Colin Fine has already answered the question but I'm going to shed some light on its history.
Words that end with [ŋ] rather than [ŋɡ] in Modern English is the result of Cluster simplification that occurred in Modern English. Word-final <ng> used to be pronounced [ŋɡ] in Old and Middle English, but due to a sound change in Modern English, it became [ŋ]. With this sound change, [ŋ] became a phoneme in English.
Examples: Hang, sing, king etc., had [ŋɡ] word-finally but they came to be pronounced with [ŋ] in Modern English (though many dialects such as West Midlands and North West England kept the pronunciation with [ŋɡ]).
[ŋg] mostly remained word-medially (in the middle of monomorphemic words):
Examples: Finger, hunger, anger, linger etc., have [ŋg] word-medially.
In most cases, word-medial <ng> in some suffixed words did not become [ŋg]. I only know two of those suffixes; -ing and -er.
Examples: Singer, singing, hanger, ringing etc.
Comparatives and superlatives have [ŋg] word-medially. They're an exception to the normal rule of thumb.
Examples: younger, longer, stronger, longest etc.
Hangover is a compound word of hang and over, 'hang' has [ŋ] so its compounds don't have [ŋɡ]. It seems that the [g] doesn't get pronounced in compound words.
(Note that the pronunciation with [ŋɡ] can still be heard in many dialects.
It's also explained briefly in Phonological variables and the sources of accent variables
Also see singer-finger split - Wikipedia