Using equivariant bifurcation theory, and on the basis of symmetry
considerations independent of the model, we classify
square and hexagonally periodic patterns that typically arise when a homeotropic
or planar isotropic nematic state becomes unstable, perhaps as a consequence of an
applied magnetic or electric field.  We relate this to a Landau - de Gennes 
model for the free energy, and derive dispersion relations in
sufficient generality to illustrate the role of up/down symmetry in determining
which patterns can arise as a stable bifurcation branch from either initial state.