Finite Group Databasebeta

← #11724470

Subgroups — #11724470

Order 864.

442 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
4#64266—312nomaximal
5#64159—31nomaximal
6C4×C3⋊(C3⋊D3)—41nomaximal
7C32×SL(2,𝔽3)C2211yescharacteristic
8C32×#41D311yescharacteristic, complemented
9#3828—31no—
10C3×#41⋊C3D3112yescomplemented
11C3⋊GL(2,𝔽3)—312no—
12C3⋊#226—312no—
13#3756—31no—
14#3755—31no—
15#3754—31no—
16#3753—31no—
17C2×C3⋊(C3⋊C8)—31no—
18C2×C3⋊(C3⋊D3)—41no—
19C33×C4—41no—
20C3⋊(C3⋊Dic3)—41no—
21#985—99no—
22#950—94no—
23C32×Q8C2×D311yescharacteristic
24C32×D4—31no—
25C2×C32×C4—31no—
26#638—61no—
27C4⋊(C3⋊D3)—31no—
28C4×C3⋊D3—61no—
29C4×C3⋊D3—1212no—
30#634—31no—
31C3×SL(2,𝔽3)C2×D3112yes—
32C3⋊(C3⋊C8)—31no—
33C3⋊(C3⋊C8)—31no—
34C2×C33—41no—
35C3⋊(C3⋊D3)—42no—
36C3×#41C3⋊D314yescomplemented
37#235—94no—
38#41⋊C3C3⋊D319yescomplemented
39GL(2,𝔽3)—99no—
40#226—99no—
41#216—94no—
42#215—94no—
43#214—94no—
44#213—94no—
45C2×C3⋊C8—94no—
46C22×C32—31no—
47C2×C3⋊D3—61no—
48C2×C3⋊D3—1212no—
49C32×C4—31no—
50C32×C4—412no—
51C32×C4S411yescharacteristic
52C3⋊Dic3—61no—
53C3⋊Dic3—1212no—
54#135—271noSylow
55C33—41noSylow
56C3×Q8C2×C3⋊D314yes—
57C3×D4—34no—
58C2×C3×C4—34no—
59#67—184no—
60D12—94no—
61C4×D3—184no—
62C4×D3—369no—
63Dic6—94no—
64SL(2,𝔽3)C2×C3⋊D319yes—
65C3⋊C8—94no—
66C3⋊C8—94no—
67C2×C32—412no—
68C2×C32—61no—
69C2×C32C2×S411yescharacteristic
70C3⋊D3—1224no—
71C3⋊D3—121no—
72#41—271no—
73#41C3⋊(C3⋊D3)11yescharacteristic, complemented
74Q16—271no—
75SD16—271no—
76SD16—271no—
77D8—271no—
78C2×C8—271no—
79C22×C3—34no—
80C2×D3—184no—
81C2×D3—369no—
82C3×C4—34no—
83C3×C4—49no—
84C3×C4C3⋊S414yes—
85Dic3—184no—
86Dic3—369no—
87C32—412no—
88C32#98511yescharacteristic, complemented
89Q8—271no—
90Q8C2×C3⋊(C3⋊D3)11yescharacteristic
91D4—31no—
92D4—271no—
93D4—541no—
94C2×C4—31no—
95C2×C4—541no—
96C8—271no—
97C8—271no—
98C2×C3—49no—
99C2×C3—64no—
100C2×C3C2×C3⋊S414yes—
101D3—3618no—
102D3—364no—
103C22—31no—
104C22—541no—
105C4—31no—
106C4—541no—
107C4C3⋊(C3⋊S4)11yescharacteristic
108C3—49no—
109C3#6426614yescomplemented
110C2—61no—
111C2—1081no—
112C2C2×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, hypercenterC4C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C22
FrattiniC2C2×C3⋊(C3⋊S4)
FittingC32×#41D3
socleC2×C32C2×S4
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C32×#41⋊C3C2semidirect
2. C3⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C3⋊(C3⋊#226)C2semidirect
8. C32×#41D3semidirect
10. C3×#41⋊C3D3semidirect
36. C3×#41C3⋊D3semidirect
38. #41⋊C3C3⋊D3semidirect
73. #41C3⋊(C3⋊D3)semidirect
88. C32#985semidirect
109. C3#64266semidirect

Maximal subgroups

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

subgroupindex
C32×#41⋊C32
C3⋊(C3⋊GL(2,𝔽3))2
C3⋊(C3⋊#226)2
#642663
#641593
C4×C3⋊(C3⋊D3)4