  We prove a Koebe distortion theorem for the average derivative of a
quasiconformal mapping between domains in the sub-Riemannian Heisenberg group
$\mathbb{H}_1$. Several auxiliary properties of quasiconformal mappings between
subdomains of $\mathbb{H}_1$ are proven, including distortion of balls
estimates and local BMO-estimates for the logarithm of the Jacobian of a
quasiconformal mapping. Applications of the Koebe theorem include diameter
bounds for images of curves, comparison of integrals of the average derivative
and the operator norm of the horizontal differential, as well as the study of
quasiconformal densities and metrics in domains in $\mathbb{H}_1$. The theorems
are discussed for the sub-Riemannian and the Kor\'anyi distances. This extends
results due to Astala--Gehring, Astala--Koskela, Koskela and
Bonk--Koskela--Rohde.
