On the Sylow $2$-subgroup of a nonabelian simple group.

Question

Let $G$ be a finite nonabelian simple group and $P$ be a Sylow $2$-subgroup of $G$.

QUESTION:

Is $P$ cyclic or quaternion group?

Answer 1

The thesis On Sylow $2$-subgroups of finite simple groups of order up to $2^{10}$ gives a list of groups which are realizable as Sylow $2$-subgroups of $G$. So we see that there are many possibilities. So $P$ need neither be cyclic nor quaternion. In fact, $Q_8$ is not realizable. Indeed, certain groups are never realizable, e.g., all abelian 2-groups with an automorphism group that is a 2-group.are not realizable.

Actually, very good answers have been given at this "duplicate":

What do Sylow 2-subgroups of finite simple groups look like?