Whereas all 230 three-dimensional space groups occur in organic crystals, out of only 17 plane groups some highly symmetric ones such as p31m have not yet been observed in two-dimensional (2D) crystals of organic molecules. Here a kagome network with p31m symmetry is reported for cobalt phthalocyanine on Cu(111). This unusual structure results from substrate-induced reduction of molecular symmetry and substrate-mediated interaction via quantum interference of surface electrons. These interactions provide additional control over the symmetry of 2D crystals of phthalocyanines and lead to a variety of other symmetries in self-assembled arrays.