Finite Group Databasebeta

← #11724344

Subgroups — #11724344

Order 864.

356 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C2×#157⋊D3C214yesmaximal, complemented
2C4⋊#1113C212yesmaximal, complemented
3C3×C4×#157C211yesmaximal, characteristic, complemented
4#64231—31nomaximal
5C2×#1113C2214yes—
6C2×C3×#157C2212yes—
7#157⋊D3—28no—
8C3×C4⋊(C3⋊D3)—22no—
9C3×C4×C3⋊D3C2211yescharacteristic
10#6295—22no—
11C2×GO+(4,𝔽2)—34no—
12#3802—32no—
13D3×D12—62no—
14C4×#157D311yescharacteristic, complemented
15C2×C3×C3⋊D3—24no—
16C2×C3×C3⋊D3C2311yescharacteristic
17#1113—28no—
18C3×#157D414yescomplemented
19C33×C4D411yescharacteristic, complemented
20C3×C3⋊Dic3D411yescharacteristic, complemented
21#874—91nomaximal
22C2×D32—34no—
23C2×D32—64no—
24C2×#157C2×D312yes—
25GO+(4,𝔽2)—68no—
26C4⋊(C3⋊D3)—22no—
27C4×C3⋊D3C2×D311yescharacteristic
28C3×D12—62no—
29C3×D12—62no—
30C3×D12—122no—
31C3×C4×D3—62no—
32#626—64no—
33#625—62no—
34C2×C33C2×D411yescharacteristic
35C3×C3⋊D3—44no—
36C3×C3⋊D3C2×D412yes—
37D3×D4—182no—
38C2×D12—92no—
39C2×D12—94no—
40C3×C42—91no—
41C2×C3⋊D3—24no—
42C2×C3⋊D3C22×D311yescharacteristic
43C2×C3×D3—62no—
44C2×C3×D3—64no—
45C2×C3×D3—64no—
46C2×C3×D3—124no—
47D32—68no—
48D32—128no—
49#157D1214yescomplemented
50C32×C4—22no—
51C32×C4—42no—
52C32×C4D1211yescharacteristic, complemented
53C3⋊Dic3D1211yescharacteristic, complemented
54C3×Dic3—62no—
55#127—271noSylow
56C33#12711yescharacteristic, complemented, Sylow
57C22×D3—94no—
58C22×D3—184no—
59C3×D4—182no—
60C2×C3×C4—92no—
61C2×C3×C4—91no—
62#67—184no—
63D12—62no—
64D12—62no—
65D12—122no—
66D12—182no—
67D12—188no—
68C4×D3—62no—
69C2×C32—22no—
70C2×C32—42no—
71C2×C32C2×D1211yescharacteristic
72C3⋊D3—44no—
73C3⋊D3C2×D1212yes—
74C3×D3—64no—
75C3×D3—124no—
76C3×D3—124no—
77C3×D3—244no—
78C2×D4—272no—
79C2×D4—274no—
80C42—91no—
81C22×C3—91no—
82C22×C3—184no—
83C2×D3—62no—
84C2×D3—64no—
85C2×D3—64no—
86C2×D3—124no—
87C2×D3—188no—
88C2×D3—368no—
89C3×C4—22no—
90C3×C4—42no—
91C3×C4—94no—
92C3×C4—91no—
93C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
94Dic3—62no—
95C32—22no—
96C32—42no—
97C32#87411yescharacteristic, complemented
98C23—274no—
99D4—182no—
100D4—542no—
101D4—548no—
102C2×C4—92no—
103C2×C4—91no—
104C2×C3—22no—
105C2×C3—42no—
106C2×C3—92no—
107C2×C3—364no—
108C2×C3C2×GO+(4,𝔽2)11yescharacteristic
109D3—64no—
110D3—124no—
111D3—124no—
112D3—244no—
113C22—91no—
114C22—184no—
115C22—548no—
116C4—94no—
117C4—91no—
118C4#157⋊D311yescharacteristic, complemented
119C3—22no—
120C3—42no—
121C3#6423111yescharacteristic, complemented
122C2—92no—
123C2—364no—
124C2C2×#157⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C23
FittingC33×C4D4
socleC2×C33C2×D4
hypercenterC4#157⋊D3
hypocenterC33#127
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×#157⋊D3C2semidirect
2. C4⋊#1113C2semidirect
3. C3×C4×#157C2semidirect
14. C4×#157D3semidirect
18. C3×#157D4semidirect
19. C33×C4D4semidirect
20. C3×C3⋊Dic3D4semidirect
49. #157D12semidirect
52. C32×C4D12semidirect
53. C3⋊Dic3D12semidirect
56. C33#127semidirect
93. C3×C4GO+(4,𝔽2)semidirect
97. C32#874semidirect
118. C4#157⋊D3semidirect
121. C3#64231semidirect

Maximal subgroups

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

subgroupindex
C2×#157⋊D32
C4⋊#11132
C3×C4×#1572
#642313
#8749