Subgroup lattice of UT(3,3)

Question

I need to draw a subgroup lattice of $ UT(3,\mathbb{Z_{3}}) $, the group of upper triangular matrices with diagonal one. How to do it? And whether there is somebody ready image? (I know that it can xgap but I have a problem with installing on windows)

Answer 1

If you cannot run xgap, there is a GAP function DotFileLatticeSubgroups that outputs a graphical description of a subgroup lattice the http://www.graphviz.org format which presumably has a viewer program for Windows. For example

gap> s:=SylowSubgroup(SL(3,3),3);
<group of 3x3 matrices of size 27 over GF(3)>
gap> DotFileLatticeSubgroups(LatticeSubgroups(s),"tmp.dot");

produces a file that creates this picture:

The first number of a subgroup is the conjugacy class, the second the number within the class. Normal subgroups just have a single number.

Answer 2

The group in question is the finite Heisenberg group $H_3(3)$ of order $27$. Its subgroup lattice can be computed with GAP, or similar programs. The group is nilpotent of class $2$. $H_3(p)$ has $p^2+2p+4$ subgroups, so $H_3(3)$ has $19$ subgroups.

Reference: The wiki-page.