Montoya, with the fastest laps of the race, charged past to claim his seventh consecutive podium.
Schumacher holds F1 records for most championships, race victories, fastest laps, pole positions and more.
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He set fastest laps once he was clear of traffic in the last stint of the race but by then he had lost too much ground.
However, the 2009 champion burst clear at the restart and pumped in a series of fastest laps to build a two-second lead just after the halfway point of the race.
As the rain held off at Hockenheim, the Red Bulls and Ferrari dominated the fastest laps leaving world champion Button and team-mate Hamilton struggling 0.634 seconds behind the front two, something that troubled both McLaren drivers.
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At the American Le Mans Series' six-hour endurance race at Mazda Raceway Laguna Seca in September, the GT3 R Hybrid walked away from the field, posting the fastest laps in the GT class and stopping only three times for fuel, compared with other competitors' five or more stops.
Toseland, who enjoyed his best performance of last season at this circuit, was among the pacesetters over the opening laps and ended up seventh fastest despite his misdemeanour - merely a result of pushing the front end too hard as he chased an improved lap time.
Hamilton was fastest in the morning in the wet and completed 113 laps as he fine tunes his new car ahead of the opening race.
The Brazilian clocked an unofficial fastest time of one minute and 12.574 seconds as he completed 102 laps, autosport.com reported, with veteran Spaniard De La Rosa posting 1:12.784.
Recognising the potential danger of that situation, governing body the FIA has said that all laps in qualifying must be completed within a specified percentage of a driver's fastest time.
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