Abelian not implies contained in abelian subgroup of maximum order
Example of a dihedral group
Further information: dihedral group:D16
Consider the dihedral group of order , specifically:
This has an Abelian subgroup of order eight: the cyclic subgroup generated by . Consider now the cyclic subgroup given as:
is a subgroup of order four.
Clearly, is self-centralizing in : . Thus, is not contained in a bigger Abelian subgroup, and hence is not contained in an Abelian subgroup of maximum order.