die Menge
amount / quantity
A definite or indefinite number or quantity of things, substances, or people.
Für den Kuchen braucht man eine große Menge Zucker.
For the cake, you need a large amount of sugar.

amount / crowd
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amount / quantity
A definite or indefinite number or quantity of things, substances, or people.
Für den Kuchen braucht man eine große Menge Zucker.
For the cake, you need a large amount of sugar.

The meaning of 'Menge' is easy to recognize from context. If it refers to things or substances, it means quantity. If it refers to people in one place, it means a crowd. In the context of numbers and symbols, it refers to a mathematical set.
Ich habe eine große Menge an Informationen gesammelt.
I collected a large amount of information.
When 'eine Menge' (a lot/a quantity) describes a quantity, it is usually followed by 'von + dative' or directly by the noun in the plural. The formal variant with the genitive is rare.
Wir haben eine Menge von alten Büchern im Keller gefunden.
We found a lot of old books in the basement.
The expression 'eine ganze Menge' is a very common intensifier and means “quite a lot” or “a considerable number.”
Das neue Auto hat eine ganze Menge Geld gekostet.
The new car cost quite a lot of money.
When 'die Menge' refers to a group of people, it is often combined with verbs that describe a collective action, e.g. 'jubeln' (to cheer), 'protestieren' (to protest), 'sich versammeln' (to gather), or 'strömen' (to stream/flow).
Nach dem Sieg jubelte die Menge auf dem Platz.
After the victory, the crowd cheered in the square.
In the meaning of 'collection of objects', 'Menge' is a technical term in mathematics and is used almost exclusively in this context.
In der Schule lernt man die Grundlagen der Mengenlehre.
At school, you learn the basics of set theory.
group
'die Gruppe' and 'die Menge' can both refer to several people or things. 'Gruppe' emphasizes that the people or things belong together or are seen as a unit. 'Menge' emphasizes the number or size more. With people, 'Menge' can also mean a large crowd gathered in one place. Learners may confuse the words because both have to do with multiple people. A simple guideline is: if the people are working together or belong together, 'Gruppe' fits. If the main idea is many people or a lot of something, 'Menge' fits better.

Vor dem Bahnhof wartete eine große Menge auf den Bus.
A large crowd was waiting for the bus in front of the train station.
Tom lernt mit einer Gruppe von Kollegen für die Prüfung.
Tom is studying with a group of colleagues for the exam.