Subgroup structure of quaternion group

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The quaternion group has four types of subgroups, when classified upto automorphism:

  1. The trivial subgroup. (1)
  2. The center, given by \pm 1. This is isomorphic to the cyclic group of order two. (1)
  3. The three four-element cyclic subgroups, generated by the elements i,j,k respectively. They are all related by outer automorphisms, though no two of them are conjugate. (In fact, they're all normal). Each is isomorphic to the cyclic group of order four. (3)
  4. The whole group. (1)

The center (type (2))

The four-element subgroups (type (3))