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 $\left|\right(QQ{)}^{6_c}(\overline{Q}\overline{Q}{)}^{{\overline{6}}_c}⟩$ configuration over the color-triplet $\left|\right(QQ{)}^{{\overline{3}}_c}(\overline{Q}\overline{Q}{)}^{3_c}⟩$ 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 $1^{+}$ $bb\overline{b}\overline{c}$ state with a mass 15719.1 MeV. It is just above the ${\eta }_b{\overline{B}}_c$ threshold. But this channel is forbidden because of the conservation of the angular momentum and parity.