These are the crystallographic groups of a cubic crystal system: 23, 432, , 3m, and . All of them contain four diagonal 3-fold axes. These axes are arranged as 3-fold axes in a cube, directed along its four space diagonals (the cube has  symmetry). These symbols are constructed the following way:
* First position – symmetrically equivalent directions of the coordinate axes x, y, and z. They are equivalent due to the presence of diagonal 3-fold axes.
* Second position  – diagonal 3 or  axes.
* Third position – diagonal directions between any two of the three coordinate axes x, y, and z. These can be 2, m, or .
All Hermann–Mauguin symbols presented above are called full symbols. For many groups they can be simplified by omitting n-fold rotation axes in  positions. This can be done if the rotation axis can be unambiguously obtained from the combination of symmetry elements presented in the symbol. For example, the short symbol for  is mmm, for  is mm, and for  is mm. In groups containing one higher-order axis, this higher-order axis cannot be omitted. For example, symbols  and  can be simplified to 4/mmm (or mm) and 6/mmm (or mm), but not to mmm; the short symbol for  is m. The full and short symbols for all 32 crystallographic point groups are given in crystallographic point groups page.
Besides five cubic groups, there are two more non-crystallographic icosahedral groups (I and I in Schoenflies notation) and two limit groups (K and K in Schoenflies notation). The Hermann–Mauguin symbols were not designed for non-crystallographic groups, so their symbols are rather nominal and based on similarity to symbols of the crystallographic groups of a cubic crystal system. Group I can be denoted as 235, 25, 532, 53. The possible short symbols for I are m, m, mm, m. The possible symbols for limit group K are ∞∞ or 2∞, and for K are ∞ or m or ∞∞m.