I need to find the normal subgroup lattice of the group $U_{6n} = \langle a,b | a^{2n} = b^ 3= 1, a^{-1}ba = b^{-1}\rangle$. To the best of my knowledge this group was introduced at first by GORDON JAMES and MARTIN LIEBECK in their book "Representations and Characters of Groups". I have recently been working on normal subgroup lattices of finite groups. I can obtain the normal subgroup lattice structure of all groups in the mentioned book other than $U_{6n}$.

What is the normal subgroup lattice of $U_{6n}$ or where can I find it?

**Edit:**

I am very thankful for the answer and the comment given by Derek Holt_ but his formula for computing the number of normal subgroups seems to be incorrect. To show this, I consider a simple GAP program (written by my student Ms. Fatemeh Moftakhar) to compute the number of normal subgroups and 2n + d(n), where d(n) denotes the number of divisors of n. My program and its output is as follows:

```
g:=function(n)
local f,a,b,r,v;
f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
r:=[a^(2*n),b^3,b*(a^(-1))*b*a];
v:=f/r;
return v;
end;
AD:=[];BD:=[];
for i in [1,2..20] do
t:=NormalSubgroups(g(i));
b:=Size(t);
Add(AD,b);
od;
for i in [1,2..20] do
t:=2*i + Size(DivisorsInt(i));
Add(BD,t);
od;
```

Output of the Program:

```
[ 3, 5, 6, 7, 6, 10, 6, 9, 9, 10, 6, 14, 6, 10, 12, 11, 6, 15, 6, 14 ],
[ 3, 6, 8, 11, 12, 16, 16, 20, 21, 24, 24, 30, 28, 32, 34, 37, 36, 42, 40, 46 ].
```

Best,