Let \(a= \{a(k)\}^\infty_{k=-\infty}\) be a complex sequence with finite support. The subdivision operator \(S_a= (a(j- 2k))^\infty_{j,k=-\infty}\) plays an important role in wavelet analysis and subdivision algorithms. Note that \(S_a\) is a bi-infinite Toeplitz-like matrix. The author shows that the spectrum of \(S_a\) in \(l_p\) \((1\leq p\leq\infty)\) is always a closed disc centered at \(0\). Moreover, all the points in the open disc of the spectrum of \(S_a\) are in the residual spectrum, except for finitely many points.