Finite Group Databasebeta

← #11722708

Subgroups — #11722708

Order 864.

184 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 30 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
1C3⋊D3×Dic6C211yesmaximal, characteristic
2#88988C211yesmaximal, characteristic
3C3×C4⋊#157C211yesmaximal, characteristic
4#63784—31nomaximal
5C2×C3×#157—21no—
6#6308C411yescharacteristic, complemented
7C3×C4×C3⋊D3C2211yescharacteristic
8C32×Dic6C411yescharacteristic, complemented
9#6293—21no—
10Dic3×C3⋊D3—21no—
11C3⋊#622—21no—
12Q8×C3⋊D3—31no—
13D3×Dic6—62no—
14C4⋊#157D311yescharacteristic
15#3787—31no—
16C2×C3×C3⋊D3D411yescharacteristic
17C3×#157—41no—
18C33×C4C2×C411yescharacteristic
19C3⋊(C3⋊Dic3)—21no—
20C3×C3⋊Dic3D411yescharacteristic
21C32×Dic3—21no—
22#810—91nomaximal
23C2×#157—21no—
24C32×Q8—31no—
25C4×C3⋊D3—61no—
26C4×C3⋊D3C2×D311yescharacteristic
27#634—31no—
28#634—62no—
29#634—61no—
30#634—122no—
31C3×C4×D3—62no—
32C3×Dic6—22no—
33#627—64no—
34D3×Dic3—64no—
35#622—61no—
36C2×C33C22⋊C411yescharacteristic
37C3×C3⋊D3SD1611yescharacteristic
38C3×C3⋊D3Q1611yescharacteristic
39D3×Q8—182no—
40C2×Dic6—91no—
41C3×C4⋊C4—91no—
42C2×C3⋊C8—91no—
43C2×C3⋊D3#6711yescharacteristic
44C2×C3×D3—62no—
45#157—41no—
46C32×C4—22no—
47C32×C4—42no—
48C32×C4—61no—
49C32×C4C4×D311yescharacteristic
50C3⋊Dic3—61no—
51C3⋊Dic3—64no—
52C3⋊Dic3—124no—
53C3⋊Dic3D1211yescharacteristic
54C3×Dic3—24no—
55C3×Dic3—62no—
56#103—271noSylow
57C33#10311yescharacteristic, complemented, Sylow
58C3×Q8—62no—
59C2×C3×C4—91no—
60C2×C3×C4—181no—
61C2×Dic3—181no—
62C4×D3—62no—
63C4×D3—184no—
64Dic6—91no—
65Dic6—182no—
66Dic6—181no—
67Dic6—184no—
68Dic6—362no—
69Dic6#15711yescharacteristic, complemented
70C3⋊C8—181no—
71C2×C32—22no—
72C2×C32—42no—
73C2×C32(C2×D3)⋊C411yescharacteristic
74C3⋊D3#21411yescharacteristic
75C3⋊D3#21611yescharacteristic
76C3×D3—62no—
77C3×D3—62no—
78C2×Q8—271no—
79C2×C8—271no—
80C4⋊C4—91no—
81C22×C3—91no—
82C2×D3—62no—
83C3×C4—22no—
84C3×C4—42no—
85C3×C4—64no—
86C3×C4—91no—
87C3×C4—361no—
88C3×C4C2×#15711yescharacteristic
89Dic3—21no—
90Dic3—62no—
91Dic3—184no—
92Dic3—181no—
93Dic3—364no—
94C32—22no—
95C32—42no—
96C32#81011yescharacteristic, complemented
97Q8—31no—
98Q8—271no—
99Q8—541no—
100C2×C4—91no—
101C2×C4—181no—
102C2×C4—541no—
103C8—541no—
104C2×C3—22no—
105C2×C3—42no—
106C2×C3—91no—
107C2×C3—91no—
108C2×C3C22⋊#15711yescharacteristic
109D3—62no—
110D3—62no—
111C22—91no—
112C4—61no—
113C4—91no—
114C4—361no—
115C4—541no—
116C4D3×#15711yescharacteristic
117C3—22no—
118C3—42no—
119C3#6378411yescharacteristic, complemented
120C2—91no—
121C2—91no—
122C2(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC2(C2×D3)⋊#157
commutator, FittingC33×C4C2×C4
Frattini, hypercenterC4D3×#157
socleC2×C33C22⋊C4
prime radicalC2×C3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C22
hypocenterC33#103
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
6. #6308C4semidirect
8. C32×Dic6C4semidirect
57. C33#103semidirect
69. Dic6#157semidirect
96. C32#810semidirect
119. C3#63784semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×Dic62
#889882
C3×C4⋊#1572
#637843
#8109