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A quick count of Geller-Grimms list (http://www.geller-grimm.de/famil.htm) gives
Nematocera: 40 familes
Brachycera 'Orthorrhapha': 30 families
Brachycera 'Cyclorrhapha': 103 families
Total: 173

Counted by hand so the number is probably off by a few. And some family divisions are not universally accepted. Since families are a 100% artificial classification scheme there will probably never be a official number of families.