Abstract The baby Skyrme model is a (2+1)-dimensional analogue of the Skyrme model, in which baryons are described by topological solitons. We introduce a version of the baby Skyrme model in which the global O(3) symmetry is broken to the dihedral group DN. It is found that the single soliton in this theory is composed of N partons that are topologically confined. The case N=3 is studied in some detail and multi-soliton solutions are computed and related to polyiamonds, which are plane figures composed of equilateral triangles joined by common edges. It is shown that the solitons may be viewed as pieces of a doubly periodic soliton lattice. It is proved, for a general baby Skyrme model on a general torus, that the condition that the energy of a soliton lattice is critical with respect to variations of the torus is equivalent to the field satisfying a virial relation and being, in a precise sense, conformal on average. An alternative model with D3 symmetry is also introduced, which has an exact explicit soliton lattice solution. Soliton solutions are computed and compared in the two D3 theories. Some comments are made regarding the extension of these ideas to the Skyrme model.