This study entails the third part of a global flare energetics project, in which Ramaty High-Energy Solar Spectroscopic Imager (RHESSI) data of 191 M and X-class flare events from the first 3.5yrs of the Solar Dynamics Observatory mission are analyzed. We fit a thermal and a nonthermal component to RHESSI spectra, yielding the temperature of the differential emission measure (DEM) tail, the nonthermal power-law slope and flux, and the thermal/nonthermal cross-over energy e_co_. From these parameters, we calculate the total nonthermal energy E_nt_ in electrons with two different methods: (1) using the observed cross-over energy e_co_ as low-energy cutoff, and (2) using the low-energy cutoff e_wt_ predicted by the warm thick-target bremsstrahlung model of Kontar et al. Based on a mean temperature of T_e_=8.6MK in active regions, we find low-energy cutoff energies of e_wt_=6.2+/-1.6keV for the warm-target model, which is significantly lower than the cross-over energies e_co_=21+/-6keV. Comparing with the statistics of magnetically dissipated energies E_mag_ and thermal energies E_th_ from the two previous studies, we find the following mean (logarithmic) energy ratios with the warm-target model: E_nt_=0.41E_mag_, E_th_=0.08E_mag_, and E_th_=0.15E_nt_. The total dissipated magnetic energy exceeds the thermal energy in 95% and the nonthermal energy in 71% of the flare events, which confirms that magnetic reconnection processes are sufficient to explain flare energies. The nonthermal energy exceeds the thermal energy in 85% of the events, which largely confirms the warm thick-target model.