  A fullerene graph is a cubic 3-connected plane graph with (exactly 12)
pentagonal faces and hexagonal faces. Let $F_n$ be a fullerene graph with $n$
vertices. A set $\mathcal H$ of mutually disjoint hexagons of $F_n$ is a sextet
pattern if $F_n$ has a perfect matching which alternates on and off each
hexagon in $\mathcal H$. The maximum cardinality of sextet patterns of $F_n$ is
the Clar number of $F_n$. It was shown that the Clar number is no more than
$\lfloor\frac {n-12} 6\rfloor$. Many fullerenes with experimental evidence
attain the upper bound, for instance, $\text{C}_{60}$ and $\text{C}_{70}$. In
this paper, we characterize extremal fullerene graphs whose Clar numbers equal
$\frac{n-12} 6$. By the characterization, we show that there are precisely 18
fullerene graphs with 60 vertices, including $\text{C}_{60}$, achieving the
maximum Clar number 8 and we construct all these extremal fullerene graphs.
