We discuss the fixed-point structure and symmetry breaking patterns of hot and dense QCD. Our study particularly addresses the phase structure at low temperature and large quark chemical potential, a region where the application of fully first-principles approaches is currently difficult at best. To this end, we employ a Fierz-complete set of four-quark interactions which are dynamically generated by the gauge degrees of freedom in the renormalization group (RG) approach underlying our study. We observe that the dense regime is dominated by diquark degrees of freedom in contrast to the dominance of pions at small quark chemical potential. This change in the dominance of the associated interaction channels is driven by a corresponding change in the fixed-point structure when the chemical potential is varied. In particular at large quark chemical potential we find that the use of a Fierz-complete set of four-quark interactions is indeed of great importance. Phenomenological implications of these findings for the critical temperature and the equation of state at large chemical potential are discussed.