Future Events vs Past Events
With both these verbs, insist and suggest, the particular inflection
you choose for that second verb which they control triggers a dramatic shift in
meaning that alternates between possible future events yet to pass and
definite past events that are already done and gone. That’s a huge change in meaning, so it’s really important that you understand how to select the version whose meaning you intend.
But there are also syntactic restrictions that apply in an exactly opposite
fashion to each of these two verbs. I’ll explain those first.
Possible Futures
With insist you can use a gerund-participial ‑ɪɴɢ clause (with that clause’s subject in either the accusative or possessive form if it’s a pronoun) but not an
infinitive clause:
- I insisted that she land here.
(She hadn’t done so yet and I wanted her to do so in the future.)
- I insisted on her landing here.
(Same meaning as #1.)
- *I insisted on her to land here. [UNGRAMMATICAL!]
But with suggest it works the other way around: you cannot use a gerund-participial ‑ɪɴɢ clause for the non-finite case, only an infinitive clause with a for-to complementizer with an overt subject, which must be in accusative case only:
- I suggested that she land here.
(She hadn’t done so yet but I’ve said she should or could do so in the future.)
- *I suggested her landing here. [UNGRAMMATICAL!]
- I suggested for her to land here.
(Same meaning as #4; this sounds a little odd to some folks, though.)
Real Pasts
Finally, if you use the past tense in the subordinate clause, this means
something completely different because now that it’s in the indicative, not
the subjunctive. It’s a factual matter now, not a propositional one.
- I insisted that she landed here.
(I’m asserting that she did indeed do so already in the past and I’ve tried to convince you of this fact.)
- I suggested that she landed here.
(I’m saying I think she might really have already have done so in the past.)
You can also use either of the other two ways of expressing
the past for both those “real past” scenarios:
- I insisted that she did land here.
(I’m asserting that she did indeed do so already in the past and I’ve tried to convince you of this fact.)
- I insisted that she has landed here.
(I’m again asserting that she did indeed do so already in the past and I’ve tried to convince you of this fact.)
- I suggested that she did land here.
(I’m saying I think she might really have already have done so in the past.)
- I suggested that she has landed here.
(I’m again saying I think she might really have already have done so in the past.)
Summary
So items 1 through 6 (well, 1 and 2 and 4 and 6, as 3 and 5 are ungrammatical) are all talking about a possible future event still to
unfold (and it might not ever happen), while 7 through 12 are all talking about a definite past event that
has already taken place.