In this thesis we compute the motivic cohomology ring (also known as Bloch's higher Chow groups) with finite coefficients for the two nonabelian groups of order $27$, thought of as affine algebraic groups over $\mathbb{C}$. Specifically, letting $\tau$ denote a generator of the motivic cohomology group $H^{0,1}(BG,\Z/3) \cong \Z/3$ where $G$ is one of these groups, we show that the motivic cohomology ring contains no $\tau$-torsion, and so can be computed as a weight filtration on the ordinary group cohomology. In the case of a prime $p > 3$, there are also two nonabelian groups of order $p^3$. We make progress toward computing the motivic cohomology in the general case as well by reducing the question to understanding the $\tau$-torsion on the motivic cohomology of a $p$-dimensional variety; we also compute the motivic cohomology of $BG$ for general $p$ modulo the $\tau$-torsion classes.