Finite Group Databasebeta

← D3×C4⋊(C3⋊(C3⋊D3)) #13049671

Subgroups — D3×C4⋊(C3⋊(C3⋊D3)) #13049671

Order 1296.

2136 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2136 conjugacy classes of subgroups — too many to draw. Only its 322 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×C4×D3C211yesmaximal, characteristic, complemented
5#10620147C212yesmaximal, complemented
6C2×C4⋊(C3⋊(C3⋊D3))—31nomaximal
7D3×C4⋊(C3⋊D3)—313nomaximal
8C2×C3⋊(C3⋊(C3⋊D3))C2212yescomplemented
9C2×C3×C3⋊(C3⋊D3)C2212yescomplemented
10C2×C33×D3C2211yescharacteristic
11D3×C3⋊(C3⋊D3)—24no—
12C34×C4C2211yescharacteristic, complemented
13C33×Dic3C2211yescharacteristic, complemented
14C22×C3⋊(C3⋊D3)—32no—
15C2×D3×C3⋊D3—326no—
16C2×C33×C4—31no—
17C4⋊(C3⋊(C3⋊D3))—32no—
18C4⋊(C3⋊(C3⋊D3))—31no—
19C4⋊(C3⋊(C3⋊D3))—313no—
20C4⋊(C3⋊(C3⋊D3))—613no—
21C4⋊(C3⋊(C3⋊D3))D311yescharacteristic, direct product factor
22C3×C4⋊(C3⋊D3)—313no—
23C32×C4×D3D3113yescomplemented
24#6292—326no—
25C2×C34C2311yescharacteristic
26C3⋊(C3⋊(C3⋊D3))—22no—
27C3×C3⋊(C3⋊D3)—22no—
28C33×D3D412yescomplemented
29C2×C4⋊(C3⋊D3)—913no—
30D3×D12—913no—
31C22×C33—31no—
32C2×C3⋊(C3⋊D3)—32no—
33C2×C3⋊(C3⋊D3)—326no—
34C2×C3⋊(C3⋊D3)—64no—
35C2×C3⋊(C3⋊D3)—626no—
36C2×C3⋊(C3⋊D3)C2×D312yescomplemented
37C2×C3×C3⋊D3—326no—
38C2×C32×D3C2×D3113yes—
39D3×C3⋊D3—652no—
40C33×C4—213no—
41C33×C4—31no—
42C33×C4C2×D3113yescomplemented
43C33×C4C2×D311yescharacteristic, complemented
44C32×Dic3C2×D3113yescomplemented
45C34C2×D411yescharacteristic, complemented, Sylow
46C22×C3⋊D3—926no—
47C2×D32—926no—
48C2×C32×C4—313no—
49C4⋊(C3⋊D3)—313no—
50C4⋊(C3⋊D3)—926no—
51C4⋊(C3⋊D3)—913no—
52C4⋊(C3⋊D3)—913no—
53C4⋊(C3⋊D3)—1852no—
54C3×D12—913no—
55C3×C4×D3C3⋊D3113yescomplemented
56#626—926no—
57C2×C33—213no—
58C2×C33—32no—
59C2×C33C22×D3113yes—
60C2×C33C22×D311yescharacteristic
61C3⋊(C3⋊D3)—22no—
62C3⋊(C3⋊D3)—62no—
63C3⋊(C3⋊D3)—626no—
64C3⋊(C3⋊D3)—1226no—
65C3×C3⋊D3—626no—
66C32×D3D12126yescomplemented
67D3×D4—271no—
68C2×D12—2713no—
69C22×C32—313no—
70C2×C3⋊D3—326no—
71C2×C3⋊D3—926no—
72C2×C3⋊D3—926no—
73C2×C3⋊D3—18104no—
74C2×C3⋊D3—1852no—
75C2×C3×D3—926no—
76C2×C3×D3C2×C3⋊D3113yes—
77D32—1852no—
78C32×C4—252no—
79C32×C4—313no—
80C32×C4D32113yescomplemented
81C32×C4C2×C3⋊D3113yescomplemented
82C3×Dic3C2×C3⋊D3113yescomplemented
83C33—213no—
84C33C2×D12113yescomplemented
85C33D3×D411yescharacteristic, complemented
86C22×D3—2726no—
87C22×D3—272no—
88C3×D4—271no—
89C2×C3×C4—313no—
