The semi-finals of the One-Day Cup will be held in Hampshire.
Tom Alsop hit 189 not out off 155 balls, his highest List A total, and was one of two players to score 132 in the match.
The Hampshire team defeated Yorkshire by 72 runs.
The holders of the tournament were knocked out by Kent, as were the other two teams in the quarter-finals.
The winner of the Kent and Notts game will progress to the semi-finals on 30 August, with the loser going to the final.
It wasn't enough to see them through to the knockout stage after Sam Northeast made 177 not out off 148 balls for Glamorgan and Billy Root hit a new club record for the fourth bowler in a List A game, but it was still a good performance.
The winner-take-all top of the table clash between the two teams was won by Sussex.
The hosts had already plundered four 300-plus scores in this competition this summer, but this time they really pushed on, thanks to the third-century stand between Alsop and Pujara.
The previous record of 399-4 was made off 10 overs in a CB40 match in 2011.
Joe Cracknell top-scored with 71 as Middlesex were skittled out for 243 to go from top of the group to fourth.
Puj has a calming effect. Everything seems to be under control. Alsop said that he understands the game and that they feed off him.
Durham lost for the seventh time in eight games to finish pointless at the bottom of the group.
Chris Rushworth reduced the Foxes to 158-8 after Durham were bowled out for 199.
A ninth-wicket stand of 30 between Beuran Hendricks and Eddie Barnes gave Leicestershire victory.
Stanley McAlindon, batting number 10, hit his maiden half-century as Durham recovered from 54-7 to win.
They needed results elsewhere to go in their favor after they raced to an eight-wicket win in just 32 overs.
Tom Lawes, the teenager who survived a drop on seven to hit a personal best 75 off 63 balls, was the hero of the day as he helped the team recover from a poor start.
The all out was never enough because of 4-44 from Toby Pettman in his second appearance, and 3-24 from the other side.
After Saturday's victory over Leicestershire, Ben and Lyndon James helped them reach 193-2 off 31.5 overs.
The help they needed came in the form of a surprise by previously winless Somerset at Edgbaston.
The ground where the T20 Blast hopes were ended by Hampshire on Finals Day was where Lewis Goldsworthy made a maiden List A century.
Goldsworthy and James Rew shared a third-century stand in 35 overs, but the Bears' batting line-up was weakened by the loss of Will Rhodes and Ethan Brookes.
Hampshire began their game on top of the group and stayed there as a consistent batting display proved enough to earn their seventh win in eight games.
The foundation was laid by Ben Brown, who made 60 off 83 balls, before Prest and Donald added 83 in eight overs, including six sixes.
Dom Bess was hit on the helmet by a throw from the outfield and Yorkshire were reduced to 34-7 and 85-5.
They were all out for 241 in the 46th over.
George Balderson hit a six and 15 fours in his 106 off 87 balls, but Kent were able to chase down the target with one ball to spare.
Josh Bohannon had made 75, while Harry Podmore and Grant Stewart claimed three of Kent's four bowling pins.
In Jack Blatherwick's second over, he sent back Ben Compton and Ollie Robinson, but AlexBlake kept Kent in the hunt with 81 off 78 balls, supported well byDarren Stevens and Stewart.
The record for the club's highest score in List A and first-class cricket is now held by Northeast after he made 410 in the Championship last month.
The home side made a good fist of the run-chase but ended on 337-9 despite Ben Cox's 84 not out off 78 balls and scores of 58 and 63 by Jake and Joe Leach.
The result put Glamorgan in third place, but their hopes of continuing their trophy defence were dashed by the Kent win.
There was a dead rubber game at Wantage Road where Luis Reece scored 106 off 125 balls for the Derby team.
A stand of 161 between Lewis McManus and Tom Taylor helped the home side post 311-7, but the visitors replied with 312-2 in 48.6 overs.
The quarter-finals are on August 26.
The semi-finals are on August 30.
The final is on September 17th at theTrent Bridge.
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