There are five
Bravais lattices
in a plane and 14 in threedimensional space. They represent the
distinct ways to fill an area or volume by repeating a single unit cell periodically and without
leaving any spaces. All Bravais lattices contain only a single type of
lattice point
(which, in the case of crystal lattices, represents an atom)  all lattice points are indistinguishable.
There are two distinguishing features between Bravais lattices, the symmetry of the unit cell and
whether the unit cell is primitive or (body, side, or face) centred, i.e. whether it
contains additional lattice points beyond the corner ones.
In the case of simple metals, where the structure consists entirely of identical atoms, one of
the Bravais lattices would be sufficient to describe the periodic structure. However, in
binary compounds
(or even more complex ones) such as salts (e.g. NaCl) a Bravais lattice alone
isn't sufficient to describe the structure  the second atom type needs to be fitted into the
unit cell. As a result, the symmetry of the unit cell may change. In the example, an
additional (orange, i.e. different) atom is fitted on the horizontal but not the vertical edges of the cell.
Because of the translational symmetry of the lattice, lattice points on an edge of the unit
cell have to be repeated on the opposite edge, so the extra atom appears both at the bottom
and at the top. The picture shows the symmetry elements in green  in the case of the
original lattice, two fourfold axes at the centre and corner of the cell, plenty of mirror
lines and a few twofold rotation axes at the intersection points of the mirrors. Adding
the extra atom breaks much of this symmetry: the fourfold axes are reduced to twofold
symmetry because they would otherwise reproduce orange atoms on the sides, and the diagonal
mirror planes vanish for the same reason.
Note that the
extra atom must be different
from the existing ones: if they were the same,
we could describe the new structure using a different lattice without adding any atoms.
In this example, the cell would split in half, and the new unit cell would be a primitive
rectangular one.
Another cause of
symmetry breaking
occurs in
molecular crystals.
Here, each lattice point
is occupied by a molecule rather than a (spherical) atom. The low symmetry of the molecule
results in much of the symmetry breaking down. The example shows the unit cell of a
(hypothetical) cubic ice crystal. Because of the shape of the water molecules occupying
each lattice point, all the rotations and most mirror planes will fail, leaving a cell with
little internal symmetry. Of course this is just a special case of the situation in the
previous example  there are two extra atoms in
general positions
(positions not coinciding with any symmetry element) in the cell.
Bravais lattice + point group = plane/space group
Because, in general, knowledge of the Bravais lattice alone is insufficient to reconstruct
a crystal structure, we need to specify the internal symmetry of the unit cell (its
point group)
along with the lattice. This combination is known as a
plane group (2D) or space group (3D).
The word group is meant here in the mathematical
sense of a set of symmetry elements and the symmetry operations acting on them. There are
17 plane groups and 230 space groups
in total.
Rotations  

1  1fold rotation point  (no symmetry)  
2  2fold rotation point  (180°)  
3  3fold rotation point  (120°)  
4  4fold rotation point  (90°)  
6  6fold rotation point  (60°)  
Reflections  
m  mirror axis  
g  glide axis  (flip & translate by a/2) 
The table lists all symmetry operations in two dimensions, along with their symbols used in the
names of symmetry groups and for graphical purposes. All symmetry elements in 2D are based on
rotations or reflections. Translations are generally taken care of by the periodicity of the
lattice rather than as a symmetry operation internal to the cell. An exception is the glide
plane, which combines a reflection with a translation of the reflected object by half a unit
cell vector parallel to the glide axis. The onefold rotation serves no purpose as such; in
group theory,
each group must have a neutral element (which does nothing). This allows
arithmetic to be carried out with the symmetry operations, e.g. turning a threefold
axis three times does exactly nothing  in group theory terms,
3+3+3=1,
while
3+3=3.
The following table shows the relationships between the 17 plane groups. The individual
pictures are taken from the Wikipedia article on
Wallpaper groups,
which contains many examples of different tilings from architecture,
arts and biology which represent these plane groups.
The groups are either based on rotations or reflections (mirror or glide axes) or a mixture of the two. Most are set in primitive cells, but two are based on the centred rectangular Bravais lattice.
The nomenclature of the groups follows a pattern (the
HermannMauguin scheme):
The first letter indicates whether the
Bravais lattice is primitive (p) or centred (c). This is followed by the predominant
symmetry element, i.e. the highestorder rotation (6, 4, 3, 2, or 1). Two intersecting
mirror axes always generate a twofold rotation point, so the '2' symbolising these rotations
is usually omitted, and any mirror (m) or glide (g) axes take precedence in these cases.
Where necessary to identify a group uniquely, a second symmetry element is specified,
which will usually be a mirror or glide plane. Finally, for structures with a threefold
axis and a mirror plane, there are two distinct groups depending on whether all rotation
points are located on a mirror line or not; these are distinguished, somewhat arbitrarily,
by including a 1fold rotation (i.e. no particular symmetry) as either second or
third symmetry element.
Note that most groups contain more symmetry elements than those specified in their HermannMauguin symbol  only as many as needed to describe the structure uniquely are usually listed.
Rotations  

