  In this paper we study various Hardy inequalities in the Heisenberg group
$\mathbb H^n$, w.r.t. the Carnot-Carath\'eodory distance $\delta$ from the
origin. We firstly show that the optimal constant for the Hardy inequality is
strictly smaller than $n^2 = (Q-2)^2/4$, where $Q$ is the homogenous dimension.
Then, we prove that, independently of $n$, the Heisenberg group does not
support a radial Hardy inequality, i.e., a Hardy inequality where the gradient
term is replaced by its projection along $\nabla_{\mathbb
  H}\delta$. This is in stark contrast with the Euclidean case, where the
radial Hardy inequality is equivalent to the standard one, and has the same
constant.
  Motivated by these results, we consider Hardy inequalities for non-radial
directions, i.e., directions tangent to the Carnot-Carath\'eodory balls. In
particular, we show that the associated constant is bounded on homogeneous
cones $C_\Sigma$ with base $\Sigma\subset \mathbb S^{2n}$, even when $\Sigma$
degenerates to a point. This is a genuinely sub-Riemannian behavior, as such
constant is well-known to explode for homogeneous cones in the Euclidean space.
