  Let $1<p\leq \infty$ and let $n\geq 2.$ It was proved independently by C.
Calder\'on, R. Coifman and G. Weiss that the dyadic maximal function
\begin{equation*}
  \mathcal{M}^{d\sigma}_Df(x)=\sup_{j\in\mathbb{Z}}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-2^jy)d\sigma(y)\right|
\end{equation*} is a bounded operator on $L^p(\mathbb{R}^n)$ where $d\sigma(y)$
is the surface measure on $\mathbb{S}^{n-1}.$ In this paper we prove an
analogue of this result on arbitrary graded Lie groups. More precisely, to any
finite Borel measure $d\sigma$ with compact support on a graded Lie group $G,$
we associate the corresponding dyadic maximal function
$\mathcal{M}_D^{d\sigma}$ using the homogeneous structure of the group. Then,
we prove a criterion in terms of the order (at zero and at infinity) of the
group Fourier transform $\widehat{d\sigma}$ of $d\sigma$ with respect to a
fixed Rockland operator $\mathcal{R}$ on $G$ that assures the boundedness of
$\mathcal{M}_D^{d\sigma}$ on $L^p(G)$ for all $1<p\leq \infty.$
