I do not know what you mean with "simple Mathematics Calculation" so I cannot follow how you arrive at 28 lattices. Thereby it has to be noted that everybody uses a different definition of the term lattice. In some context the term is used simply for points (e.g. atom position) of an arbitrary periodic structure, in other context, the term is used only for translation lattices in the strict sense. This causes a lot of confusion, if one discovers apparent contractions if you mix up different definitions of the word lattice.

The Bravais lattices are "types" of translation lattices distinct by symmetry. A translation lattice assembles all (all in a group theory sense; completeness) shifts which map a 3d-periodic structure on itself. The points of atoms in an hcp structure (I do not use the word lattice here!) simply are not the points representing a translation lattice. E.g. the vector (1/3 2/3 1/2) connecting two atoms in the hcp structure is not a translation vector.

Hello Mohammadamin Fakharipasandi

There are 7 Crystals systems, and these "subdivisions":

Primitive, Body Centered, Face-Centered and Base-Centered.

We would have 28 Bravais Lattice if every crystal system had all "subdivisions". But some of them don't, because they are equivalent to some of the official 14 Bravais Lattice. The only Bravais lattice for Hexagonal is the Primitive one, because Body-Centered, Face-Centered and Base-Centered Hexagonal are equivalent to some other lattice.

But "Close Packed" doesn't refer to Bravais Lattice specifically. Hexagonal Close Packed is a Hexagonal Primitive cell, check this site:

https://msestudent.com/hexagonal-close-packed-hcp-unit-cell/

Magnesium can crystallize in HCP, but it's a Hexagonal Primitive cell.

Feel free to reply if you need some help.

Best regards,

Ricardo Tadeu

There are 14 types of Bravais lattice, others can be reduced to these basic 14 combinations.

In short:

A Bravais lattice is defined as a set of points given by the vectors:

R = n1a1+ n2a2+ n3a3

with the conditions: n1,n2,n3 are integer (pos/neg/zero) and a1 a2 a3 are non coplanar vectors. These conditions are fulfilled only for 5 2D and 14 3D lattices. The hcp structure simply does not fit the above conditions.