Finite Group Databasebeta

← #13049672

Subgroups — #13049672

Order 1296.

2136 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 2136 conjugacy classes of subgroups — too many to draw. Only its 268 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×D3×C3⋊(C3⋊D3)C212yesmaximal, complemented
2C4×C3⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
3C3×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
4C33×D12C211yesmaximal, characteristic, complemented
5#10620145C212yesmaximal, complemented
6D4×C3⋊(C3⋊D3)—31nomaximal
7#89026—313nomaximal
8C2×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C3×C3⋊(C3⋊D3)C2212yescomplemented
10C2×C33×D3C2212yescomplemented
11D3×C3⋊(C3⋊D3)—24no—
12C34×C4C2211yescharacteristic, complemented
13C3⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
14C22×C3⋊(C3⋊D3)—32no—
15C2×D3×C3⋊D3—326no—
16C33×D4—31no—
17#6312—32no—
18C4⋊(C3⋊(C3⋊D3))D311yescharacteristic, complemented
19C4×C3⋊(C3⋊D3)—31no—
20C4×C3⋊(C3⋊D3)—313no—
21C4×C3⋊(C3⋊D3)—613no—
22C3×C4⋊(C3⋊D3)—313no—
23C32×D12D3113yescomplemented
24#6290—326no—
25C2×C34C2311yescharacteristic
26C3⋊(C3⋊(C3⋊D3))D412yescomplemented
27C3×C3⋊(C3⋊D3)—22no—
28C33×D3—22no—
29D4×C3⋊D3—913no—
30#3802—913no—
31C22×C33—32no—
32C2×C3⋊(C3⋊D3)—31no—
33C2×C3⋊(C3⋊D3)—313no—
34C2×C3⋊(C3⋊D3)—64no—
35C2×C3⋊(C3⋊D3)—613no—
36C2×C3⋊(C3⋊D3)C2×D312yescomplemented
37C2×C3×C3⋊D3—326no—
38C2×C32×D3C2×D3126yescomplemented
39D3×C3⋊D3—652no—
40C33×C4—213no—
41C33×C4C2×D3113yescomplemented
42C33×C4C2×D311yescharacteristic, complemented
43C3⋊(C3⋊Dic3)—31no—
44C3⋊(C3⋊Dic3)—313no—
45C3⋊(C3⋊Dic3)—613no—
46C34C2×D411yescharacteristic, complemented, Sylow
47C22×C3⋊D3—926no—
48C2×D32—926no—
49C32×D4—313no—
50#638—926no—
51C4⋊(C3⋊D3)—313no—
52C4×C3⋊D3—913no—
53C4×C3⋊D3—913no—
54C4×C3⋊D3—1852no—
55C3×D12—913no—
56C3×D12C3⋊D3113yescomplemented
57#625—926no—
58C2×C33—213no—
59C2×C33—62no—
60C2×C33C22×D3113yes—
61C2×C33C22×D311yescharacteristic
62C3⋊(C3⋊D3)—22no—
63C3⋊(C3⋊D3)—32no—
64C3⋊(C3⋊D3)—326no—
65C3⋊(C3⋊D3)—626no—
66C3×C3⋊D3—626no—
67C32×D3—226no—
68D3×D4—2713no—
69D3×D4—271no—
70C22×C32—326no—
71C2×C3⋊D3—326no—
72C2×C3⋊D3—913no—
73C2×C3⋊D3—913no—
74C2×C3⋊D3—1852no—
75C2×C3⋊D3—1852no—
76C2×C3×D3—926no—
77C2×C3×D3C2×C3⋊D3126yescomplemented
78D32—1852no—
79C32×C4—252no—
80C32×C4D32113yescomplemented
81C32×C4C2×C3⋊D3113yescomplemented
82C3⋊Dic3—913no—
83C3⋊Dic3—913no—
84C3⋊Dic3—1852no—
85C33—213no—
86C33D3×D4113yescomplemented
87C33D3×D411yescharacteristic, complemented
88C22×D3—2726no—
89C22×D3—272no—
90C3×D4—313no—
91C3×D4—271no—
