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We apply a method of Bender [H. Bender. Finite groups with large subgroups.  Illinois J. Math .  18  (1974), 223–228.] to determine the order of a group  G  of  J  3 -type. Moreover, we determine the local  p -structure of  G  for every prime  p  dividing the order of  G . The results of this paper are obtained by exploiting the action of  G  on its geometry [B. Baumeister. A geometry for groups of  J  3 -type.  Arch. Math.  ( Basel )  88  (2007), 491–499.] and by sophisticated use of elementary group theory.

A method of Bender applied to groups of J 3 -type