The magnetoelectric effects in multiferroic materials enable the mutual control of electric polarization by a magnetic field and magnetization by an electric field. Nonvolatile electric-field control of magnetization is extremely important for information storage applications, but has been rarely realized in single-phase multiferroic materials. Here we demonstrate the prominent direct and converse magnetoelectric effects in the Y-type hexaferrite BaSrCoZnFe11AlO22 single crystal. The electric polarization due to conical magnetic structure can be totally reversed by a small magnetic field, giving rise to large magnetoelectric coefficients of 6000 and 4000 ps/m at 100 and 200 K, respectively. The ab-plane magnetization can be controlled by electric fields with a large hysteresis, leading to nonvolatile change of magnetization. In addition, the reversal of magnetization by electric fields is also realized at 200 K. These diverse magnetoelectric effects with large coefficients highlight the promise of hexaferrites as potential multiferroic materials.