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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 ananalogue of this result on arbitrary graded Lie groups. More precisely, to anyfinite 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 thegroup Fourier transform $\widehat{d\sigma}$ of $d\sigma$ with respect to afixed Rockland operator $\mathcal{R}$ on $G$ that assures the boundedness of$\mathcal{M}_D^{d\sigma}$ on $L^p(G)$ for all $1<p\leq \infty.$

Boundedness of the dyadic maximal function on graded Lie groups