Rubber does not conduct electricity. In fact, it is generally known as an effective insulator and is widely used in gear designed to prevent electrocution. The only solids that conduct electricity are metals and graphite.
Conductors of electricity such as metals all possess the quality of having free-flowing electrons in the outer shells of their component atoms that are not tied to any particular atom. These free electrons react when an electrical charge is applied by conducting the electricity across the body of the conducting substance. Rubber and other insulators do not possess any such free electrons and accordingly do not conduct electricity.