We search for $J/psi$ radiative decays into a weakly interacting neutral particle, namely an invisible particle, using the $J/psi$ produced through the process $psi(3686)topi^+pi^-J/psi$ in a data sample of $(448.1pm2.9)times 10^6$ $psi(3686)$ decays collected by the BESIII detector at BEPCII. No significant signal is observed. Using a modified frequentist method, upper limits on the branching fractions are set under different assumptions of invisible particle masses up to 1.2 $mathrm{ Gekern -0.1em V}/c^2$. The upper limit corresponding to an invisible particle with zero mass is 7.0$times 10^{-7}$ at the 90% confidence level.
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