An inbred maize line, designated PH51H, the plants and seeds of inbred maize line PH51H, methods for producing a maize plant, either inbred or hybrid, produced by crossing the inbred maize line PH51H with itself or with another maize plant, and hybrid maize seeds and plants produced by crossing the inbred line PH51H with another maize line or plant and to methods for producing a maize plant containing in its genetic material one or more transgenes and to the transgenic maize plants produced by that method. This invention also relates to inbred maize lines derived from inbred maize line PH51H, to methods for producing other inbred maize lines derived from inbred maize line PH51H and to the inbred maize lines derived by the use of those methods.