Using simplest sextic fields, the author shows that there are real cyclotomic fields of class numbers greater than their conductors. More precisely, he proves the following theorem: Assume that \(\Delta_m=m^2+3m+9\equiv1\pmod4\) is square-free, \(m\geq-1\). Let \(t_m\) denote the number of distinct prime factors of \(\Delta_m\). Then the class number of the maximal real subfield \(\mathbb Q(\zeta_{\Delta_m})^+\) of the cyclotomic field of conductor \(\Delta_m\) satisfies \[ h_{\mathbb Q(\zeta_{\Delta_m})^+}\geq\frac1{5e} \frac{\Delta_m^2}{3^{t_m}\log^6(4\Delta_m)}. \] In particular, it holds that \(h_{\mathbb Q(\zeta_{\Delta_m})^+}>\Delta_m\) for \(m\geq24\cdot10^6\). To show that this theorem covers infinitely many cases, the author proves that \(\#\{1\leq m\leq x;\,m^2+3m+9\equiv1\pmod4\) is square-free\} is asymptotic to \(cx\), where \[ c=\tfrac13 \prod_{p\equiv1\pmod3} \bigl(1-\tfrac2{p^2}\bigr)= 0.311\dots \]