$\frac{SU(2)}{N}= Q$. Examples of finite groups $N$ and $Q$?

Let us consider
1 \to N \to SU(2) \to Q \to 1

(1). Can we find some examples of $N$ and $Q$ so that either $N$ or $Q$ contain finite groups?

We know $SU(2)/Z_2=SO(3)$, so we can choose $N=Z_2$ and $Q=SO(3)$ as an example. Here $Z_n$ is $Z/nZ$ as a finite group of order $n$.

Since $SU(2)$ contains the quaternion $H_8$ as a subgroup (correct?), then $SU(2) \supset H_8$. We know $H_8/Z_2=(Z_2)^2$ and $H_8/Z_4=Z_2$. Do we have something similar by replacing $H_8$ by $SU(2)$? Does it make sense to consider:

$$SU(2)/Z_2=SO(3) \text{ ( see the above )}$$
What are these $N$ and $Q$ if they make senses?

(3) If $H_8$ is a normal subgroup of $SU(2)$ (is this true?), then we can ask
1 \to H_8 \to SU(2) \to Q \to 1
What is $Q=SU(2)/H_8=?$

Partial answers are welcome!

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