A novel inbred maize line designated PHC78 and seed, plants and plant parts thereof. Methods for producing a maize plant that comprise crossing inbred maize line PHC78 with another maize plant. Methods for producing a maize plant containing in its genetic material one or more traits introgressed into PHC78 through backcross conversion and/or transformation, and to the maize seed, plant and plant part produced thereby. Hybrid maize seed, plant or plant part produced by crossing the inbred line PHC78 or a trait conversion of PHC78 with another maize line. Inbred maize lines derived from inbred maize line PHC78, methods for producing other inbred maize lines derived from inbred maize line PHC78 and the inbred maize lines and their parts derived by the use of those methods.