Do all verbs form the conjunctive II with an umlaut?

Question

As far I know we normally form the conjunctive II with the preterite (past) form of the verb. Coming from the preterite, we replace a with ä, u with ü and o with ö and also add an extra e if possible. For example:

1. gehen – gingen – sind gegangen.
   Präteritum: Ich ging
   Konjuktiv II: Ich ginge

2. arbeiten – arbeiteten – haben gearbeitet
   Präteritum: Ich arbeitete
   Konjuktiv II : Ich arbeitete (as there is no a, o, u to replace and it has already got an e at the end)

According to this rule besuchen and machen should be conjugated as following:

1. besuchen – besuchten – haben besucht
   Präteritum: Ich besuchte
   Konjuktiv II: Ich besüchte – which seems to be wrong; the correct conjugation is besuchte.

2. machen – machten – haben gemacht
   Präteritum: Ich machte
   Konjuktiv II: Ich mächte – wrong, correct conjugation is machte.

Could anyone please explain why are we not replacing u with ü in besuchen and a with ä in machen?

Answer 1

The rule you cited only applies to strong verbs (also called irregular verbs):

lesen: las, läse
biegen: bog, böge
schaffen: schuf, schüfe
laufen: lief, liefe

It is not applicable to weak verbs (regular verbs), which form the past tenses with -t-. For them, the Konjunktiv II is normally identical to the Präteritum:

machen: machte, machte
besuchen: besuchte, besuchte

(Note that the rule does not cover all cases of strong verbs, either; e.g., sterben has forms starb and stürbe.)