Bravais lattice
Crystals are grouped into seven crystal systems, according to characteristic symmetry. The characteristic symmetry is the minimum symmetry of that system. The characteristic symmetry of a crystal is a combination of one or more rotations and inversions.
A lattice is a regular array of points. Each point must have the same number of neighbors as every other point and the neighbors must always be found at the same distances and directions. All points are in the same environment. A Bravais Lattice is a three dimensional lattice. A Bravais Lattice tiles space without any gaps or holes. There are 14 ways in which this can be accomplished. Lattices are characterized by translation symmetry.
Bravais Lattices contain seven crystal systems and four lattice centering types. The seven crystal systems along with their characteristic elements of symmetry are presented below.
No.  Type  Description 
1  Primitive  Lattice points on corners only. Symbol: P. 
2  Face Centered  Lattice points on corners as well as centered on faces. Symbols: A (bc faces); B (ac faces); C (ab faces). 
3  AllFace Centered  Lattice points on corners as well as in the centers of all faces. Symbol: F. 
4  BodyCentered  Lattice points on corners as well as in the center of the unit cell body. Symbol: I. 
The names below are linked to PDB files of Bravais Lattices. Each cell is easily created from simple geometry, and laws of sines and cosines. For clarity the centered points are colored differently from the other points, but in fact, are identical by translational symmetry. For the hexagonal cell, the lattice is shown with and without the neighboring unit cells. The file containing the neighboring unit cells illustrates the hexagonal geometry. Three hexagonal unit cells form a hexagonal shape.
No.  Bravais Lattice Type 
Coordinate Description 
Crystal System/ Characteristic Symmetry 
1  Primitive Cubic (P) 
a = b = c a = b = g = 90 
Cubic
Four 3fold axes along a+b+c, –a+b+c, a–b+c, –a–b+c. 
2  Face Centered Cubic (F) 
a = b = c a = b = g = 90 

3  Body Centered Cubic (I) 
a = b = c a = b = g = 90 

4  Primitive Orthorhombic (P) 
a =/= b =/= c a = b = g = 90 
Orthorhombic
Three mutually perpendicular 2fold rotation or rotatoryinversion axes along a, b, and c. 
5  Face Centered Orthorhombic (C) 
a =/= b =/= c a = b = g = 90 

6  Face Centered Orthorhombic (F) 
a =/= b =/= c a = b = g = 90 

7  Body Centered Orthorhombic (I) 
a =/= b =/= c a = b = g = 90 

8  Primitive Tetragonal (P) 
a = b =/= c a = b = g = 90 
Tetragonal
A single 4fold rotation or rotatoryinversion axis along c. 
9  Body Centered Tetragonal (I) 
a = b =/= c a = b = g = 90 

10  Simple Monoclinic (P) 
a =/= b =/= c a = g = 90, b =/= 90 
Monoclinic
A single 2fold rotation or rotary inversion axis along b. 
11  BFace Centered Monoclinic (C) 
a =/= b =/= c a = g = 90, b =/= 90 

12  Hexagonal (P)
one cell three cells 
a = b =/= c a = b = 90, g = 120 
AHexagonal
A single 6fold rotation or rotatoryinversion axis along c. 
13  Triclinic (P) 
a =/= b =/= c a =/= b =/= g =/= 90 
Triclinic
Identity or inversion in any direction. 
14  Primative Rhombohedral (P) 
a = b = c a = b = g =/= 90 
Trigonal
Threefold axis along c. 