Brauer-Fowler theorem on existence of subgroup of order greater than the cube root of the group order

Statement
Suppose $$G$$ is a finite group whose order is an even number greater than 2. Then, $$G$$ has a proper subgroup $$H$$ such that:

$$|G| < |H|^3$$

Further, if the center $$Z(G)$$ is an odd-order group, then we can choose $$H$$ to be the centralizer $$C_G(x)$$ of some non-identity strongly real element $$x$$ of $$G$$.

Similar facts

 * Dihedral trick
 * Finite group having at least two conjugacy classes of involutions has order less than the cube of the maximum of orders of centralizers of involutions
 * 2-Sylow subgroup is TI implies it is normal or there is exactly one conjugacy class of involutions

Opposite facts

 * Finite simple non-abelian group has order greater than product of order of proper subgroup and its centralizer
 * Every proper abelian subgroup of a finite simple non-abelian group has order less than its square root