οἶδα, ‘know’ is an indispensible and almost inscrutable perfect:

so important, it has it’s own principal parts—οἶδα, εἴσομαι, ᾔδη--

and puzzling because its forms are hard to construe and easily confused with other verbs (≠ εἰμί, ἔσομαι, etc.).

We start with the root of εἶδον, ‘wid’ (as in Latin video)

The indicative has two markers (as usual): no –κ- but (1) reduplication and (2) o-grade.

But the reduplication is entirely concealed because the initial consonant was the weak –w- (digamma):

so original ‘wo-wid-a’→ οἶδα.

This explains the odd subjunctives and optatives: what looks like augment is really the reduplication (without o-grade):

‘we-wid-o’ etc. → εἰδῶ

The infinitive: ‘we-wid-enai’ → εἰδέναι.

The pluperfect adds augment to that stem ἠιδ-η = ᾔδη.

The (present) perfect also distinguishes singular from plural by contrasting strong and weak stems:

singular has the o-grade, οἶδα, οἶσθα οἶδε; the plural doesn’t, ἴσμεν, ἴστε, ἴσασι.

As for all those ‘sneaky little verbs’ easily confused with οἶδα, refer to 307.

Note, for example, the singular imperatives, ‘be’ and ‘know’ are identical: ἴσθι;

but the plurals distinguish, ἴστε ≠ ἔστε