It is not true for . That is the non-abelian group which is not or .

It has three cyclic Sylow 2-subgroups. Therefore they are not normal.

EDIT: I am confused now, you said "... subgroup of a group of order 2^n". If the group has order 2^n then the Sylow 2-subgroup is the entire group itself. So it is definitely normal. But I do not think you really meant this.