A search for fermionic top quark partners $T$ of charge $2/3$ is presented. The search is carried out in proton-proton collisions corresponding to an integrated luminosity of $19.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected at a center-of-mass energy of $\sqrt{s}=8\text{ }\text{ }\mathrm{TeV}$ with the CMS detector at the LHC. The $T$ quarks are assumed to be produced strongly in pairs and can decay into $tH$, $tZ$, and $bW$. The search is performed in five exclusive channels: a single-lepton channel, a multilepton channel, two all-hadronic channels optimized either for the $bW$ or the $tH$ decay, and one channel in which the Higgs boson decays into two photons. The results are found to be compatible with the standard model expectations in all the investigated final states. A statistical combination of these results is performed and lower limits on the $T$ quark mass are set. Depending on the branching fractions, lower mass limits between 720 and 920 GeV at 95% confidence level are found. These are among the strongest limits on vectorlike $T$ quarks obtained to date.