  We study inequalities related to the heat kernel for the hypoelliptic
sublaplacian on an H-type Lie group. Specifically, we obtain precise pointwise
upper and lower bounds on the heat kernel function itself. We then apply these
bounds to derive an estimate on the gradient of solutions of the heat equation,
which is known to have various significant consequences including logarithmic
Sobolev inequalities. We also present a computation of the heat kernel, and a
discussion of the geometry of H-type groups including their geodesics and
Carnot-Carath\'eodory distance functions.
