We map out the possible ordered states in bilayer graphene at the neutrality point by extending the previous renormalization group treatment of many-body instabilities to finite temperature, trigonal warping and externally applied perpendicular electric field. We were able to analytically determine all outcomes of the RG flow equations for the 9 four-fermion coupling constants. While the full phase diagram exhibits a rich structure, we confirm that when forward scattering dominates, the only ordering tendency with divergent susceptibility at finite temperature is the nematic. At finite temperature this result is stable with respect to small back and layer imbalance scattering; further increasing their strength leads to the layer antiferromagnet. We also determine conditions for other ordered states to appear and compare our results to the special cases of attractive and repulsive Hubbard models where exact results are available.