The quaternion group is one of the two non-Abelian groups of the five total finite groups of order 8. It is formed by the quaternions 
, 
, 
, and 
, denoted 
or 
.
 |  |  |  | 1 |  |  |  | |
 | 1 |  |  |  |  |  |  |  |
 |  |  |  |  |  | 1 |  |  |
 |  |  |  |  |  |  | 1 |  |
 |  |  |  |  |  |  |  | 1 |
| 1 |  |  |  |  | 1 |  |  |  |
 |  | 1 |  |  |  |  |  |  |
 |  |  | 1 |  |  |  |  |  |
 |  |  |  | 1 |  |  |  |  |
The multiplication table for 
is illustrated above, where rows and columns are given in
the order 
,
, 
, 
, 1, 
, 
, 
, as in the table above.
The cycle graph of the quaternion group is illustrated above.
The quaternion group has conjugacy classes 
, 
, 
, 
, and 
. Its subgroups are 
, 
, 
, 
, 
, and 
, all of which are normal
subgroups.
