Finite Group Databasebeta

← #11722390

Subgroups — #11722390

Order 864.

208 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 41 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×#41⋊C9C211yesmaximal, characteristic, complemented
2C3⋊#3689C211yesmaximal, characteristic, complemented
3C3⋊#3688C211yesmaximal, characteristic, complemented
4#64159—31nomaximal
5#63691—33nomaximal
6C4×C3⋊D9—41nomaximal
7C3×Q8⋊C9C2211yescharacteristic
8C32×#41D311yescharacteristic
9#3828—31no—
10#3756—31no—
11#3755—31no—
12#3754—31no—
13#3753—31no—
14C2×C3⋊(C3⋊C8)—31no—
15#41⋊C9D313yescomplemented
16#3689—33no—
17#3688—33no—
18C2×C3⋊D9—41no—
19C3×C4×C9—41no—
20C3⋊Dic9—41no—
21#950—93no—
22#950—91no—
23C32×Q8C2×D311yescharacteristic
24C32×D4—31no—
25C2×C32×C4—31no—
26#638—61no—
27C4⋊(C3⋊D3)—31no—
28C4×C3⋊D3—61no—
29#634—31no—
30C3⋊(C3⋊C8)—31no—
31C3⋊(C3⋊C8)—31no—
32C4×D9—123no—
33Q8⋊C9C2×D313yes—
34C2×C3×C9—41no—
35C3⋊D9—42no—
36C3×#41D913yescomplemented
37C3×#41C3⋊D311yescharacteristic
38#235—93no—
39#235—91no—
40#216—93no—
41#216—91no—
42#215—93no—
43#215—91no—
44#214—93no—
45#214—91no—
46#213—93no—
47#213—91no—
48C2×C3⋊C8—93no—
49C2×C3⋊C8—91no—
50C22×C32—31no—
51C2×C3⋊D3—61no—
52C32×C4—31no—
53C32×C4S411yescharacteristic
54C3⋊Dic3—61no—
55C2×D9—123no—
56C4×C9—43no—
57Dic9—123no—
58#135—271noSylow
59C3×C9—41noSylow
60C3×Q8C2×D913yes—
61C3×Q8C2×C3⋊D311yescharacteristic
62C3×D4—33no—
63C3×D4—31no—
64C2×C3×C4—33no—
65C2×C3×C4—31no—
66#67—181no—
67#67—183no—
68D12—91no—
69D12—93no—
70C4×D3—183no—
71C4×D3—181no—
72Dic6—91no—
73Dic6—93no—
74C3⋊C8—91no—
75C3⋊C8—93no—
76C3⋊C8—91no—
77C3⋊C8—93no—
78C2×C32—61no—
79C2×C32C2×S411yescharacteristic
80C3⋊D3—121no—
81C2×C9—43no—
82D9—126no—
83#41—271no—
84#41C3⋊D911yescharacteristic, complemented
85Q16—271no—
86SD16—271no—
87SD16—271no—
88D8—271no—
89C2×C8—271no—
90C22×C3—31no—
91C22×C3—33no—
92C2×D3—181no—
93C2×D3—183no—
94C3×C4—31no—
95C3×C4—33no—
96C3×C4(C22⋊C9)⋊C213yes—
97C3×C4C3⋊S411yescharacteristic
98Dic3—181no—
99Dic3—183no—
100C32#98511yescharacteristic
101C9—43no—
102Q8—271no—
103Q8C2×C3⋊D911yescharacteristic
104D4—31no—
105D4—271no—
106D4—541no—
107C2×C4—31no—
108C2×C4—541no—
109C8—271no—
110C8—271no—
111C2×C3—61no—
112C2×C3—63no—
113C2×C3C2×(C22⋊C9)⋊C213yes—
114C2×C3C2×C3⋊S411yescharacteristic
115D3—361no—
116D3—363no—
117C22—31no—
118C22—541no—
119C4—31no—
120C4—541no—
121C4C3⋊((C22⋊C9)⋊C2)11yescharacteristic
122C3#6369113yescomplemented
123C3#6426611yescharacteristic
124C2—61no—
125C2—1081no—
126C2C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
subgroupG/H
center, hypercenterC4C3⋊((C22⋊C9)⋊C2)
commutator, cosocle, hypocenterC3×Q8⋊C9C22
FrattiniC2×C3C2×C3⋊S4
FittingC32×#41D3
socleC2×C32C2×S4
LCSG ▷ C3×Q8⋊C9
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×#41⋊C9C2semidirect
2. C3⋊#3689C2semidirect
3. C3⋊#3688C2semidirect
15. #41⋊C9D3semidirect
36. C3×#41D9semidirect
84. #41C3⋊D9semidirect
122. C3#63691semidirect

Maximal subgroups

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

subgroupindex
C3×#41⋊C92
C3⋊#36892
C3⋊#36882
#641593
#636913
C4×C3⋊D94