How do I call the functional group of this alkane?
Under the 2004 provisional recommendations by IUPAC, one possible systematic name is 4-(2-methylpropan-2-yl)octane, which indicates that the longest chain in the substituent is three carbons long, is connected to the main carbon chain via the second carbon (-2-yl), and has a methyl group on the second carbon (2-methyl).

4-(1,1-dimethylethyl)octane is also possible, and indicates that there is an ethyl substituent on the fourth carbon in octane, which is further substituted with two methyl groups on the first carbon where it branches off (see section P-29.4).

However, the preferred IUPAC name is 4-tert-butyloctane; the tert-butyl group is used to describe a $\ce{-C(CH3)3}$ group and is preferred over either of the two above systematic names (see section P-56.2.2.2).

I don't have access to the 2013 recommendations but I believe it should not be different.