Upright square Simple 
diagonal square Centered 

In mathematics, the square lattice is a type of lattice in a twodimensional Euclidean space. It is the twodimensional version of the integer lattice, denoted as .^{[1]} It is one of the five types of twodimensional lattices as classified by their symmetry groups;^{[2]} its symmetry group in IUC notation as p4m,^{[3]} Coxeter notation as [4,4],^{[4]} and orbifold notation as *442.^{[5]}
Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as the upright square lattice and diagonal square lattice; the latter is also called the centered square lattice.^{[6]} They differ by an angle of 45°. This is related to the fact that a square lattice can be partitioned into two square sublattices, as is evident in the colouring of a checkerboard.
The square lattice's symmetry category is wallpaper group p4m. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. An upright square lattice can be viewed as a diagonal square lattice with a mesh size that is √2 times as large, with the centers of the squares added. Correspondingly, after adding the centers of the squares of an upright square lattice one obtains a diagonal square lattice with a mesh size that is √2 times as small as that of the original lattice. A pattern with 4fold rotational symmetry has a square lattice of 4fold rotocenters that is a factor √2 finer and diagonally oriented relative to the lattice of translational symmetry.
With respect to reflection axes there are three possibilities:
p4, [4,4]^{+}, (442)  p4g, [4,4^{+}], (4*2)  p4m, [4,4], (*442) 

Wallpaper group p4, with the arrangement within a primitive cell of the 2 and 4fold rotocenters (also applicable for p4g and p4m). Fundamental domain

Wallpaper group p4g. There are reflection axes in two directions, not through the 4fold rotocenters. Fundamental domain

Wallpaper group p4m. There are reflection axes in four directions, through the 4fold rotocenters. In two directions the reflection axes are oriented the same as, and as dense as, those for p4g, but shifted. In the other two directions they are linearly a factor √2 denser. Fundamental domain

The square lattice class names, Schönflies notation, HermannMauguin notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table below.
Geometric class, point group  Arithmetic class 
Wallpaper groups  

Schön.  Intl  Orb.  Cox.  
C_{4}  4  (44)  [4]^{+}  None  p4 (442) 

D_{4}  4mm  (*44)  [4]  Both  p4m (*442) 
p4g (4*2) 