A novel inbred maize variety designated PHHHN and seed, plants and plant parts thereof. Methods for producing a maize plant that comprise crossing inbred maize variety PHHHN with another maize plant. Methods for producing a maize plant containing in its genetic material one or more traits introgressed into PHHHN 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 variety PHHHN or a trait conversion of PHHHN with another maize variety. Inbred maize varieties derived from inbred maize variety PHHHN, methods for producing other inbred maize varieties derived from inbred maize variety PHHHN and the inbred maize varieties and their parts derived by the use of those methods.