1  1fold rotation axis  (no symmetry)  
2  2fold rotation axis  (180°)  
2_{1}  2fold screw axis  (180° & translate by a/2)  
3  3fold rotation axis  (120°)  
3_{1}  3fold screw axes  (120° & translate by a/3)  
3_{2}  (120° & translate by 2a/3)  
4  4fold rotation axis  (90°)  
4_{1}  4fold screw axes  (90° & translate by a/4)  
4_{2}  (90° & translate by a/2)  
4_{3}  (90° & translate by 3a/4)  
6  6fold rotation axis  (60°)  
6_{1}  6fold screw axes  (60° & translate by a/6)  
6_{2}  (60° & translate by a/3)  
6_{3}  (60° & translate by a/2)  
6_{4}  (60° & translate by 2a/3)  
6_{5}  (60° & translate by 5a/6)  
Rotoinversions  
1  centre of inversion  (point inversion only)  
3  3fold inversion axis  (120° & inversion)  
4  4fold inversion axis  (90° & inversion)  
6  6fold inversion axis  (60° & inversion)  
Reflections  
m  mirror plane  
a,b,c  axial glide planes  (flip & translate by (a,b,c)/2)  
n  diagonal glides  (flip & translate by half a face diagonal in each direction) 

d  diamond glides  (flip & translate by a quarter of a face diagonal each) 
In three dimensions, a number of additional symmetry operations arise. First, there are
screw axes
in addition to the plain rotations we know from the twodimensional
case. A screw axis is a combination of a rotation and a translation along the direction
of the axis of rotation. The translation part is a movement along the axis by an integer
multiple of steps 1/n of the lattice parameter in length, where n is the
order of the rotation. The two operations occur together, i.e. no lattice point
is generated at the intermediate position (after the rotation, but before the translation).
Threedimensional space also supports inversion at a point (a
centre of inversion,
above).
This is similar to a focal point  all points are mapped to positions directly across the
centre, at the same distance. As a result, shaped objects such as molecules are mapped
upsidedown instead of simply mirrored.
A centre of inversion is a special case of a
rotoinversion axis,
i.e. a
rotation combined with an inversion  in the case of the pure centre of inversion, the
rotation axis is just onefold. A twofold rotoinversion axis is equivalent to a mirror
plane perpendicular to the axis; it is not listed as a separate symmetry element for this
reason. Three, four and sixfold rotoinversion axes all occur in 3D space and are
independent symmetry operations. As was the case with screw axes, an intermediate lattice
point (after the rotation, but before the inversion) is not generated.
Some additional possibilities also arise among the reflections: in addition to the
axial glide planes
corresponding to the glide axis in 2D, there are glide planes with a
translation component along all three face diagonals of the unit cell simultaneously, by either
a quarter or half of the length of that diagonal.
number of  2D  3D 

Bravais lattices  5  14 
point groups  10  32 
plane/space groups  17  230 
As in the 2D case, the symmetry operations are collected in point groups. The number of distinct
groups is again finite  there are now 32. When combined with 14 Bravais lattices, this results
in 230 distinct space groups. Every crystalline material has one of these 230 structures. All
230 space groups are listed in the
International Tables
from the International Union of Crystallography. The Tables show each space group with a graphical
representation of their symmetry elements, special
positions (points located on one or more of the symmetry elements) and general positions (where
a point at x,y,z is repeated within the unit cell due to the symmetry of the structure). Systematic
absences of diffraction lines due to nonprimitive lattices or symmetry operations including an
internal translation (glide planes and screw axes) are also shown.
[sample
from the International Tables  see e.g. space group P2_{1}/c on p.184]
Having introduced point, plane and space groups, we will make a small detour to see how group theory treats symmetry elements and their relationships.