Finite Group Databasebeta

← #11722784

Subgroups — #11722784

Order 864.

164 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 29 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
1#89040C211yesmaximal, characteristic
2C3×C4⋊#157C211yesmaximal, characteristic
3#88808C211yesmaximal, characteristic
4#63730—31nomaximal
5C2×C3×#157—21no—
6C3×C4×C3⋊D3C2211yescharacteristic
7C3×#634—21no—
8#6296C412yescomplemented
9#6294—21no—
10C3×C3⋊(C3⋊C8)—21no—
11#3797—31no—
12D3×Dic6—61no—
13C4⋊#157D311yescharacteristic
14#3710—31no—
15D3×C3⋊C8—61no—
16C2×C3×C3⋊D3D411yescharacteristic
17C3×#157—41no—
18C33×C4D411yescharacteristic
19C3×C3⋊Dic3—22no—
20C3×C3⋊Dic3C2×C411yescharacteristic
21#810—91nomaximal
22C2×#157—21no—
23C4×C3⋊D3C2×D311yescharacteristic
24#634—21no—
25C3×C4×D3—61no—
26C3×C4×D3—61no—
27C3×Dic6—61no—
28C3×Dic6—61no—
29C3×Dic6—121no—
30#627—32no—
31#627—62no—
32#624—61no—
33D3×Dic3—62no—
34C3⋊(C3⋊C8)—21no—
35C3×C3⋊C8—61no—
36C3×C3⋊C8—61no—
37C3×C3⋊C8—121no—
38C2×C33C22⋊C411yescharacteristic
39C3×C3⋊D3SD1611yescharacteristic
40C3×C3⋊D3Q1611yescharacteristic
41D3×Q8—181no—
42C2×Dic6—91no—
43C3×C4⋊C4—91no—
44C2×C3⋊C8—91no—
45D3×C8—181no—
46C2×C3⋊D3#6711yescharacteristic
47C2×C3×D3—61no—
48C2×C3×D3—61no—
49#157—41no—
50C32×C4—21no—
51C32×C4—21no—
52C32×C4—41no—
53C32×C4—41no—
54C32×C4D1211yescharacteristic
55C3⋊Dic3—22no—
56C3⋊Dic3C4×D311yescharacteristic
57C3×Dic3—61no—
58C3×Dic3—61no—
59C3×Dic3—62no—
60C3×Dic3—62no—
61C3×Dic3—122no—
62#103—271noSylow
63C33#10311yescharacteristic, complemented, Sylow
64C3×Q8—181no—
65C2×C3×C4—91no—
66C2×C3×C4—181no—
67C2×Dic3—181no—
68C4×D3—61no—
69C4×D3—61no—
70C4×D3—182no—
71Dic6—61no—
72Dic6—61no—
73Dic6—92no—
74Dic6—121no—
75Dic6—182no—
76C3×C8—181no—
77C3⋊C8—61no—
78C3⋊C8—61no—
79C3⋊C8—121no—
80C2×C32—21no—
81C2×C32—21no—
82C2×C32—41no—
83C2×C32—41no—
84C2×C32(C2×D3)⋊C411yescharacteristic
85C3⋊D3#21411yescharacteristic
86C3⋊D3#21611yescharacteristic
87C3×D3—61no—
88C3×D3—61no—
89C3×D3—61no—
90C3×D3—61no—
91C2×Q8—271no—
92C2×C8—271no—
93C4⋊C4—91no—
94C22×C3—91no—
95C2×D3—61no—
96C2×D3—61no—
97C3×C4—21no—
98C3×C4—21no—
99C3×C4—41no—
100C3×C4—41no—
101C3×C4—91no—
102C3×C4—182no—
103C3×C4—361no—
104C3×C4GO+(4,𝔽2)11yescharacteristic
105Dic3—61no—
106Dic3—61no—
107Dic3—62no—
108Dic3—62no—
109Dic3—122no—
110C32—21no—
111C32—21no—
112C32—41no—
113C32—41no—
114C32#81011yescharacteristic, complemented
115Q8—181no—
116Q8—272no—
117C2×C4—91no—
118C2×C4—181no—
119C2×C4—541no—
120C8—181no—
121C2×C3—21no—
122C2×C3—21no—
123C2×C3—41no—
124C2×C3—41no—
125C2×C3—91no—
126C2×C3—91no—
127C2×C3D32⋊C411yescharacteristic
128D3—61no—
129D3—61no—
130D3—61no—
131D3—61no—
132C22—91no—
133C4—91no—
134C4—182no—
135C4—361no—
136C4#157⋊D311yescharacteristic
137C3—21no—
138C3—21no—
139C3—41no—
140C3—41no—
141C3#6373011yescharacteristic, complemented
142C2—91no—
143C2—91no—
144C2#8893911yescharacteristic
subgroupG/H
centerC2#88939
commutatorC3×C3⋊Dic3C2×C4
Frattini, hypercenterC4#157⋊D3
FittingC33×C4D4
socleC2×C33C22⋊C4
prime radicalC2×C3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C22
hypocenterC33#103
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
8. #6296C4semidirect
63. C33#103semidirect
114. C32#810semidirect
141. C3#63730semidirect

Maximal subgroups

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

subgroupindex
#890402
C3×C4⋊#1572
#888082
#637303
#8109