The Poynting vector is not well behaved with respect to special relativity.
It is not part of any four-vector.
Of course the same can be said of the E-vector and the B-vector; they
are not well behaved either.
The Poynting vector can be expressed as the spacelike rows of the timelike
column of a 4x4 matrix "T". The other row in that column is the Poynting
energy density. In particular, if we have observer with 4-velocity u, then
T(u) is 4-vector representing the energy density and energy-flow density.