  In the framework of an extended chromomagnetic model, we systematically study
the mass spectrum of the $S$-wave $qQ\bar{Q}\bar{Q}$ tetraquarks. Their mass
spectra are mainly determined by the color interaction. For the
$qc\bar{c}\bar{c}$, $qb\bar{c}\bar{c}$ and $qb\bar{b}\bar{b}$ tetraquarks, the
color interaction favors the color-sextet
$\ket{(qQ)^{6_{c}}(\bar{Q}\bar{Q})^{\bar{6}_{c}}}$ configuration over the
color-triplet $\ket{(qQ)^{\bar{3}_{c}}(\bar{Q}\bar{Q})^{3_{c}}}$ one. But for
the $qc\bar{b}\bar{b}$ tetraquarks, the color-triplet configuration is favored.
We find no stable states which lie below the thresholds of two pseudoscalar
mesons. The lowest axial-vector states with the $qQ\bar{b}\bar{b}$ flavor
configuration may be narrow. They lie just above the thresholds of two
pseudoscalar mesons, but cannot decay into these channels because of the
conservation of the angular momentum and parity.
