Finite Group Databasebeta

← #11724491

Subgroups — #11724491

Order 864.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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⋊(#41⋊C3))C212yesmaximal, complemented
3#64366C311yesmaximal, characteristic, complemented
4C3⋊#994—34nomaximal
5C3×#634—41nomaximal
6C32×SL(2,𝔽3)C2211yescharacteristic, complemented
7C32×#41C2×C311yescharacteristic
8#3832C2×C312yes—
9D4×C3⋊D3—32no—
10#3828—31no—
11C2×C4⋊(C3⋊D3)—31no—
12C3×#41⋊C3—24no—
13C3×#41⋊C3D314yescomplemented
14C3⋊(#41⋊C3)—38no—
15C33×C4—41no—
16C3×C3⋊Dic3—42no—
17#1009—34no—
18#994—91no—
19C22×C3⋊D3—32no—
20C32×Q8C22×C311yescharacteristic
21C32×D4—31no—
22C2×C32×C4—31no—
23#638—32no—
24C4⋊(C3⋊D3)—32no—
25C4⋊(C3⋊D3)—31no—
26C4×C3⋊D3—32no—
27#634A411yescharacteristic
28C3×Dic6—124no—
29C3×SL(2,𝔽3)—24no—
30C3×SL(2,𝔽3)C2×D314yescomplemented
31C2×C33—41no—
32C3×#41C3×D314yes—
33#239—38no—
34D3×D4—98no—
35#235—94no—
36C2×D12—94no—
37#41⋊C3—24no—
38#41⋊C3—92no—
39#41⋊C3C3⋊D311yescharacteristic, complemented
40C22×C32—31no—
41C2×C3⋊D3—32no—
42C2×C3⋊D3—122no—
43C32×C4—31no—
44C32×C4—44no—
45C32×C4—84no—
46C32×C4C2×A411yescharacteristic
47C3⋊Dic3C2×A412yes—
48C3×Dic3—128no—
49H(5,𝔽2)—91noSylow
50C33—41noSylow
51C22×D3—98no—
52C3×Q8—361no—
53C3×Q8C2×C3×D314yes—
54C3×D4—34no—
55C2×C3×C4—34no—
56#67—98no—
57D12—98no—
58D12—94no—
59C4×D3—98no—
60Dic6—34no—
61SL(2,𝔽3)—24no—
62SL(2,𝔽3)C2×C3⋊D311yescharacteristic, complemented
63C2×C32—44no—
64C2×C32—61no—
65C2×C32—84no—
66C2×C32C22×A411yescharacteristic
67C3⋊D3—62no—
68#41—92no—
69#41—271no—
70#41C3×C3⋊D311yescharacteristic
71C2×D4—271no—
72C2×D4—272no—
73C22×C3—34no—
74C2×D3—98no—
75C2×D3—368no—
76C3×C4—34no—
77C3×C4—41no—
78C3×C4—84no—
79C3×C4—362no—
80C3×C4D3×A414yes—
81Dic3—38no—
82C32—44no—
83C32—84no—
84C32#99411yescharacteristic, complemented
85C23—272no—
86Q8—91no—
87Q8C2×C3×C3⋊D311yescharacteristic
88D4—31no—
89D4—272no—
90D4—272no—
91D4—271no—
92C2×C4—31no—
93C2×C4—272no—
94C2×C3—41no—
95C2×C3—64no—
96C2×C3—84no—
97C2×C3C2×D3×A414yes—
98D3—188no—
99C22—31no—
100C22—272no—
101C22—1082no—
102C4—31no—
103C4—92no—
104C4A4×C3⋊D311yescharacteristic
105C3—41no—
106C3—84no—
107C3C3⋊#99414yescomplemented
108C2—61no—
109C2—542no—
110C2C2×A4×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4×C3⋊D3
commutator, cosocle, hypocenterC32×Q8C22×C3
FittingC32×#41C2×C3
socleC2×C32C22×A4
hypercenterC4A4×C3⋊D3
LCSG ▷ C32×Q8
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C32×#41⋊C3C2semidirect
2. C3⋊(C3⋊(#41⋊C3))C2semidirect
3. #64366C3semidirect
6. C32×SL(2,𝔽3)C22semidirect
13. C3×#41⋊C3D3semidirect
30. C3×SL(2,𝔽3)C2×D3semidirect
39. #41⋊C3C3⋊D3semidirect
62. SL(2,𝔽3)C2×C3⋊D3semidirect
84. C32#994semidirect
107. C3C3⋊#994semidirect

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⋊(#41⋊C3))2
#643663
C3⋊#9943
C3×#6344