In order to investigate the group-based potential games with infeasible profiles, the group-based potential games with incomplete-profile is proposed. Firstly, using the semi-tensor product theory, a method is provided to verify whether a finite game is a (strongly) group-based potential game with incomplete-profile by checking whether a linear equation has a solution. Then, the relationship between the group-based potential games and the group-based potential games with incomplete-profile is studied. Finally, an algorithm is proposed to find the smallest set of infeasible profiles, such that the finite game will be a group-based potential game in the remaining profiles.