90#67—272no—
91D12—913no—
92D12—2726no—
93D12—2713no—
94D12—271no—
95D12—5413no—
96C4×D3C3⋊(C3⋊D3)11yescharacteristic, complemented
97C2×C32—252no—
98C2×C32—326no—
99C2×C32C2×D32113yes—
100C2×C32C22×C3⋊D3113yes—
101C3⋊D3—626no—
102C3⋊D3—1826no—
103C3⋊D3—1826no—
104C3⋊D3—36104no—
105C3×D3—1826no—
106C3×D3C4⋊(C3⋊D3)126yescomplemented
107C2×D4—811noSylow
108C22×C3—313no—
109C22×C3—272no—
110C2×D3—926no—
111C2×D3—2726no—
112C2×D3—272no—
113C2×D3—5426no—
114C2×D3—5452no—
115C2×D3—544no—
116C2×D3C2×C3⋊(C3⋊D3)11yescharacteristic
117C3×C4—213no—
118C3×C4—313no—
119C3×C4D3×C3⋊D3113yescomplemented
120C3×C4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
121Dic3C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
122C32—252no—
123C32D3×D12113yescomplemented
124C32C2×C4⋊(C3⋊D3)113yescomplemented
125C23—812no—
126D4—271no—
127D4—812no—
128D4—811no—
129C2×C4—31no—
130C2×C3—213no—
131C2×C3—326no—
132C2×C3—542no—
133C2×C3C2×D3×C3⋊D3113yes—
134C2×C3C22×C3⋊(C3⋊D3)11yescharacteristic
135D3—1826no—
136D3—5426no—
137D3—542no—
138D3—10826no—
139D3C4⋊(C3⋊(C3⋊D3))12yesdirect product factor
140C22—31no—
141C22—272no—
142C22—812no—
143C22—1624no—
144C4—31no—
145C4D3×C3⋊(C3⋊D3)11yescharacteristic, complemented
146C3—213no—
147C3D3×C4⋊(C3⋊D3)113yescomplemented
148C3C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic, complemented
149C2—32no—
150C2—542no—
151C2—1622no—
152C2C2×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×C4×D3C2semidirect
5. #10620147C2semidirect
8. C2×C3⋊(C3⋊(C3⋊D3))C22semidirect
9. C2×C3×C3⋊(C3⋊D3)C22semidirect
12. C34×C4C22semidirect
13. C33×Dic3C22semidirect
21. C4⋊(C3⋊(C3⋊D3))D3direct (N × H)
23. C32×C4×D3D3semidirect
28. C33×D3D4semidirect
36. C2×C3⋊(C3⋊D3)C2×D3semidirect
42. C33×C4C2×D3semidirect
43. C33×C4C2×D3semidirect
44. C32×Dic3C2×D3semidirect
45. C34C2×D4semidirect
55. C3×C4×D3C3⋊D3semidirect
66. C32×D3D12semidirect
80. C32×C4D32semidirect
81. C32×C4C2×C3⋊D3semidirect
82. C3×Dic3C2×C3⋊D3semidirect
84. C33C2×D12semidirect
85. C33D3×D4semidirect
96. C4×D3C3⋊(C3⋊D3)semidirect
106. C3×D3C4⋊(C3⋊D3)semidirect
119. C3×C4D3×C3⋊D3semidirect
120. C3×C4C2×C3⋊(C3⋊D3)semidirect
121. Dic3C2×C3⋊(C3⋊D3)semidirect
123. C32D3×D12semidirect
124. C32C2×C4⋊(C3⋊D3)semidirect
139. D3C4⋊(C3⋊(C3⋊D3))direct (N × H)
145. C4D3×C3⋊(C3⋊D3)semidirect
147. C3D3×C4⋊(C3⋊D3)semidirect
148. C3C2×C4⋊(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×C4×D32
#106201472
C2×C4⋊(C3⋊(C3⋊D3))3
D3×C4⋊(C3⋊D3)3