  We answer a question posed by D\'enes 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.
