Dihedral groups are all realizable in the plane. You have probably seen a dihedral group and didn't realize it. Some flowers have petals that make dihedral groups. Some examples are shown below. This is a D1 group.
All dihedral groups contain a reflection.
This is a D3 group. It contains three reflections and a rotation of order 3. Some people call these wheel patterns. This is a D12. Patterns like these often appear in stained glass windows.