Finite Group Databasebeta

← C4×D3×C3⋊(C3⋊D3) #13049669

Subgroups — C4×D3×C3⋊(C3⋊D3) #13049669

Order 1296.

2136 conjugacy classes of subgroups; only the 332 normal subgroups are shown.

normal subgroups only

This group has 2136 conjugacy classes of subgroups — too many to draw. Only its 332 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)C211yesmaximal, characteristic
2C4×C3⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
3C3×C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C33×C4×D3C211yesmaximal, characteristic, complemented
5#10620144C211yesmaximal, characteristic, complemented
6Dic3×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
7D3×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
8C2×C4×C3⋊(C3⋊D3)—31nomaximal
9C4×D3×C3⋊D3—313nomaximal
10C2×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C33×D3C2211yescharacteristic
13D3×C3⋊(C3⋊D3)C414yesdirect product factor
14C34×C4C2211yescharacteristic, complemented
15C3⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
16C3×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
17C33×Dic3C2211yescharacteristic, complemented
18C22×C3⋊(C3⋊D3)—31no—
19C2×D3×C3⋊D3—313no—
20C2×C33×C4—31no—
21C2×C3⋊(C3⋊Dic3)—31no—
22C4×C3⋊(C3⋊D3)—31no—
23C4×C3⋊(C3⋊D3)—31no—
24C4×C3⋊(C3⋊D3)—313no—
25C4×C3⋊(C3⋊D3)—31no—
26C4×C3⋊(C3⋊D3)—613no—
27C4×C3⋊(C3⋊D3)D311yescharacteristic, direct product factor
28C3×C4×C3⋊D3—313no—
29C32×C4×D3D3113yescomplemented
30#6289—313no—
31Dic3×C3⋊D3—313no—
32D3×C3⋊Dic3—313no—
33C2×C34C2311yescharacteristic
34C3⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
35C3×C3⋊(C3⋊D3)C2×C412yescomplemented
36C33×D3C2×C412yescomplemented
37C2×C4×C3⋊D3—913no—
38C4×D32—913no—
39C22×C33—31no—
40C2×C3⋊(C3⋊D3)—31no—
41C2×C3⋊(C3⋊D3)—313no—
42C2×C3⋊(C3⋊D3)—34no—
43C2×C3⋊(C3⋊D3)—613no—
44C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
45C2×C3×C3⋊D3—313no—
46C2×C32×D3C2×D3113yes—
47D3×C3⋊D3—352no—
48C33×C4—213no—
49C33×C4—31no—
50C33×C4C2×D3113yescomplemented
51C33×C4C2×D311yescharacteristic, complemented
52C3⋊(C3⋊Dic3)—31no—
53C3⋊(C3⋊Dic3)—313no—
54C3⋊(C3⋊Dic3)—613no—
55C3⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
56C3×C3⋊Dic3—313no—
57C32×Dic3C2×D3113yescomplemented
58C34C22×C411yescharacteristic, complemented, Sylow
59C22×C3⋊D3—913no—
60C2×D32—913no—
61C2×C32×C4—313no—
62C2×C3⋊Dic3—913no—
63C4×C3⋊D3—313no—
64C4×C3⋊D3—913no—
65C4×C3⋊D3—913no—
66C4×C3⋊D3—913no—
67C4×C3⋊D3—913no—
68C4×C3⋊D3—1852no—
69C3×C4×D3—913no—
70C3×C4×D3C3⋊D3113yescomplemented
71#624—913no—
72D3×Dic3—913no—
73D3×Dic3—913no—
74C2×C33—213no—
75C2×C33—32no—
76C2×C33C22×D3113yes—
77C2×C33C22×D311yescharacteristic
78C3⋊(C3⋊D3)—32no—
79C3⋊(C3⋊D3)—326no—
80C3⋊(C3⋊D3)—626no—
81C3⋊(C3⋊D3)C4×D312yesdirect product factor
82C3×C3⋊D3—326no—
83C32×D3C4×D3126yescomplemented
84C2×C4×D3—2713no—
85C2×C4×D3—271no—
86C22×C32—313no—
87C2×C3⋊D3—313no—
88C2×C3⋊D3—952no—
89C2×C3⋊D3—913no—
90C2×C3⋊D3—913no—
91C2×C3⋊D3—1852no—
92C2×C3×D3—913no—
93C2×C3×D3C2×C3⋊D3113yes—
94D32—952no—
95C32×C4—252no—
96C32×C4—313no—
97C32×C4D32113yescomplemented
98C32×C4C2×C3⋊D3113yescomplemented
99C3⋊Dic3—313no—
100C3⋊Dic3—913no—
101C3⋊Dic3—913no—
102C3⋊Dic3—1852no—
103C3×Dic3—913no—
104C3×Dic3C2×C3⋊D3113yescomplemented
105C33—213no—
106C33C2×C4×D3113yescomplemented
