The following page was created by Prof Pedersen, and "orphaned." Years ago, I was given permission to post and maintain it here. Feel free to use it however you like but not Prof Schaps' page is much more extensive.
<a, b; a^{2} = b^{2} = (ab)^{2} = 1>The Cayley table of the group is (putting c = ab):
 1 a b cA matrix representation is the four 2x2 matrices
+
1  1 a b c
a  a 1 c b
b  b c 1 a
c  c b a 1
[1 0] [1 0] [1 0] [1 0]A permutation representation is the following four elements of S_{4}:
[0 1], [0 1], [ 0 1], [ 0 1]
(1), (1, 2)(3, 4), (1, 3)(2, 4) and (1, 4)(2, 3).Its lattice of subgroups is (in the notation of the Cayley table)
V
/  \
<a> <b> <c>
\  /
{1}
<s,t; s^{2} = t^{2} = 1, sts = tst>Another presentation (with s <> (1, 2, 3), t <> (1, 2)) is
<s,t; s^{3} = t^{2} = 1, ts = s^{2} t>In terms of this second presentation, with 2 = s^{2}, u = ts and v = ts^{2}, the Cayley table is
 1 s 2 t u vThis shows S_{3} is isomorphic to D_{3}, the dihedral group of degree 3, that is, the symmetries of an equilateral triangle (this never happens for n > 3). The lattice of subgroups of S_{3} is
+
1  1 s 2 t u v
s  s 2 1 v t u
2  2 1 s u v t
t  t u v 1 s 2
u  u v t 2 1 s
v  v t u s 2 1
S_{3}The first three proper subgroups have order two, while <s> has order three and is the only normal one.
/ /  \
<t> <u> <v> <s>
\ \  /
{1}
<s, t; s^{4} = t^{2} = e; ts = s^{3} t>In terms of these generators (s corresponds to rotation by pi/2 and t to a reflection about an axis through a vertex), the eight elements are 1,s,s^{2},s^{3},t,ts,ts^{2} and ts^{3}. Using the notation 2 = s^{2}, 3 = s^{3},_{ }t2 = ts^{2} and t3 = ts^{3}, the Cayley table is
 1 s 2 3 t ts t2 t3Its subgroup lattice is
+
1  1 s 2 3 t ts t2 t3
s  s 2 3 1 t3 t ts t2
2  2 3 1 s t2 t3 t ts
3  3 1 s 2 ts t2 t3 t
t  t ts t2 t3 1 s 2 3
ts ts t2 t3 t 3 1 s 2
t2 t2 t3 t ts 2 3 1 s
t3 t3 t ts t2 s 2 3 1
D_{4}Of these, the proper normal subgroups are the three of order four and <s^{2}> of order two.
/  \
{1,s^{2},t,ts^{2}} <s> {1,s^{2},st,ts}
/  \  /  \
<ts^{2}> <t> <s^{2}> <st> <ts>
\ \  / /
{1}
<s, t; s^{4} = 1, s^{2} = t^{2}, sts = t>Q can be realized as consisting of the eight quaternions 1, 1, i, i, j, j, k, k, where i is the imaginary square root of 1, and j and k also obey j^{2} = k^{2} = 1. These quaternions multiply according to clockwise movement around the figure
iFor example, ij = k and ji = k (negative because anticlockwise).
/ \
k  j
s = [i 0] t = [0 i]The subgroup lattice of Q is
[0 i] [i 0]
QAll of these subgroups are normal in Q.
/  \
<s> <st> <t>
\  /
<s^{2}>

{1}
A_{4}The only proper normal subgroup is <(1,2)(3,4),(1,3)(2,4)>.
/ \ \ \ \
<(1,2)(3,4),(1,3)(2,4)> <(1,2,3)> <(1,2,4)> <(1,3,4)> <(2,3,4)>
/  \  / / /
<(1,2)(3,4)> <(1,3)(2,4)> <(1,4)(2,3)>  / / /
\ \ \ / / / /
{1}
<s, t; s^{6} = 1, s^{3} = t^{2}, sts = t>T is the semidirect product of C_{3} by C_{4} by the map g : C_{4} > Aut(C_{3}) given by g(k) = a^{k}, where a is the automorphism a(x) = x.
<x,y; x^{4} = y^{3} = 1, yxy = x>In terms of these generators, using AB for x^{A} y^{B}, the Cayley table for T is
 00 10 20 30 01 02 11 21 31 12 22 32A 2x2 matrix representation of this group over the complex numbers is given by
+
1 = 00 00 10 20 30 01 02 11 21 31 12 22 32
x = 10 10 20 30 00 11 12 21 31 01 22 32 02
x^{2} = 20 20 30 00 10 21 22 31 01 11 32 02 12
x^{3} = 30 30 00 10 20 31 32 01 11 21 02 12 22
y = 01 01 12 21 32 02 00 10 22 30 11 20 31
y^{2} = 02 02 11 22 31 00 01 12 20 32 10 21 30
xy = 11 11 22 31 02 12 10 20 32 00 21 30 01
x^{2}y = 21 21 32 01 12 22 20 30 02 10 31 00 11
x^{3}y = 31 31 02 11 22 32 30 00 12 20 01 10 21
xy^{2} = 12 12 21 32 01 10 11 22 30 02 20 31 00
x^{2}y^{2} = 22 22 31 02 11 20 21 32 00 12 30 01 10
x^{3}y^{2} = 32 32 01 12 21 30 31 02 10 22 00 11 20
[0 i] [w 0 ]where i is a square root of 1 and w is nonreal cube root of 1, for example w = e^{2\pi i/3}.
x <> [i 0] y <> [0 w^{2}]
<s,t; s^{8} = t^{2} = 1, st = ts^{3} >
<s,t; s^{8} = t^{2} = 1, st = ts^{5} >
GThis is the same subgroup lattice structure as for the lattice of subgroups of C_{8} x C_{2}, although the groups are of course nonisomorphic.
/  \
<s^{2},t> <s> <st>
/  \  /
<s^{4},t> <s^{2}t> <s^{2}>
/  \  /
<t> <s^{4}t> <s^{4}>
\  /
{1}
< s,t; s^{4} = t^{4} = 1, st = ts^{3} >SmallGroup id: ???????
<a,b,c; a^{4} = b^{2} = c^{2} = 1, cbca^{2}b = 1, bab = a, cac = a>
<x,y,z; x^{2} = y^{3} = z^{3} = 1, yz = zy, yxy = x, zxz = x>
<s,t; s^{4} = t^{5} = 1, tst = s>
<s,t; s^{4} = t^{5} = 1, ts = st^{2}>This is the Galois group of x^{5} 2 over the rationals, and can be represented as the subgroup of S_{5} generated by (2, 3, 5, 4) and (1, 2, 3, 4, 5).
<s,t; s^{9} = t^{3} = 1, st = ts^{4} >
<x,y,z; x^{3} = y^{3} = z^{3} = 1, yz = zyx, xy = yx, xz = zx>Reference: Burnside, p. 145.
<x,y,; x^{7} = y^{4} = 1, yx =x^{1}y>