Details of selection
This is a simple choice based number the number of rotations. The only tricky issue is when there two points, one with a rotation of order 2 and another with a rotation of order 4 (say).
There are a large number in this collection with a pattern based upon a square:
The 12-pointed star is a centre of the rotation of order 4, but the middle of the four diamonds is the centre of a rotation of order 2. Hence this pattern is counted amongst the patterns with rotations of order 4.
We have the same situations with rotations of order 3 and 6 with:
The 6-pointed star is a centre of a rotation of order 6, while the hexagon is the centre of a rotation of order 3. Hence this pattern is counted amongst the patterns with rotations of order 6.