I'm with Bill that it's definitely a subjunctive clause.

If you go by the traditional definition of finiteness as simply related 
to verb inflection, this should be a non-finite clause (despite the 
misleading name of "present subjunctive" sometimes used.) This is the 
base form of the verb. Notice, for example, the lack of tense concord 
that would be mandatory with a different verb in the matrix clause 
(e.g., "they said that the Greek colonies of Lydia *recognized* ...")

But note that both Quirk et alia and Huddleston & Pullum classify 
subjunctives (and imperatives) as finite forms.

The logic of doing so is that finiteness is a gradient scale rather than 
a strictly binary category. Subjunctive and imperative verbs don't have 
all the finite properties of indicative verbs, but they have more than 
infinitives do. There's a nice table of the various criteria and how 
they stack up in section 3.52 of /A Comprehensive Grammar of the English 
Language/.

Castilleja, Janet wrote:
> Hi
>  
> What do you folks make of this sentence?  Is the clause that begins 'that the Greek colonies..." finite or non-finite?  I think it is non-finite, but I'm wavering a little. 
>  
>  
> When they reached the coast of Asia   Minor, they insisted that the Greek colonies of Lydia recognize the Persian Kings as their over- Lords and pay them a stipulated tax.
>  
> Thanks
>  
> Janet Castilleja
> Heritage University
> 
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