We designed four bicyclic nitramines and three cage nitramines by incorporating -N(NO2)-CH2-N-(NO2)-, -N(NO2), and -O- linkages based on the HMX (1,3,5,7-tetranitro-1,3,5,7-tet razocane) framework. Then, their electronic structure, heats of formation, energetic properties, strain energy, thermal stability, and impact sensitivity were systematically studied using density functional theory (DFT). Compared to the parent compound HMX, all the title compounds have much higher density, better detonation properties, and better oxygen balance. Among them, four compounds have extraordinary high detonation properties (D > 9.70 km/s and P > 44.30 GPa). Moreover, most of the title compounds exhibit better thermal stability and lower impact sensitivity than CL-20 2,4,6,8,10,12-hexanitro-2,4,6,8,10,12-hexaazaisowurtzitane) or HNHAA (hexanitro-hexaazaadamantane). Thus, all of the seven new nitramine compounds are promising candidates for high energy density compounds. In particular, five compounds exhibit a best combination of better oxygen balance, good thermal stability, excellent detonation properties superior to or comparable to CL-20 or HNHAA, and lower impact sensitivity than CL-20 or HNHAA. The results indicate that our unusual design strategy that constructing bicyclic or cage nitramines based on the HMX framework by incorporating the intramolecular linkages is very useful for developing novel energetic compounds with excellent detonation performance and low sensitivity.