  We have studied the charmonium and bottomonium hybrid states with various
$J^{PC}$ quantum numbers in QCD sum rules. At leading order in $\alpha_s$, the
two-point correlation functions have been calculated up to dimension six
including the tri-gluon condensate and four-quark condensate. After performing
the QCD sum rule analysis, we have confirmed that the dimension six condensates
can stabilize the hybrid sum rules and allow the reliable mass predictions. We
have updated the mass spectra of the charmonium and bottomonium hybrid states
and identified that the negative-parity states with $J^{PC}=(0, 1, 2)^{-+},
1^{--}$ form the lightest hybrid supermultiplet while the positive-parity
states with $J^{PC}=(0, 1)^{+-}, (0, 1, 2)^{++}$ belong to a heavier hybrid
supermultiplet.
