  Let $U(q)$ be a Sylow $p$-subgroup of the Chevalley groups $D_4(q)$ where $q$
is a power of a prime $p$. We describe a construction of all complex
irreducible characters of $U(q)$ and obtain a classification of these
irreducible characters via the root subgroups which are contained in the center
of these characters. Furthermore, we show that the multiplicities of the
degrees of these irreducible characters are given by polynomials in $q-1$ with
nonnegative coefficients.
