Fermi-Löwdin orbital self-interaction corrected density functional theory (FLO-SIC DFT) is applied to C3H5, NO3-, O3, and CH3. In general our results indicate that FLO-SIC does favor symmetric setups for molecules with nontrivial chemical bonding. Further we discuss two types of possible symmetry breaking. In the case of O3 a ground state density that breaks symmetry can be found for the symmetric molecular geometry that may be caused by insufficient treatment of correlation energy. The CH3 radical presents a second type of symmetry breaking were the lowest energy geometry becomes distorted. The latter highlights the importance of further development of approximate DFT functionals as well as further extensions of the FLO-SIC method to overcome such nonphysical artifacts.