Diagrid structural systems have emerged as innovative and adaptable structural systems for tall buildings. These structures contain diagonal truss elements throughout the height of the structure. The present study investigated the effect of an inner steel moment frame core and an inner concentrically-braced core on a diagrid tube system under lateral seismic loading. The effect of inner cores on ductility and failure of the structures was assessed. A 36-story composite diagrid structural system with three different arrangements of diagonal angles was considered and the response spectrum analysis and nonlinear static analysis were conducted. The diagrid structures with inner cores indicated that most of the transferring lateral force in all stories is tolerated by the diagrid system; only in the last story was the shear absorbed by the internal core. Analytical results show that as the slope of braces increased, the lateral strength decreased. Diagrid structures with an inner braced core indicated greater ductility and strength than diagrid structures with inner moment frame cores. Moreover, the results indicate that the diagrid tube with an inner core produced no significant increase in ductility. To avoid premature buckling of braces, perimeter diagonal braces should be replaced with buckling-restrained elements.