In this article, we report on the computation of the next-to-leading order (NLO) QCD corrections to $pp\to {\mu }^{-}{\overline{\nu }}_{\mu }{e}^{+}{\nu }_e\overline{b}b\overline{b}b$ at the LHC, which is an irreducible background to $pp\to t\overline{t}H(\to b\overline{b})$. This is the first time that a full NLO computation for a $2\to 8$ process with 6 external strongly interacting partons is made public. No approximations are used, and all off-shell and interference effects are taken into account. Cross sections and differential distributions from the full computation are compared to results obtained by using a double-pole approximation for the top quarks. The difference between the full calculation and the one using the double-pole approximation is in general below 5% but can reach 10% in some regions of phase space.