Let $A$ be an affine plane of finite order $n$ and $T$ a group of translations, not all with the same direction, of $A$. If $|T| \ge n$, then $n$ is a power of a prime. If $|T| < n^{2}$ and the non-trivial subgroups with fixed directions all have the same order $k > 1$, then the number of directions for which $T$ contains no non-trivial translations is either 2 or $> \sqrt{n}$ in the first case $|T| = n$ is a power of $h = 2$.