insist作“反复说”、“一定要”、“坚决要求”解,可以是及物动词,也可以是不及物动词。
一、
作不及物动词用时,后跟on或upon引起的短语
1) insist on/ upon+名词   
They insisted on a definite answer.
I insisted on a replacement for the broken part on my bike.      
Our English teacher insists upon the importance of the correct pronunciation.

2) insist on / upon +动名词(或动名词的复合结构)   
She insisted on seeing the manager as soon as possible.  
He insisted on a second message being sent (不用to be sent) to them.
We insisted upon his staying with us for another week.

注意: insist on (doing) sth. , stick to (doing) sth. 与persist in (doing) sth.的区别:
这三个短语都有“坚持”的意思,但侧重点不同。
insist on (doing) sth.是"坚持"某种做法、意见、主张。如:
They insisted on leaving that day.  他们坚持要在那天走。
stick to (doing ) sth. 是"坚持"原则、计划、诺言、决定、真理、路线等。如:
He always sticks to his promise / his plan. 他总是坚守自己的诺言/计划。
如果表示"继续努力从事于某项工作",stick to后也可接动名词。如:
If you stick to practising English, you'll make great progress. 要是你每天坚持练习英语,就会取得很大进步。
而persist in (doing )sth.指不作任何改变地坚持某事,可能是一件好事,也可能是一件坏事。常包含“固执己见”的意思,但也可指意志的坚定。如:
She persists in her opinion.  
He persists in his sparetime studies.
She persisted in wearing that old-fashioned dress.
The old woman persists in taking half an hour's exercise every day. (表示意志的坚定,不用insist on来代替)

二、 insist 作及物动词的用法
insist的此种用法只能接从句作宾语。
1)insist作“坚决要求”、“一定要”,其宾语从句常用与将来事实相反的虚拟语气,即should+动词原形,其中should可以省略。如:
Her father insists that she (should) learn music after she leaves school.  
2) insist作"坚持说"、"坚持认为"时,用陈述语气。如:
Alice insisted that she had done nothing wrong.
3) insist that...有时与insist on / upon互换。如上面1)、2)中的两个例句可分别改写为:
Her father insists on her learning music after she leaves school.
Alice insisted on her having done nothing wrong.