  We study vortex-lattice phases for a Bose gas trapped in a rotating
optical-lattice near the superfluid--Mott-insulator transition. We find a
series of abrupt structural phase transitions where vortices are pinned with
their cores only on plaquettes or only on sites. We discuss connections between
these vortex structures and the Hofstadter-butterfly spectrum of free particles
on a rotating lattice.
