Enumerating the Derangements of an $n$-Cube via M\"obius Inversion

In $\mathcal L$, the semilattice of faces of an $n$-cube, we count the number of automorphisms of $\mathcal L$ that fix a given subalgebra -- either pointwise or as a subalgebra. By using M\"obius inversion we get a formula for the number of derangements on the $n$-cube in terms of the M\"obius function on the lattice of MR-subalgebras. We compute this M\"obius function.