Transfer systems are combinatorial objects with connections to both equivariant homotopy theory and model categories. (In particular, transfer systems on the subgroup lattice of a group, G, determine both the N-infinity operads associated to G and the possible model structures on the subgroup lattice.) In work with Balchin, Barrero, Scheirer, Sulyma, Wisdom, Zapata Castro, and work with Evans, we investigate several questions related to the generation of transfer systems. This talk will focus on our results related to the complexity of a finite group, G, which we define to be the maximal size of a minimal generating set for a transfer system on the subgroup lattice of G. In order to determine the complexity of several families of groups, we introduce rainbow diagrams, a new representation of appropriately symmetric generating sets for transfer systems.