  In recent work, Wang and the third author defined a class of 'extremal'
vertex operator algebras (VOAs), consisting of those with at least two simple
modules and conformal dimensions as large as possible for the central charge.
In this article we show that there are exactly 15 character vectors of extremal
VOAs with two simple modules. All but one of the 15 character vectors is
realized by a previously known VOA. The last character vector is realized by a
new VOA with central charge 33.
