You can use the subjunctive after these verbs:
Propose recommend insist demand
• I suggest (that) you take a vacation.
• They insisted (that) we have dinner with them.
• I insisted (that) he have dinner with me.
• He demanded (that) she apologize to him.
You can use the subjunctive for the present, past, or future:
I insist you come with us.
They insist I go with them.
Other structures are possible after insist and suggest:
• They insisted on my having dinner with them.
• She suggested that he buy some new clothes. (not suggested him to buy)
• What do you suggest I do? (not suggest me to do)
Should is sometimes used instead of the subjunctive.
• She suggested that he should buy some new clothes.
• The doctor recommended that I should rest for a few days.