Firstly, be clear on the difference between POINT groups, which include reflection, rotation and inversion (and hence leave one point unchanged), and SPACE groups, which include the above plus translational symmetry elements.

Schönflies notation was originally developed to describe point group symmetry. It's used more often by chemists and spectroscopists interested in molecular symmetries. The large letter refers to the family: C for rotational axes; S for rotation-reflection axes; D, T, O and I for dihedral, tetrahedral, octahedral and icosahedral symmetries. These may be modified by a number to indicate either the order of the primary rotation axis (C,S) or the number of mirror planes (D,T,O,I) and either an h or a v to indicate the presence of 'horizontal' (perpendicular to the primary axis) or 'vertical' (containing the primary axis) mirror planes. The IUCr publishes a list of equivalences between Schönflies and Hermann-Mauguin point groups here:

http://link.springer.com/referenceworkentry/10.1107/97809553602060000524

(click on the HTML link if you can't get access.)

Schönflies notation has been widened to apply to space group symmetry, but not in any particularly meaningful way. Within each point group, the different space groups are simply numbered in the order in which they appear in IUCr space group tables. The best list of equivalences I know of is maintained by the team behind the software Spacegroup, and can be found starting on the 5th page of the user manual:

http://elk.sourceforge.net/spacegroup.pdf