The naming of groups does indeed come mostly from Latin, sometimes with Italian influence.
The etyomologies for the first nine are:
Number Group name Latin Italian Meaning
1 solo solus solo "alone"
2 duo duo due "two" - this is the Latin word
3 trio tres tre "three" - tres, but on the pattern of duo
4 quartet quarto quartetto "fourth" (as opposed to "four")
5 quintet quintus quinto "fifth"
6 sextet sex sei "six" - from the Latin (old English: sestet)
7 septet septem sette "seven"
8 octet octo otto "eight"
9 nonet novem nove "nine"
It is odd how 4 and 5 have their root in the ordinal number ("fourth, fifth") while the others count directly (two, three, six, etc). Also - the "-et" ending can have the connotation of "a piece written for..." (e.g. "duet").