The Prefect of Police
We're just given a few lines to this guy because he doesn't have a name or even a single line of dialogue in the story. What's interesting about him is that he's a foil, or point of contrast, to Dupin. Still, if the last line in whole story is spent being mean to the Prefect, he must be important somehow.
The Prefect of Police (read: chief of police) seems to be some kind of friend of Dupin's, because he gets Dupin in to see the crime scene. But he also seems to resent Dupin's holier-than-thou, uppity attitude, telling Dupin at the end that it's good for people to mind their own business. This is after Dupin has solved the case that the Prefect couldn't, so Dupin's feeling pretty smug. Dupin writes off the Prefect's complaints as sour grapes, saying that the Prefect has "cunning" (123) but no imagination.
In other words, getting back to the distinction between the chess player and the whist player that we lay out in our "Summary," the Prefect is an example of the analytical mind, which often manages to get things right if it has all the facts before it. What he doesn't have is the additional imagination of minds like Dupin's. So, in effect, Dupin is using the Prefect of Police as a way of concluding the argument the narrator presents in the first paragraphs of the story (further evidence that Dupin and the narrator are functionally the same guy. See the narrator's "Character Analysis" for more details).
Dupin says, of the Prefect of Police, that "In his wisdom is no stamen. It is all head and no body, like the pictures of the Goddess Laverna" (123). What does he mean? The stamen is that part of the flower that produces pollen to fertilize the pistil (check out this article if you're a budding botanist). Laverna is a minor Roman goddess, the patron deity of thieves.
What matters to Poe in both of these cases is that they represent, for him, the presence of only half of a necessary system: a pistil with no stamen, and a head with no body. According to Dupin, the Prefect is plenty smart, but he lacks that extra factor, the mental equivalent of the fertility of the flower or the body of the goddess, that would make him great like Dupin. No wonder the Prefect's feeling kind of angry by the end of the story, if this is what Dupin thinks of him.