  We study the interplay between large-spin, spin-orbit coupling, and
superfluidity for bosons in a two dimensional optical lattice, focusing on the
spin-1 spin-orbit coupled system recently realized at the Joint Quantum
Institute [Campbell et. al., arXiv:1501.05984]. We find a rich quantum phase
diagram, where, in addition to the conventional phases ---superfluid and
insulator--- contained in the spin-$1$ Bose-Hubbard model, there are new
lattice symmetry breaking phases. For weak interactions, the interplay between
two length scales, the lattice momentum and the spin-orbit wave-vector induce a
phase transition from a uniform superfluid to a phase where bosons
simultaneously condense at the center and edge of the Brillouin zone at a
non-zero spin-orbit strength. This state is characterized by spin density wave
order, which arises from the spin-$1$ nature of the system. Interactions
suppress spin density wave order, and favor a superfluid \textit{only} at the
Brillouin zone edge. This state has spatially oscillating mean field order
parameters, but a homogeneous density. We show that the spin density wave
superfluid phase survives in a two dimensional harmonic trap, and thus
establish that our results are directly applicable to experiments on $^{87}$Rb,
$^7$Li, and $^{41}$K.
