Recent theoretical studies have identified carbon-nitrogen cages that are potentially stable high energy density materials (HEDM). One such molecule is an N6C6H6 cage in which a six-membered ring of nitrogen is bonded to C3H3 triangles on both sides. This molecule is based on the structure of the most stable N-12 cage, with six carbon atoms substituted into the structure. In the current study, several N6C6H6 isomers (including the previously studied cage) are examined by theoretical calculations to determine which is actually the most stable. Stability will be evaluated from two points of view: (1) thermodynamic stability of one isomer versus another and (2) kinetic stability of each isomer as determined by the energetics of bond breaking. Density functional theory (B3LYP), perturbation theory (MP2 and MP4), and coupled-cluster theory (CCSD(T)) are used in this study, along with the correlation-consistent basis sets of Dunning. Trends in thermodynamic and kinetic stability are discussed.