A search for the decays of the Higgs and $Z$ bosons to $J/\psi\gamma$ and $\Upsilon(nS)\gamma$ ($n=1,2,3$) is performed with $pp$ collision data samples corresponding to integrated luminosities of up to $20.3\mathrm{fb}^{-1}$ collected at $\sqrt{s}=8\mathrm{TeV}$ with the ATLAS detector at the CERN Large Hadron Collider. No significant excess of events is observed above expected backgrounds and 95\% CL upper limits are placed on the branching fractions. In the $J/\psi\gamma$ final state the limits are $1.5\times10^{-3}$ and $2.6\times10^{-6}$ for the Higgs and $Z$ bosons, respectively, while in the \Upsilon(1S,2S,3S)\,\gamma final states the limits are $(1.3,1.9,1.3)\times10^{-3}$ and $(3.4,6.5,5.4)\times10^{-6}$, respectively.

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