A matroidal family is a nonempty set ℱ of connected finite graphs such that for every arbitrary finite graph G the edge sets of the subgraphs of G which are isomorphic to an element of ℱ form a matroid on the edge set of G. In the present paper the question whether there are any matroidal families besides the four previously described by Simões-Pereira is answered affirmatively. It is shown that for every natural number n ⩾ 2 there is a matroidal family that contains the complete graph with n vertices. For n = 4 this settles Simões-Pereira's conjecture that there is a matroidal family containing all wheels.