107C33C2×C4×D311yescharacteristic, complemented
108C22×D3—2713no—
109C22×D3—271no—
110C2×C3×C4—313no—
111C2×C3×C4—271no—
112C2×Dic3—2713no—
113C2×Dic3—271no—
114C4×D3—913no—
115C4×D3—2713no—
116C4×D3—2713no—
117C4×D3—271no—
118C4×D3—2713no—
119C4×D3—271no—
120C4×D3—271no—
121C4×D3—5413no—
122C4×D3C3⋊(C3⋊D3)11yescharacteristic, direct product factor
123C2×C32—252no—
124C2×C32—326no—
125C2×C32C2×D32113yes—
126C2×C32C22×C3⋊D3113yes—
127C3⋊D3—326no—
128C3⋊D3—926no—
129C3⋊D3—926no—
130C3⋊D3—18104no—
131C3×D3—926no—
132C3×D3C4×C3⋊D3126yescomplemented
133C22×C4—811noSylow
134C22×C3—313no—
135C22×C3—271no—
136C2×D3—913no—
137C2×D3—2752no—
138C2×D3—2713no—
139C2×D3—271no—
140C2×D3—274no—
141C2×D3—5413no—
142C2×D3C2×C3⋊(C3⋊D3)11yescharacteristic
143C3×C4—213no—
144C3×C4—313no—
145C3×C4—271no—
146C3×C4D3×C3⋊D3113yescomplemented
147C3×C4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
148Dic3—913no—
149Dic3—2713no—
150Dic3—271no—
151Dic3—5413no—
152Dic3C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
153C32—252no—
154C32C4×D32113yescomplemented
155C32C2×C4×C3⋊D3113yescomplemented
156C23—811no—
157C2×C4—31no—
158C2×C4—271no—
159C2×C4—811no—
160C2×C4—811no—
161C2×C4—811no—
162C2×C4—811no—
163C2×C3—213no—
164C2×C3—326no—
165C2×C3—272no—
166C2×C3C2×D3×C3⋊D3113yes—
167C2×C3C22×C3⋊(C3⋊D3)11yescharacteristic
168D3—926no—
169D3—2726no—
170D3—272no—
171D3—5426no—
172D3C4×C3⋊(C3⋊D3)12yesdirect product factor
173C22—31no—
174C22—271no—
175C22—814no—
176C22—811no—
177C4—31no—
178C4—271no—
179C4—811no—
180C4D3×C3⋊(C3⋊D3)11yescharacteristic, direct product factor
181C3—213no—
182C3C4×D3×C3⋊D3113yescomplemented
183C3C2×C4×C3⋊(C3⋊D3)11yescharacteristic, complemented
184C2—32no—
185C2—272no—
186C2—812no—
187C2C2×D3×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, hypercenterC4D3×C3⋊(C3⋊D3)
commutator, hypocenterC34C22×C4
FrattiniC2C2×D3×C3⋊(C3⋊D3)
FittingC34×C4C22
socle, cosocleC2×C34C23
prime radicalC2×D3×C3⋊(C3⋊D3)C2
LCSG ▷ C34
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C4×C3⋊(C3⋊(C3⋊D3))C2semidirect
3. C3×C4×C3⋊(C3⋊D3)C2semidirect
4. C33×C4×D3C2semidirect
5. #10620144C2semidirect
6. Dic3×C3⋊(C3⋊D3)C2semidirect
7. D3×C3⋊(C3⋊Dic3)C2semidirect
13. D3×C3⋊(C3⋊D3)C4direct (N × H)
14. C34×C4C22semidirect
15. C3⋊(C3⋊(C3⋊Dic3))C22semidirect
16. C3×C3⋊(C3⋊Dic3)C22semidirect
17. C33×Dic3C22semidirect
27. C4×C3⋊(C3⋊D3)D3direct (N × H)
29. C32×C4×D3D3semidirect
34. C3⋊(C3⋊(C3⋊D3))C2×C4semidirect
35. C3×C3⋊(C3⋊D3)C2×C4semidirect
36. C33×D3C2×C4semidirect
50. C33×C4C2×D3semidirect
51. C33×C4C2×D3semidirect
55. C3⋊(C3⋊Dic3)C2×D3semidirect
57. C32×Dic3C2×D3semidirect
58. C34C22×C4semidirect
70. C3×C4×D3C3⋊D3semidirect
81. C3⋊(C3⋊D3)C4×D3direct (N × H)
83. C32×D3C4×D3semidirect
97. C32×C4D32semidirect
98. C32×C4C2×C3⋊D3semidirect
104. C3×Dic3C2×C3⋊D3semidirect
106. C33C2×C4×D3semidirect
107. C33C2×C4×D3semidirect
122. C4×D3C3⋊(C3⋊D3)direct (N × H)
132. C3×D3C4×C3⋊D3semidirect
146. C3×C4D3×C3⋊D3semidirect
147. C3×C4C2×C3⋊(C3⋊D3)semidirect
152. Dic3C2×C3⋊(C3⋊D3)semidirect
154. C32C4×D32semidirect
155. C32C2×C4×C3⋊D3semidirect
172. D3C4×C3⋊(C3⋊D3)direct (N × H)
180. C4D3×C3⋊(C3⋊D3)direct (N × H)
182. C3C4×D3×C3⋊D3semidirect
183. 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
#106201442
Dic3×C3⋊(C3⋊D3)2
D3×C3⋊(C3⋊Dic3)2
C2×C4×C3⋊(C3⋊D3)3
C4×D3×C3⋊D33