I couldn’t find any six-group games, but I did uncover a five-group 20-point win I got against @alemitrani back in 2018.
The first group founded was 
with (2) C4. I must have been in an interesting period of my opening development because I played (4) as a corner enclosure.
The next, unnumbered group was O17 with (6), which as you can see eventually died. However, its extension at (9) K17 survived and become the 
group.
The third group was 
with (12) R6, approaching the lower right.
Black’s (19) M17 split from the O17 stones.
With (20) O17, invading the corner, I started another group; this group also died, but by (32) its southern remnant had conglomerated into .
(36) F17 was initially an extension from , but it turned out to be the foundation of 
. With (41-5) Black split those two groups.
During the running fight on the top side, was actually able to connect on (76) to , forming the supergroup which lasted for about another 75 moves. At this point, then, there were three groups: , , and .
On (98), I tried to break into Black’s central territory with L4, the root of the final group, 
. Black split with (101); however, after the fighting was resolved I still had the opportunity to connect to .
For a time it was possible for to connect to , but Black prevented this with his split on (137).
On (156) I apparently tried to play some very fancy endgame, but all that came of it was that was resplit on (159).
On (202) I committed to the separate existence of and , defining the final five groups.
I might easily have finished with only two groups, in the west and in the east, but in the game I ended up, as you see, with five.
By the way, it’s Firefox that has the cute pixelly emoji. I wish Discourse retained them.
9.5-point five-group win from September.
In that game it’s obvious that I took four corners, lived on the right side and invaded the centre twice, fleeing to the north and south.
To take four corners and win, you typically do have to finish with at least five groups.
4.5-point loss from 2018, five groups.
As you can tell, I haven’t found a sixer yet.