==Subgroups==

{{further|[[Subgroup structure of ~~prime-cube order ~~unitriangular matrix group:~~U~~UT(3,p)]]}}

~~Here is the complete list of subgroups~~{{#lst: ~~# The trivial ~~subgroup ~~(1)~~~~# The center, which is a group ~~structure of ~~order <math>p</math>. In ~~unitriangular matrix ~~terms, this is the subgroup comprising matrices <math>a_{ij}</math> with <math>a_{12} = a_{23} = 0</math>. ~~group:UT(~~1)~~~~# Subgroups of order <math>p</math> generated by non-central elements. These are not normal~~3, ~~and occur in conjugacy classes of size <math>p</math>. (<math>p(~~p~~+1~~)~~</math>)~~~~# Subgroups of order <math>p^2</math> containing the center. These are the inverse images via the quotient map by the center, of subgroups of order <math>p</math> in the [[inner automorphism group]]. (<math>p + 1</math>)~~~~# The whole group. (1)~~ ~~{{normal subgroups}}~~ ~~The subgroups in (1), (2), (4) and (5) above are normal.~~ ~~{{characteristic subgroups~~|summary}} ~~The subgroups in (1), (2) and (5) above are normal. In other words, there are only three characteristic subgroups. Some notable facts:~~ ~~* The group is [[characteristic-comparable group|characteristic-comparable]]: any two characteristic subgroups can be compared~~~~* More generally, any characteristic subgroup and any normal subgroup can be compared.~~~~* The characteristic subgroups are precisely the subgroups that occur in the [[derived series]], [[upper central series]] and [[lower central series]].~~

==Linear representation theory==