We investigate the quality (Q) factor and the mode dispersion of single-defect nanocavities based on a triangular-lattice GaAs photonic-crystal (PC) membrane, which contain InAs quantum dots (QDs) as a broadband emitter. To obtain a high Q factor for the dipole mode, we modulate the radii and positions of the air holes surrounding the nanocavity while keeping sixfold symmetry. A maximum Q of 17 000 is experimentally demonstrated with a mode volume of V=0.39(λ/n)3. We obtain a Q/V of 44 000(n/λ)3, one of the highest values ever reported with QD-embedded PC nanocavities. We also observe ten cavity modes within the first photonic band gap for the modulated structure. Their dispersion and polarization properties agree well with the numerical results.