  We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of
factorizable Markov maps acting on a von Neumann algebra $M$ equipped with a
normal faithful state can be dilated by a group of Markov $*$-automorphisms
analogous to the case of a single factorizable Markov operator, which is an
optimal result. We also give a version of this result for strongly continuous
semigroups of operators acting on noncommutative $\mathrm{L}^p$-spaces and
examples of semigroups to which the results of this paper can be applied. Our
results implies the boundedness of the McIntosh's $\mathrm{H}^\infty$
functional calculus of the generators of these semigroups on the associated
noncommutative $\mathrm{L}^p$-spaces generalising some previous work from
Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson
semigroups which are even new in the case of $\mathbb{R}^n$.
