Small Partial Latin Squares that Cannot be Embedded in a Cayley Table

We answer a question posed by Déenes and Keedwell that is equivalent to the following. For each order n what is the smallest size of a partial latin square that cannot be embedded into the Cayley table of any group of order n? We also solve some variants of this question and in each case classify the smallest examples that cannot be embedded. We close with a question about embedding of diagonal partial latin squares in Cayley tables.