The fourteen Bravais lattices can be grouped into seven crystal systems by using the symmetry properties of unit cells, as well as by the relations between the sides and angles of the unit cell. Consider two unit cells as shown below.

Figure 16.1 Unit cells. Sides are a, b and c and the angles are (between b and c in the bc plane), (in the ac plane,                    between a and c) and (between a and b in the ab plane. (a) cubic unit cell, (b) non-cubic unit cell

In a cubic crystal system (formed from cubic unit cells placed at the lattice points) there are four C₃ axes placed in a tetrahedral arrangement. In Fig 16.1(a) the line joining the points 3 and 5, for example is a C₃ axis. What this means is that if the unit cell/crystal is rotated by 120^o, 240^o and 360^o (three angles, multiples of 120^o), we get an arrangement which is indistinguishable from the original arrangement. Having only a C₁ axis is as good as having no symmetry at all because every object has a C₁ axis of symmetry, i.e., if you rotate it with respect to any axis by 360^o, you will recover the original arrangement. A triclinic crystal has no symmetry or has only a C₁ symmetry axis.

The symmetry elements of the seven crystal systems are given in Table 16.1