Finite Group Databasebeta

← #11722391

Subgroups — #11722391

Order 864.

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

normal subgroups only

This group has 226 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⋊#3689C212yesmaximal, complemented
3#64158—31nomaximal
4#63692—33nomaximal
5C4⋊(C3⋊D9)—41nomaximal
6C3×Q8⋊C9C2211yescharacteristic, complemented
7C32×#41D311yescharacteristic
8C2×C4⋊(C3⋊D3)—31no—
9#3755—32no—
10#3753—32no—
11#3748—31no—
12#41⋊C9D313yescomplemented
13#3689—36no—
14C2×C3⋊D9—42no—
15C3×C4×C9—41no—
16#949—93no—
17#949—91no—
18C22×C3⋊D3—61no—
19C32×Q8C2×D311yescharacteristic
20C32×D4—31no—
21C2×C32×C4—31no—
22C4⋊(C3⋊D3)—32no—
23C4⋊(C3⋊D3)—61no—
24C3⋊(C3⋊C8)—32no—
25D36—123no—
26Q8⋊C9C2×D313yescomplemented
27C2×C3×C9—41no—
28C3⋊D9—82no—
29C3×#41D913yescomplemented
30C3×#41C3⋊D311yescharacteristic
31C2×D12—93no—
32C2×D12—91no—
33#215—96no—
34#215—92no—
35#213—96no—
36#213—92no—
37#208—93no—
38#208—91no—
39C22×C32—31no—
40C2×C3⋊D3—62no—
41C2×C3⋊D3—122no—
42C32×C4—31no—
43C32×C4S411yescharacteristic
44C2×D9—126no—
45C4×C9—43no—
46AGL(1,ℤ/8)—271noSylow
47C3×C9—41noSylow
48C22×D3—181no—
49C22×D3—183no—
50C3×Q8C2×D913yes—
51C3×Q8C2×C3⋊D311yescharacteristic
52C3×D4—33no—
53C3×D4—31no—
54C2×C3×C4—33no—
55C2×C3×C4—31no—
56D12—92no—
57D12—96no—
58D12—183no—
59D12—181no—
60C3⋊C8—92no—
61C3⋊C8—96no—
62C2×C32—61no—
63C2×C32C2×S411yescharacteristic
64C3⋊D3—122no—
65C2×C9—43no—
66D9—246no—
67#41C3⋊D911yescharacteristic, complemented
68C2×D4—271no—
69SD16—272no—
70D8—272no—
71M16—271no—
72C22×C3—31no—
73C22×C3—33no—
74C2×D3—182no—
75C2×D3—186no—
76C2×D3—362no—
77C2×D3—366no—
78C3×C4—31no—
79C3×C4—33no—
80C3×C4(C22⋊C9)⋊C213yes—
81C3×C4C3⋊S411yescharacteristic
82C32#98611yescharacteristic
83C9—43no—
84C23—541no—
85Q8C2×C3⋊D911yescharacteristic
86D4—31no—
87D4—272no—
88D4—541no—
89C2×C4—31no—
90C8—272no—
91C2×C3—61no—
92C2×C3—63no—
93C2×C3C2×(C22⋊C9)⋊C213yes—
94C2×C3C2×C3⋊S411yescharacteristic
95D3—362no—
96D3—366no—
97C22—31no—
98C22—542no—
99C22—1082no—
100C4—31no—
101C4C3⋊((C22⋊C9)⋊C2)11yescharacteristic
102C3#6369213yescomplemented
103C3#6426711yescharacteristic
104C2—61no—
105C2—1082no—
106C2C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
subgroupG/H
centerC2C2×C3⋊((C22⋊C9)⋊C2)
commutator, cosocle, hypocenterC3×Q8⋊C9C22
FrattiniC2×C3C2×C3⋊S4
FittingC32×#41D3
socleC2×C32C2×S4
hypercenterC4C3⋊((C22⋊C9)⋊C2)
LCSG ▷ C3×Q8⋊C9
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×#41⋊C9C2semidirect
2. C3⋊#3689C2semidirect
6. C3×Q8⋊C9C22semidirect
12. #41⋊C9D3semidirect
26. Q8⋊C9C2×D3semidirect
29. C3×#41D9semidirect
67. #41C3⋊D9semidirect
102. C3#63692semidirect

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
#641583
#636923
C4⋊(C3⋊D9)4