The perfect tense of modal verbs requires forms of the auxiliary haben in the Front Bracket (F.B.) and the infinitives of both the main verb and modal (='double infinitive') in the End Bracket (E.B.).
 

The auxiliary haben is used with the "double infinitive" even when the main verb would take the auxiliary sein if standing alone.
 

The perfect and simple past tenses of the modal are interchangeable.

PERFECT TENSE
 

= SIMPLE PAST TENSE
 

Where lassen is used as a modal, it also forms the perfect tense with a "double infinitive."
 

Word Order and the Double Infinitive Construction

When a clause containing a double infinitive is put in a subordinate clause, the front bracket is put immediately in front of the end bracket, i.e. the brackets are not flipped.

MAIN CLAUSE WORD ORDER
 

SUBORDINATE WORD ORDER
 

Examples of Modal Verbs in the Past Tense

• Wir haben eine schöne Wohnung in Bensheim finden können und haben gerade einen herrlichen Kurzurlaub im Harz erlebt.
• Wir haben mit dem Zug fahren können.