This kind of question has a long history (pre-computer), but it's always been difficult to get much intuition about solvable groups just from looking at a big list. In the case of simple groups the old search by Marshall Hall and others for those of order less than a million was a more sensible fishing expedition: such groups are sparse but interesting, while ruling out certain orders could be suggestive. Computers (and GAP in particular) have improved numerical searches a lot, but there is only so much to be learned this way.
For a sense of older times, there is the ambitious book project:
MR0168631 (29 #5889),
Hall, Marshall, Jr.; Senior, James K. ⋆The groups of order 2^n (n ≤ 6).
The Macmillan Co., New York; Collier-Macmillan, Ltd., London 1964 225 pp.
For modern times, there is a 2002 survey with plenty of references:
MR1935567 (2003h:20042), Besche, Hans Ulrich (D-AACH-DM); Eick, Bettina (D-BRNS-G); O’Brien, E. A. (NZ-AUCK), A millennium project: constructing small groups. Internat. J. Algebra Comput. 12 (2002), no. 5, 623–644.