Abstract Higher-order topological insulators hosting intriguing topologically protected hinge or corner states are of significant research interest. However, materials that possess higher-order topological hinge states associated with gapless bulk Dirac phases still need to be explored. Using first-principles calculations with hybrid exchange functional, we explore the electronic structure and topological properties of Y₃InC and a few of its sister compounds, totaling 16 bulk materials. A symmetry-protected triple point phase, with dominated d-t_2g character, is observed in Y₃InC without spin-orbit coupling. Interestingly, the spin-orbit coupling induces a twin Dirac node phase in the bulk Y₃InC. Furthermore, the computed Z₄ topological invariant reveals the higher-order topological nature of investigated materials. To demonstrate the gapless hinge states, we conduct edge state calculations using a rod-shaped geometry of Y₃InC. Remarkably, Y₃InC is identified to host multi-Dirac nodes in the bulk and surface phases together with the higher-order hinge states. These results lay the groundwork for further experimental and theoretical investigations into cubic antiperovskite materials for higher-order topological phases.