A brief preface here that as I'm working in historical context, I will mainly be using the past tense, but Manchus have not ceased existing or recognising clan and family structures, so bear in mind that much of what I say below can also be said in the present tense as well.

'Clan' in English is a somewhat clumsy way of translating the Manchu concepts of ᠮᡠᡴᡠᠨ mukūn and ᡥᠠᠯᠠ hala. Finding hard and fast definitions can be difficult, in no small part because there were changes over time in how the term was used. This is reflected, for instance, in Norman's Concise Manchu-English Dictionary:

mukūn 1. clan, extended family, kindred 2. fleet 3. herd, flock.
hala 1. clan, family, family name 2. kind, style, sort

The situation is not helped by the existence of terms like mukūn hala and halamukūn, which seem to have been interchangeable terms effectively synonymous with just mukūn. In general terms, though, it is clear that in Manchu usage, a hala consisted of multiple mukun. In general terms, mukūn would be used to refer to a particular lineage and/or extended family, while hala would refer to all those sharing some kind of family name irrespective of actual relation. The basics of hala and mukun are laid out in Shirokogoroff's 1924 monograph Social Organisation of the Manchus:

The Manchu clan [hala] is a group of persons united by the consciousness of their common origin from a male ancestor and through male ancestors, also united by the recognition of their blood relationship, having common clan spirits and recognising a series of taboos, the principal of which is the interdiction of marriage between members of a clan, i.e. exogamy.

The number of hala was not particularly high, and with hala members spread geographically across the Qing empire, the 'clan' unit in any functional sense was the mukūn, which was typically associated with a lineage in a particular locale. Of course, the taboo against endogamy still applied, so two people from different mukūn under the same hala would still not be permitted to marry. However, mukūn rarely if ever actually referred to themselves by particular distinct names, and were instead somewhat of an amorphous common recognition.

Shirogokoff also notes the existence of an intermediate division, ᡤᠠᡵᡤᠠᠨ gargan (literally, 'branches'), but argues that such arrangements seem to have been relatively informal and to have been subsumed when the mukūn came into being, and by 1924 only applied to the ᡤᡳᠣᡵᠣ Gioro hala, whose ᠠᡳᠰᡳᠨ Aisin ('golden') gargan made up the imperial family. Pamela Crossley, however, notes that mukūn are attested before hala in the older Manchu sources that had been uncovered since Shirogokoff's time, but as yet I am not aware of any more detailed modern study of the evolution of Manchu clan affiliations.

Shirogokoff found 42 identifiable Manchu hala by 1924, some with as few as a single mukūn, but this reflects conditions in a late, post-imperial state of affairs. The official genealogies of Banner clans produced in the Yongzheng and Qianlong reigns include, according to Giovanni Stary, 642 of Manchu and other Tungus clans, 235 Mongol, 43 Korean, and 247 Chinese, the latter including subdivisions of 40 'Watchtower' and 50 'Fushun' Chinese clans.

Mukūn of a hala would often, but not always, have a geographical association, and these are preserved in Stary's dictionary of Manchu names. So, for instance, we can talk of Adaha of ᡠᠯᠠ ᠪᠠ ᡳ ᡤᡡᠸᠠᠯᡤᡳᠶᠠ Ula ba-i Gūwalgiya (Gūwalgiya of Ula) and Adaha of ᠸᠠᡵᡴᠠ ᠪᠠ ᡳ ᡤᡡᠸᠠᠯᡤᡳᠶᠠ Warka ba-i Gūwalgiya (Gūwalgiya of Warka), two geographically distinguished mukūn of the same hala.

It is important to add here that Manchu naming is often somewhat mistakenly handled in English-language materials by those unfamiliar with Manchu. While it is common to, for instance, refer to Qing emperors as, for instance Aisin Gioro Hung Li/Aixin Jueluo Hongli, Manchus did not use their hala and mukūn affiliations in their names. In proper Manchu, he would simply be Hung Li.

It can be difficult to find good, up-to-date resources on the matter of how clan organisation and affiliation worked. The preface to Stary's Dictionary of Manchu Names is a good start, and has references to Shirogokoff's monograph and Crossley's 1991 book, The Manchus. Also useful (if you can get it) is Crossley's PhD thesis on the evolution of Manchu clans through the Qing period.