Quaternion Group/Subgroups

Subgroups of the Quaternion Group
Let $Q$ denote the quaternion group, whose group presentation is given as:

The subsets of $Q$ which form subgroups of $Q$ are:

From Quaternion Group is Hamiltonian we have that all of these subgroups of $Q$ are normal.

Proof
Consider the Cayley table for $Q$:

We have that:
 * $a^4 = e$

and so $\gen a = \set {e, a, a^2, a^3}$ forms a subgroup of $Q$ which is cyclic.

We have that:
 * $b^2 = a^2$

and so $\gen b = \set {e, b, a^2, a^2 b}$ forms a subgroup of $Q$ which is cyclic.

We have that:
 * $\paren {a b}^2 = a^2$

and so $\gen {a b} = \set {e, a b, a^2, a^3 b}$ forms a subgroup of $Q$ which is cyclic.

We have that:
 * $\paren {a^2}^2 = e$

and so $\gen {a^2} = \set {e, a^2}$ forms a subgroup of $Q$ which is also a subgroup of $\gen a$, $\gen b$ and $\gen {a b}$.

That exhausts all elements of $Q$.

Any subgroup generated by any $2$ elements of $Q$ which are not both in the same subgroup as described above will generate the whole of $Q$.

Also see

 * Quaternion Group is Hamiltonian