A bstract We investigate finite abelian groups which can represent symmetries of the N -Higgs-doublet models with quarks. We build a general formalism based on the powerful method of the Smith normal form and obtain an analytic upper bound on the order of abelian symmetry groups for any N . We investigate in detail the case N = 2 and rederive known results in a more compact and intuitive fashion. We also study the NHDM with the maximal finite abelian symmetry for all small N cases up to N = 5, and show that in each case all Yukawa textures compatible with such symmetry originate from a unique basic structure. This work opens the way to a systematic exploration of phenomenology of the NHDM with a desired symmetry, and illustrates the power of the Smith normal form technique.