  In this paper we present the NLO QCD + NLO EW corrections to the $ZZZ$
production with subsequent $Z$-boson leptonic decays at the LHC by adopting the
improved narrow width approximation, which takes into account off-shell
contributions and spin correlations. Integrated cross sections at $13$, $14$,
$33$ and $100~{\rm TeV}$ hadron colliders and various kinematic distributions
are presented. Our numerical results show that the jet emission correction
accounts for a large part of the total QCD correction, especially in the high
energy region. By applying a proper cut to the jet transverse momentum, e.g.,
$p_{{\rm \,T,jet}} > p_{{\rm \,T,jet}}^{{\rm \,\,\,cut}}=50~{\rm GeV}$, the jet
emission correction can be reduced and the cross section is less dependent on
the factorization/renormalization scale. This work reveals that both the NLO
QCD and the NLO EW corrections are significant. For example, the NLO QCD, NLO
EW and NLO QCD + EW relative corrections in the inclusive event selection
scheme at the $13~{\rm TeV}$ LHC can reach $63.2\%$, $-9.6\%$ and $47.5\%$,
respectively, which means that neither the NLO QCD nor the NLO EW correction is
negligible in precision calculations.
