We explore the feasibility of the discrete flavor symmetry $S_3$ to explain the pattern of neutrino masses and mixings. In the flavor basis, the neutrino mass matrix is taken to be invariant under $S_3$ symmetry at the zeroth order. The effects of breaking $S_3$ symmetry by matrices invariant under different $S_2$ subgroups of $S_3$ are studied. The resulting retrocirculant perturbation matrix leads to the perturbed $S_3$ neutrino mass matrix having a trimaximal eigenvector as suggested by the solar neutrino data. It is found that after the third order perturbation the neutrino mixings only depend on the perturbation parameter and are consistent with the current experimental data leading to very small deviations from tribimaximal mixing. These perturbations in the $S_3$ invariant neutrino mass matrix result in interesting interplay between the solar and the reactor neutrino mixing angles. We also get a strongly suppressed range of effective Majorana mass which lies well within the reach of the ongoing experiments.