
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.

crowd
A large group of people gathered in one place.
Nach dem Konzert strömte die Menge aus dem Stadion.
After the concert, the crowd streamed out of the stadium.

set
In mathematics, a collection of specific, distinct objects into a whole.
Die Menge der geraden Zahlen ist unendlich.
The set of even numbers is infinite.
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
Gruppe and Menge can both refer to several people gathered together. Gruppe tends to suggest a recognizable or connected unit, while Menge mainly emphasizes a large gathering in one place. Gruppe fits a smaller number of people with an identifiable connection; Menge is more suitable for many people gathered together.

Vor dem Eingang wartete eine große Menge auf den Beginn des Konzerts.
A large crowd waited outside the entrance for the concert to begin.
Vor dem Eingang wartete eine Gruppe von Besuchern auf ihre Führung.
A group of visitors waited outside the entrance for their guided tour.