B3LYP and CCSD(T) (single-point) calculations are performed on structures and interconversion transition states for five C2N2 isomers, i.e., the linear structures NCCN 1, CNCN 2, CNNC 3, and CCNN 4, and the NNC three-membered ring structure with exocyclic C-C bonding 5. A schematic potential-energy surface of the five C2N2 isomers is established. It is shown that the elusive and long searched for isomer 3 lies in a deep potential well against isomerization to the well-studied isomers 1 and 2, while the two transition states, which have been predicted to connect the isomers 1 and 3, are actually associated with the interexchange of the two cyano groups in isomer 2. It is also shown that the experimentally unknown but expectedly important isomer 4 may take an isomerization pathway to the isomer 1 via the intermediate 5, and the isomer 4 also lies in a deep potential well. Both the isomers 3 and 4 may be thermally as stable as the isomer 2. The results presented in this paper may provide useful information for the identification of the two isomers 3 and 4.