We present a low resolution lattice model for which we can exhaustively generate all possible compact backbone conformations for small proteins. Using simple structural and energetic criteria, for a variety of proteins, we can select for lattice structures that have significant similarities with their known native structures. Our energetic parameters are based on pairwise amino acid contact frequencies in a database of experimentally determined structures. A key step in our method involves the threading of a sequence onto every lattice model, such that a locally optimal pattern of tertiary interactions is formed. We evaluate our results against statistics collected for structures covering all of conformational space, and against statistics collected for permuted sequences. Despite the low resolution of the model, our low energy structures contain many native features. These results indicate that. the overall pattern of hydrophobicity of a sequence significantly constrains the range of folds that sequence is likely to adopt.