We study global conformal Assouad dimension in the Heisenberg group
H
n
\mathbb {H}^n
. For each
α
∈
{
0
}
∪
[
1
,
2
n
+
2
]
\alpha \in \{0\}\cup [1,2n+2]
, there is a bounded set in
H
n
\mathbb {H}^n
with Assouad dimension
α
\alpha
whose Assouad dimension cannot be lowered by any quasiconformal map of
H
n
\mathbb {H}^n
. On the other hand, for any set
S
S
in
H
n
\mathbb {H}^n
with Assouad dimension strictly less than one, the infimum of the Assouad dimensions of sets
F
(
S
)
F(S)
, taken over all quasiconformal maps
F
F
of
H
n
\mathbb {H}^n
, equals zero. We also consider dilatation-dependent bounds for quasiconformal distortion of Assouad dimension. The proofs use recent advances in self-similar fractal geometry and tilings in
H
n
\mathbb {H}^n
and regularity of the Carnot–Carathéodory distance from smooth hypersurfaces.