Direct Numerical Simulations are performed to investigate the laminar-turbulent tran- sition in a boundary layer on a sharp cone at Mach 6. The motivation for this research is to make a contribution towards understanding the nonlinear stages of transition and the final breakdown to turbulence in hypersonic boundary layers. Towards this end, three breakdown mechanisms were considered, namely, the second-mode fundamental (K-type), subharmonic (N-/H-type) and oblique breakdown. The simulations were carried out for the laboratory conditions of the hypersonic transition experiments conducted at Purdue University. Several small and medium scale simulations were carried out to explore the parameter space for fundamental and subharmonic resonance. These simulations indi- cated that for the chosen experimental conditions, the fundamental resonance was much stronger than subharmonic resonance. Subsequently a set of highly resolved fundamental and oblique breakdown simulations were performed. The nonlinear interactions observed during the breakdown process are discussed in great detail in this paper. A detailed de- scription of the flow structures that arise due to these nonlinear interactions is provided and an analysis of the skin friction and heat transfer is also presented. These controlled transition simulations clearly demonstrate that both mechanisms, fundamental and oblique breakdown, may indeed be viable paths to complete breakdown to turbulence in hypersonic boundary layers at Mach 6.