All ice crystals (of ordinary 'Ice 1h' that is) have the same basic hexagonal symmetry. The fundamental building block is a unit cell containing four water molecules. Ice crystals can be thought of as made of untold stacks of these unit cells. The stacking naturally produces the shapes that we see in the microscopic and macro world.
At right are a simple hexagonal column and plate with the four crystallographic axes marked. These small crystals make halos.
At lower right is a much larger snowflake crystal. The hexagonal stacking is the same but we see dendritic growths along the three 'a' axis directions and the basal face has the imperfections that large crystals are prone to.