92#67—2726no—
93#67—272no—
94D12—913no—
95D12C3⋊(C3⋊D3)11yescharacteristic, complemented
96C4×D3—2713no—
97C4×D3—271no—
98C4×D3—5413no—
99C2×C32—252no—
100C2×C32—626no—
101C2×C32C2×D32113yes—
102C2×C32C22×C3⋊D3113yes—
103C3⋊D3—626no—
104C3⋊D3—926no—
105C3⋊D3—926no—
106C3⋊D3—18104no—
107C3×D3—226no—
108C3×D3—1826no—
109C2×D4—811noSylow
110C22×C3—326no—
111C22×C3—272no—
112C2×D3—926no—
113C2×D3—2713no—
114C2×D3—271no—
115C2×D3—5413no—
116C2×D3—544no—
117C2×D3—5452no—
118C2×D3C2×C3⋊(C3⋊D3)12yescomplemented
119C3×C4—213no—
120C3×C4D3×C3⋊D3113yescomplemented
121C3×C4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
122Dic3—2713no—
123Dic3—271no—
124Dic3—5413no—
125C32—252no—
126C32#3802113yescomplemented
127C32D4×C3⋊D3113yescomplemented
128C23—812no—
129D4—31no—
130D4—271no—
131D4—812no—
132C2×C4—811no—
133C2×C3—213no—
134C2×C3—626no—
135C2×C3—542no—
136C2×C3C2×D3×C3⋊D3113yes—
137C2×C3C22×C3⋊(C3⋊D3)11yescharacteristic
138D3—22no—
139D3—1826no—
140D3—2726no—
141D3—272no—
142D3—5426no—
143C22—32no—
144C22—272no—
145C22—811no—
146C22—1624no—
147C4—811no—
148C4D3×C3⋊(C3⋊D3)11yescharacteristic, complemented
149C3—213no—
150C3#89026113yescomplemented
151C3D4×C3⋊(C3⋊D3)11yescharacteristic, complemented
152C2—62no—
153C2—542no—
154C2—812no—
155C2C2×D3×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×C3⋊(C3⋊D3)
commutator, socle, cosocleC2×C34C23
FittingC34×C4C22
hypercenterC4D3×C3⋊(C3⋊D3)
hypocenterC34C2×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×C3⋊(C3⋊D3)C2semidirect
2. C4×C3⋊(C3⋊(C3⋊D3))C2semidirect
3. C3×C4⋊(C3⋊(C3⋊D3))C2semidirect
4. C33×D12C2semidirect
5. #10620145C2semidirect
9. C2×C3×C3⋊(C3⋊D3)C22semidirect
10. C2×C33×D3C22semidirect
12. C34×C4C22semidirect
13. C3⋊(C3⋊(C3⋊Dic3))C22semidirect
18. C4⋊(C3⋊(C3⋊D3))D3semidirect
23. C32×D12D3semidirect
26. C3⋊(C3⋊(C3⋊D3))D4semidirect
36. C2×C3⋊(C3⋊D3)C2×D3semidirect
38. C2×C32×D3C2×D3semidirect
41. C33×C4C2×D3semidirect
42. C33×C4C2×D3semidirect
46. C34C2×D4semidirect
56. C3×D12C3⋊D3semidirect
77. C2×C3×D3C2×C3⋊D3semidirect
80. C32×C4D32semidirect
81. C32×C4C2×C3⋊D3semidirect
86. C33D3×D4semidirect
87. C33D3×D4semidirect
95. D12C3⋊(C3⋊D3)semidirect
118. C2×D3C2×C3⋊(C3⋊D3)semidirect
120. C3×C4D3×C3⋊D3semidirect
121. C3×C4C2×C3⋊(C3⋊D3)semidirect
126. C32#3802semidirect
127. C32D4×C3⋊D3semidirect
148. C4D3×C3⋊(C3⋊D3)semidirect
150. C3#89026semidirect
151. C3D4×C3⋊(C3⋊D3)semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C3⋊(C3⋊D3)2
C4×C3⋊(C3⋊(C3⋊D3))2
C3×C4⋊(C3⋊(C3⋊D3))2
C33×D122
#106201452
D4×C3⋊(C3⋊D3)3
#890263