The conformal loop ensemble CLE(k) is a conformally invariant ensemble of random non-crossing loops in a proper simply connected domain in the complex plane. In this work we study the geometry of the CLE gasket, the set of points not surrounded by any loop in the CLE. We show that the Hausdorff dimension of the gasket is bounded from below by 2-(8-k)(3k-8)/(32k) when 4