Also 16 of the 26 sporadic groups are of GF(2)-type, suggesting that in some sense sporadic groups are somehow related to special properties of the field with 2 elements.
Timmesfeld (1978) showed roughly that groups of GF(2)-type can be subdivided into 8 types.
The groups of each of these 8 types were classified by various authors.
They consist mainly of groups of Lie type with all roots of the same length over the field with 2 elements, but also include many exceptional cases, including the majority of the sporadic simple groups.
Smith (1979, p.279) gives a table of simple groups containing a large extraspecial 2-group.