We study the processes $e^+e^− \rightarrow \gamma ISR J / \psi$ , where $J / \psi \rightarrow \pi^+ \pi^− \pi^0$, and $J / \psi \rightarrow K^+ K^− \pi^0$ , and $J / \psi \rightarrow K^0_S K^\pm \pi^\mp$ using a data sample of $519     fb^{− 1}$ recorded with the BABAR detector operating at the SLAC PEP-II asymmetric-energy $e^+e^−$ collider at center-of-mass energies at and near the $\Upsilon (nS) (n = 2 , 3 , 4 )$ resonances. We measure the ratio of branching fractions $\mathscr R_1 = \frac{\mathscr B ( J / \psi \rightarrow K^+ K^− \pi^0 )}{\mathscr B ( J / \psi \rightarrow \pi^+ \pi^− \pi^0 )}$ and $\mathscr R_2 = \frac{\mathscr B ( J / \psi \rightarrow K_0^S+ K^\pm \pi^\mp )}{\mathscr B ( J / \psi \rightarrow \pi^+ \pi^− \pi^0 )}$. We perform Dalitz plot analyses of the three $J / \psi$ decay modes and measure fractions for resonances contributing to the decays. We also analyze the $J / \psi \rightarrow \pi^+ \pi^− \pi^0$ decay using the Veneziano model. We observe structures compatible with the presence of $\rho$ (1450) in all three J / $\psi$ decay modes and measure the relative branching fraction: $\mathscr R(\rho(1450)) = \frac{\mathscr B (\rho( 1450 ) \rightarrow K^+ K^−)}{B ( \rho (1450 ) \rightarrow \pi^+ \pi^− )}$ = 0.307 $\pm$ 0.084 ( stat ) $\pm$ 0.082 (sys).