Counter Example: Abelian Subgroup Not Normal

[Bachelier]

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Can someone provide us with a counter example to an abelian (sub)group that is not normal.

I'm thinking something in the center of a group.

 

[micromass]

Can you find a subgroup in $D_6$?? (the dihedral group of order 6, aka the symmetry group of a triangle). Can you find a subgroup that is not normal?? Is this subgroup abelian?