Finite Group Databasebeta

← #13049685

Subgroups — #13049685

Order 1296.

2760 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 2760 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
1C22×C3⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
2C2×D3×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#6312C211yesmaximal, characteristic, complemented
4C33×#67C211yesmaximal, characteristic, complemented
5#10620147C211yesmaximal, characteristic, complemented
6#10620146C211yesmaximal, characteristic, complemented
7#10620144C211yesmaximal, characteristic, complemented
8D4×C3⋊(C3⋊D3)—31nomaximal
9#89039—313nomaximal
10C22×C34C2211yescharacteristic, complemented
11C2×C3⋊(C3⋊(C3⋊D3))—22no—
12C2×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
14C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
15C2×C33×D3C2211yescharacteristic, complemented
16D3×C3⋊(C3⋊D3)—22no—
17C3×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C33×Dic3C2211yescharacteristic, complemented
19C22×C3⋊(C3⋊D3)—31no—
20C22×C3⋊(C3⋊D3)—313no—
21C22×C3⋊(C3⋊D3)—31no—
22C22×C3⋊(C3⋊D3)—613no—
23C2×D3×C3⋊D3—313no—
24C33×D4—31no—
25#6312—31no—
26#6312D311yescharacteristic, complemented
27C4⋊(C3⋊(C3⋊D3))—31no—
28C4×C3⋊(C3⋊D3)—31no—
29C3×#638—313no—
30C32×#67D3113yescomplemented
31#6292—313no—
32#6291—313no—
33#6289—313no—
34C2×C34—21no—
35C2×C34C2311yescharacteristic
36C3⋊(C3⋊(C3⋊D3))—21no—
37C3⋊(C3⋊(C3⋊D3))D412yescomplemented
38C3×C3⋊(C3⋊D3)—21no—
39C33×D3—21no—
40D4×C3⋊D3—913no—
41#3811—913no—
42C22×C33—213no—
43C22×C33—31no—
44C22×C33C2×D3113yescomplemented
45C22×C33C2×D311yescharacteristic, complemented
46C2×C3⋊(C3⋊D3)—31no—
47C2×C3⋊(C3⋊D3)—313no—
48C2×C3⋊(C3⋊D3)—31no—
49C2×C3⋊(C3⋊D3)—313no—
50C2×C3⋊(C3⋊D3)—62no—
51C2×C3⋊(C3⋊D3)—62no—
52C2×C3⋊(C3⋊D3)—626no—
53C2×C3⋊(C3⋊D3)—613no—
54C2×C3⋊(C3⋊D3)—613no—
55C2×C3⋊(C3⋊D3)—652no—
56C2×C3⋊(C3⋊D3)C2×D311yescharacteristic, complemented
57C2×C3×C3⋊D3—313no—
58C2×C32×D3C2×D3113yescomplemented
59D3×C3⋊D3—626no—
60C33×C4—31no—
61C3⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
62C3×C3⋊Dic3—313no—
63C32×Dic3C2×D3113yescomplemented
64C34C2×D411yescharacteristic, complemented, Sylow
65C22×C3⋊D3—913no—
66C22×C3⋊D3—913no—
67C22×C3⋊D3—913no—
68C22×C3⋊D3—1852no—
69C2×D32—913no—
70C32×D4—313no—
71#638—313no—
72#638—913no—
73C4⋊(C3⋊D3)—913no—
74C4×C3⋊D3—913no—
75C3×#67—913no—
76C3×#67C3⋊D3113yescomplemented
77#626—913no—
78#626—913no—
79#624—913no—
80C2×C33—226no—
81C2×C33—213no—
82C2×C33—21no—
83C2×C33—213no—
84C2×C33—61no—
85C2×C33C22×D3113yes—
86C2×C33C22×D311yescharacteristic
87C3⋊(C3⋊D3)—21no—
88C3⋊(C3⋊D3)—32no—
89C3⋊(C3⋊D3)—326no—
90C3⋊(C3⋊D3)—626no—
91C3⋊(C3⋊D3)—626no—
92C3⋊(C3⋊D3)—613no—
93C3⋊(C3⋊D3)—61no—
94C3×C3⋊D3—613no—
95C32×D3—213no—
96D3×D4—2713no—
97D3×D4—271no—
98C22×C32—252no—
99C22×C32—313no—
100C22×C32D32113yescomplemented
101C22×C32C2×C3⋊D3113yescomplemented
102C2×C3⋊D3—313no—
103C2×C3⋊D3—913no—
104C2×C3⋊D3—913no—
105C2×C3⋊D3—913no—
106C2×C3⋊D3—913no—
107C2×C3⋊D3—1852no—
108C2×C3⋊D3—1852no—
109C2×C3⋊D3—1826no—
110C2×C3⋊D3—1826no—
111C2×C3⋊D3—1826no—
112C2×C3⋊D3—18208no—
113C2×C3×D3—913no—
114C2×C3×D3C2×C3⋊D3113yescomplemented
115D32—1826no—
116C32×C4—313no—
117C3⋊Dic3—313no—
118C3×Dic3—913no—
119C3×Dic3C2×C3⋊D3113yescomplemented
120C33—213no—
121C33D3×D4113yescomplemented
122C33D3×D411yescharacteristic, complemented
123C22×D3—2713no—
