As there are different tables floating around, I wanted to compare them and explore the different view on the syllables' initials and finals. There is not only a difference in the order of the parts, but also in how the syllables are placed under which initial and which final. E.g. the final of yu is pronounced like nü, though the "dots" (diaresis) are missing, so this syllable is placed under final ü in some tables.

Started as Pinyin syllable sets compared the interest in differnt visualisations grew, so here you can find a bigger collection of tables, each with special characteristics.

Without wanting to say too much - here is a list of different views on the syllables of Mandarin Chinese:

Simple grouping according to written form of Pinyin

Compare the Article Pinyin on Wikipedia (de).

Missing syllables: hng, hm, n, ng, m, ê

Discrimination of some parts pronounced differently, following ISO 7098

Missing syllables: hng, hm, n, ng, m, ê

See below for ê.

More consequent view, initials y, w grouped under diphthongs and triphthong without initial character

As found in Praktisches Chinesisch, Band I, Kommerzieller Verlag, Beijing 2001, ISBN 7-100-01675-4.

Missing syllables: yo, yai, hng, hm, n, ng, m, ê

y- and w- initials are pronounced as i and u except with yi, wu and thus can be put under the empty initial row (except yo as there is no io).

-i indicates different pronuciation for zi, zhi and others in contrast to yi, ti....

e is put under two columns, as 俄 is pronounced [ɤ] but 欸 is pronounced [ɛ] which is an exception. ê is left out as it serves the same purpose as e in this context: to mark the irregular pronuncitation as [ɤ].

Forms like yu, qu and others should actually be written yü, qü as to follow the design for nü, lü but as u is written wu and there is no [tɕʰu] the diaresis are ommited.

Transposed table following Pinyin.info

Missing syllables: yo, yai, er, hng, hm, n, ng, m, ê

Compare this table to http://www.pinyin.info/rules/initials_finals.html where there is also a short explanation on different (re-)groupings.

Extended table including all syllables

This table includes all syllables and furthermore distinguishes between the different pronunciations of '-i' in zhi and zi. The finals iu, ui, un are shown as iou, uei, uen as to emphasise their pronunciation and the fact, that the forms with prepended y- and w- actually belong to this class.