I'm interested in the symmetries of two-dimensional patterns that have two sides. In other words, what discrete groups can be formed from the three-dimensional Euclidean isometries which preserve a plane?
Is there a name for this type of group? Where can I find an enumeration of them?
These are sometimes called “layer groups”, and there are 80 of them. The layer groups, Frieze groups and rod groups are together known in crystallography as “subperiodic groups”
There is a Wikipedia article about layer groups about them with several references, among them the International Tables for Crystallography, Volume E: Subperiodic groups.