Finite Group Databasebeta

← #10593166

Subgroups — #10593166

Order 576.

141 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 141 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×AGL(1,𝔽9)C211yesmaximal, characteristic
2C2×#3777C211yesmaximal, characteristic
3C2×#3773C211yesmaximal, characteristic
4C22×#157—21no—
5C22×#157C2211yescharacteristic
6C2×AGL(1,𝔽9)—21no—
7C2×AGL(1,𝔽9)—21no—
8C2×AGL(1,𝔽9)C412yescomplemented
9C2×#624—21no—
10#3777—21no—
11#3777C412yescomplemented
12#3773—21no—
13#3773C412yescomplemented
14C22×C3⋊D3D411yescharacteristic
15C2×#157—22no—
16C2×#157—41no—
17C2×#157—41no—
18C2×#157C2×C411yescharacteristic
19C2×#157C2×C411yescharacteristic
20C2×#157C2×C411yescharacteristic
21C2×#157D411yescharacteristic
22C2×#157D411yescharacteristic
23C2×#157Q811yescharacteristic
24AGL(1,𝔽9)—22no—
25C2×C3×Dic3—41no—
26#624—22no—
27#624—41no—
28#340—91nomaximal, Sylow
29C2×C4×D3—121no—
30C22×C32SD1611yescharacteristic
31C2×C3⋊D3C22⋊C411yescharacteristic
32C2×C3⋊D3C22⋊C411yescharacteristic
33C2×C3⋊D3C4⋊C411yescharacteristic
34C2×C3⋊D3D811yescharacteristic
35C2×C3⋊D3SD1611yescharacteristic
36C2×C3⋊D3Q1611yescharacteristic
37#157—42no—
38#157C4211yescharacteristic
39#157C22⋊C411yescharacteristic
40#157C4⋊C411yescharacteristic
41#157C4⋊C411yescharacteristic
42C3×Dic3—42no—
43C22×C8—91no—
44C2×C4⋊C4—91no—
45C2×C4⋊C4—91no—
46C22×D3—121no—
47C2×C3×C4—121no—
48C2×Dic3—41no—
49C4×D3—122no—
50C4×D3—122no—
51C2×C32#10211yescharacteristic
52C2×C32#10311yescharacteristic
53C2×C32#10611yescharacteristic
54C3⋊D3#9511yescharacteristic
55C3⋊D3#10211yescharacteristic
56C3⋊D3#10311yescharacteristic
57C3⋊D3#10711yescharacteristic
58C22×C4—91no—
59C22×C4—181no—
60C22×C4—181no—
61C2×C8—92no—
62C2×C8—181no—
63C2×C8—181no—
64C4⋊C4—92no—
65C4⋊C4—92no—
66C4⋊C4—181no—
67C4⋊C4—181no—
68C22×C3—41no—
69C2×D3—121no—
70C2×D3—121no—
71C2×D3—121no—
72C2×D3—121no—
73C2×D3—121no—
74C2×D3—121no—
75C3×C4—122no—
76Dic3—42no—
77C32#34011yescharacteristic, complemented, Sylow
78C23—91no—
79C2×C4—91no—
80C2×C4—91no—
81C2×C4—91no—
82C2×C4—91no—
83C2×C4—91no—
84C2×C4—91no—
85C2×C4—121no—
86C2×C4—182no—
87C2×C4—182no—
88C2×C4—361no—
89C2×C4—361no—
90C2×C4—361no—
91C8—182no—
92C2×C3—41no—
93C2×C3—41no—
94C2×C3—41no—
95D3—121no—
96D3—121no—
97D3—121no—
98D3—121no—
99C22—91no—
100C22—91no—
101C22—91no—
102C22—91no—
103C22—91no—
104C22—91no—
105C22AΓL(1,𝔽9)11yescharacteristic
106C4—91no—
107C4—91no—
108C4—91no—
109C4—91no—
110C4—122no—
111C4—362no—
112C3—41no—
113C2—91no—
114C2—91no—
115C2—91no—
116C2—91no—
117C2GO+(4,𝔽2)⋊C411yescharacteristic
118C2PSU(3,𝔽2)⋊C411yescharacteristic
119C2AGL(1,𝔽9)⋊C411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22AΓL(1,𝔽9)
commutator#157C42
Fitting, socleC22×C32SD16
prime radicalC22×C3⋊D3D4
cosocleC22×#157C22
hypocenterC32#340
LCSG ▷ #157 ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C22

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
8. C2×AGL(1,𝔽9)C4semidirect
11. #3777C4semidirect
13. #3773C4semidirect
77. C32#340semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×AGL(1,𝔽9)2
C2×#37772
C2×#37732
#3409