I need to determine the point group of the following patterns:
I think the one on the left is $D_1$ and the one on the right is $D_2$
To determine which wallpaper group you have, first identify the unit cell and symmetry operations for your images:
Then use the Wallpaper group diagram:
Both images have a maximum of 2-fold rotations and simple reflections. The left image has no perpendicular reflections, making it pmg. The right image has perpendicular reflections with an off-axis rotocenter, making it cmm.
Both of these latices have $D_2$ point groups at the center of the cell, as well as lower-order point groups at other points.