# Normal subgroup but not characteristic subgroup

I know every characteristic subgroup is normal subgroup but converse is not true. I find an example in Klien 4- group.

But I have seen on internet that in Q$$_8$$, each of cyclic subgroup of order 4 is normal (as index 2) but none of these is characteristic.

There are 3 subgroups of order 4 in Q$$_8$$, how can I show these 3 subgroups are not characteristic?

There is an automorphism $$\phi$$ of $$Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$$ which cyclically permutes $$i,j,k$$: $$\phi(\pm 1)=\pm 1, \phi(\pm i)=\pm j, \phi(\pm j)=\pm k, \phi(\pm k)=\pm i$$ So $$\phi$$ does not preserve any of the three (cyclic) order-4 subgroups $$\{1,i,-1,-i\},\{1,j,-1,-j\}, \{1,k,-1,-k\}$$ of $$Q_8$$.