About how many lines can one rotate a regular hexagon through some angle x, O<x<306 deg. so that the hexagon again occupies its original position
I've some difficulties to understand your problem:
In IR² you can only rotate something around a point. So I presume that you would like to reflect the hexagon over a straight line.
There are 6 such lines: Three lines passing through opposite vertices, three lines passing through midpoints of sides.
I've attached an image to show you what I mean.