Trees · Superchallenge problem

The octahedron graph has vertices $1,\ldots,6$ split into three opposite pairs $\{1,4\}$, $\{2,5\}$, $\{3,6\}$, with every edge present EXCEPT within a pair (so it is $4$-regular). Its Laplacian eigenvalues are $0$ (multiplicity $1$), $4$ (multiplicity $3$), and $6$ (multiplicity $2$). Find the number of spanning trees of the octahedron.