Thank you very much Professor Holt. Indeed, also I have realized that $A_{8}$ has two classes of involutions, the centralizer of the central involution is isomorphic to $C_{2}^{4} \rtimes A_{4}$ while the other involution has centralizer isomorphic to $D_{8}\times A_{4}$. Indeed, it is easy to see that my conjecture is false by this example too.

@Agol : finite rank and locally finite does not imply finite, for example Prüfer $p$ groups...

I didn't mean "proper subgroups are finite", I mean "every finitely generated subgroup is finite". So, Tarski monsters are not example. Thanks.

yes, a group has finite rank $r$ if every finitely generated subgroup is generated by at most $r$ elements.