A finite group $G$ with conjugacy class nX is said to be $nX$-complementary generated if, for any arbitrary $x\in G-\{1\}$, there is an element $y\in nX$ such that $G=⟨x,y⟩$. In this paper we prove that the Conway's group $C{o}_1$ is $nX$-complementary generated for all $n\in {\mathrm{\Pi }}_e(C{o}_1)$. Here ${\mathrm{\Pi }}_e(C{o}_1)$ denotes the set of all element orders of the group $C{o}_1$.