  In this work, we systematically study the mass spectrum of the fully heavy
tetraquark in an extended chromomagnetic model, which includes both color and
chromomagnetic interactions. Numerical results indicate that the energy level
is mainly determined by the color interaction, which 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. The
chromomagnetic interaction mixes the two color configurations and gives small
splitting. The ground state is always dominated by the color-sextet
configuration. We find no stable state below the lowest heavy quarkonium pair
thresholds. Most states may be wide since they have at least one $S$-wave decay
channel into two $S$-wave mesons. One possible narrow state is the $1^{+}$
$bb\bar{b}\bar{c}$ state with a mass $15719.1~\text{MeV}$. It is just above the
$\eta_{b}\bar{B}_{c}$ threshold. But this channel is forbidden because of the
conservation of the angular momentum and parity.
