  QCD sum rules are used to study the mass spectrum of bottom-charm hybrid
$\bar bGc$ systems. The correlation functions and the spectral densities are
calculated up to dimension six condensates at leading order of $\alpha_s$ for
several $J^P$ quantum numbers. After performing the QCD sum rule analysis, we
predict the masses of the $J^P=0^{-}, 0^{+}, 1^{-}, 1^{+}, 2^{-}, 2^{+}$
bottom-charm hybrids. These mass predictions show a similar supermultiplet
structure as the bottomonium and charmonium hybrids. Using the QCD sum-rule
mass predictions we analyze the possible hadronic decay patterns of the $\bar
cGc$, $\bar bGc$ and $\bar bGb$ hybrids including the open-flavour and
hidden-flavour mechanisms.