124C22×D3—2713no—
125C22×D3—271no—
126C22×D3—271no—
127C22×D3—5413no—
128C3×D4—313no—
129C3×D4—271no—
130#67—913no—
131#67—2713no—
132#67—271no—
133#67C3⋊(C3⋊D3)11yescharacteristic, complemented
134D12—2713no—
135D12—271no—
136C4×D3—2713no—
137C4×D3—271no—
138C2×C32—2104no—
139C2×C32—213no—
140C2×C32—213no—
141C2×C32—252no—
142C2×C32—613no—
143C2×C32C2×D32113yes—
144C2×C32C22×C3⋊D3113yes—
145C3⋊D3—613no—
146C3⋊D3—926no—
147C3⋊D3—926no—
148C3⋊D3—18104no—
149C3⋊D3—18104no—
150C3⋊D3—1813no—
151C3⋊D3—1813no—
152C3×D3—213no—
153C3×D3—1813no—
154C2×D4—811noSylow
155C22×C3—213no—
156C22×C3—313no—
157C22×C3—271no—
158C22×C3D3×C3⋊D3113yescomplemented
159C22×C3C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
160C2×D3—913no—
161C2×D3—2713no—
162C2×D3—271no—
163C2×D3—2713no—
164C2×D3—271no—
165C2×D3—5413no—
166C2×D3—5413no—
167C2×D3—542no—
168C2×D3—5426no—
169C2×D3—5426no—
170C2×D3—542no—
171C2×D3—5452no—
172C2×D3C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
173C3×C4—313no—
174C3×C4—271no—
175Dic3—913no—
176Dic3C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
177C32—252no—
178C32#3811113yescomplemented
179C32D4×C3⋊D3113yescomplemented
180C23—811no—
181C23—811no—
182D4—31no—
183D4—271no—
184D4—811no—
185D4—811no—
186C2×C4—811no—
187C2×C3—226no—
188C2×C3—213no—
189C2×C3—21no—
190C2×C3—213no—
191C2×C3—613no—
192C2×C3—541no—
193C2×C3C2×D3×C3⋊D3113yes—
194C2×C3C22×C3⋊(C3⋊D3)11yescharacteristic
195D3—21no—
196D3—1813no—
197D3—2726no—
198D3—272no—
199D3—5426no—
200D3—5426no—
201D3—5413no—
202D3—541no—
203C22—31no—
204C22—271no—
205C22—811no—
206C22—811no—
207C22—1622no—
208C22—1622no—
209C22D3×C3⋊(C3⋊D3)11yescharacteristic, complemented
210C4—31no—
211C4—271no—
212C3—213no—
213C3#89039113yescomplemented
214C3D4×C3⋊(C3⋊D3)11yescharacteristic, complemented
215C2—21no—
216C2—61no—
217C2—541no—
218C2—812no—
219C2—1621no—
220C2C2×D3×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×C3⋊(C3⋊D3)
commutator, socle, cosocleC2×C34C23
FittingC22×C34C22
hypercenterC22D3×C3⋊(C3⋊D3)
hypocenterC34C2×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊(C3⋊(C3⋊D3))C2semidirect
2. C2×D3×C3⋊(C3⋊D3)C2semidirect
3. C3×#6312C2semidirect
4. C33×#67C2semidirect
5. #10620147C2semidirect
6. #10620146C2semidirect
7. #10620144C2semidirect
10. C22×C34C22semidirect
13. C2×C3⋊(C3⋊(C3⋊D3))C22semidirect
14. C2×C3×C3⋊(C3⋊D3)C22semidirect
15. C2×C33×D3C22semidirect
17. C3×C3⋊(C3⋊Dic3)C22semidirect
18. C33×Dic3C22semidirect
26. #6312D3semidirect
30. C32×#67D3semidirect
37. C3⋊(C3⋊(C3⋊D3))D4semidirect
44. C22×C33C2×D3semidirect
45. C22×C33C2×D3semidirect
56. C2×C3⋊(C3⋊D3)C2×D3semidirect
58. C2×C32×D3C2×D3semidirect
61. C3⋊(C3⋊Dic3)C2×D3semidirect
63. C32×Dic3C2×D3semidirect
64. C34C2×D4semidirect
76. C3×#67C3⋊D3semidirect
100. C22×C32D32semidirect
101. C22×C32C2×C3⋊D3semidirect
114. C2×C3×D3C2×C3⋊D3semidirect
119. C3×Dic3C2×C3⋊D3semidirect
121. C33D3×D4semidirect
122. C33D3×D4semidirect
133. #67C3⋊(C3⋊D3)semidirect
158. C22×C3D3×C3⋊D3semidirect
159. C22×C3C2×C3⋊(C3⋊D3)semidirect
172. C2×D3C2×C3⋊(C3⋊D3)semidirect
176. Dic3C2×C3⋊(C3⋊D3)semidirect
178. C32#3811semidirect
179. C32D4×C3⋊D3semidirect
209. C22D3×C3⋊(C3⋊D3)semidirect
213. C3#89039semidirect
214. 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
C22×C3⋊(C3⋊(C3⋊D3))2
C2×D3×C3⋊(C3⋊D3)2
C3×#63122
C33×#672
#106201472
#106201462
#106201442
D4×C3⋊(C3⋊D3)3
#890393