The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups.
The simple N-groups were classified by Thompson (1968, 1970, 1971, 1973, 1974, 1974b) in a series of 6 papers totaling about 400 pages.
(The Tits group was overlooked in Thomson's original announcement in 1968, but Hearn pointed out that it was also a simple N-group.)
More generally Thompson showed that any non-solvable N-group is a subgroup of Aut(G) containing G for some simple N-group G. Gorenstein & Lyons (1976) generalized Thompson's theorem to the case of groups where all 2-local subgroups are solvable.
The primes dividing the order of the group are divided into four classes π1, π2, π3, π4 as follows The proof is subdivided into several cases depending on which of these four classes the prime 2 belongs to, and also on an integer e, which is the largest integer for which there is an elementary abelian subgroup of rank e normalized by a nontrivial 2-subgroup intersecting it trivially.