  We provide the first classification of three-photon eigenstates in a finite
periodic array of two-level atoms coupled to a waveguide. We focus on the
strongly subwavelength limit and show the hierarchical structure of the
eigenstates in the complex plane. The main characteristic eigenstates are
explored using entanglement entropy as a distinguishing feature. We show that
the rich interplay of effects from order, chaos to localisation found in
two-photon systems extends naturally to three-photon systems. There also exist
interaction-induced localised states unique to three-photon systems such as
bound trimers, corner states and trimer edge states.
