Given two 3-uniform hypergraphs F and G = (V, E), we say that G has an F-covering if we can cover V with copies of F. The minimum codegree of G is the largest integer d such that every pair of vertices from V is contained in at least d triples from E. Define c(2)(n, F) to be the largest minimum codegree among all n-vertex 3-graphs G that contain no F-covering. Determining c(2)(n, F) is a natural problem intermediate (but distinct) from the well-studied Turan problems and tiling problems. In this paper, we determine c(2)(n, K-4) (for n > 98) and the associated extremal configurations (for n > 998), where K-4 denotes the complete 3-graph on 4 vertices. We also obtain bounds on c(2)(n, F) which are apart by at most 2 in the cases where F is K-4(-) (K-4 with one edge removed), K-5(-), and the tight cycle C-5 on 5 vertices.