We have carried out an extensive search for stable polymorphs of carbon nitride with C3N5 stoichiometry using the minima hopping method. Contrary to the widely held opinion that stacked, planar, graphite-like structures are energetically the most stable carbon nitride polymorphs for various nitrogen contents, we find that this does not apply for nitrogen-rich materials owing to the high abundance of N-N bonds. In fact, our results disclose novel morphologies with moieties not previously considered for C3N5. We demonstrate that nitrogen-rich compounds crystallize in a large variety of different structures due to particular characteristics of their energy landscapes. The newly found low-energy structures of C3N5 have band gaps within good agreement with the values measured in experimental studies.