topological dynamics.
