Let n be an any integer.It is well known that there are infinitely many imaginary quadratic fields with ideal class group having a subgroup isomorphic to \Z/n\Z×\Z/n\Z.For real quadratic fields,less is known.We will prove that there exist infinitely many real quadratic number fields with ideal class group having a subgroup isomorphic to \Z n ×\Z n.We will also prove that there exist infinitely many imaginary quadratic number fields with ideal class group having a subgroup isomorphic to \Z n ×\Z n ×\Z n.