Finite Group Databasebeta

← #13163462

Subgroups — #13163462

Order 1728.

2352 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 2352 conjugacy classes of subgroups — too many to draw. Only its 351 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
1#11724684C211yesmaximal, characteristic, complemented
2#11723990C211yesmaximal, characteristic, complemented
3#11723949C211yesmaximal, characteristic, complemented
4#11723915C211yesmaximal, characteristic, complemented
5C33×#104C211yesmaximal, characteristic, complemented
6#11722193C211yesmaximal, characteristic, complemented
7#11722175C211yesmaximal, characteristic, complemented
8#10592521—313nomaximal
9C33×#41C2211yescharacteristic
10#89080—21no—
11D4×C3⋊(C3⋊D3)—22no—
12D4×C3⋊(C3⋊D3)—21no—
13#89076—21no—
14#89076C2211yescharacteristic
15C2×C4⋊(C3⋊(C3⋊D3))—21no—
16C2×C4⋊(C3⋊(C3⋊D3))—22no—
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
19C33×M16C2211yescharacteristic, complemented
20C33×C42C2211yescharacteristic, complemented
21#88862—21no—
22#88860—21no—
23#88855C2211yescharacteristic, complemented
24#88852—21no—
25#88851—21no—
26#64366—313no—
27#64158—313no—
28#64117—313no—
29#64083—313no—
30C32×#104D3113yescomplemented
31#63650—313no—
32#63632—313no—
33C22×C3⋊(C3⋊D3)—22no—
34C22×C3⋊(C3⋊D3)—21no—
35C22×C3⋊(C3⋊D3)D412yes—
36C33×Q8D411yescharacteristic, complemented
37C33×D4—21no—
38C33×D4D411yescharacteristic, complemented
39C2×C33×C4—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42#6312—22no—
43#6312—21no—
44C4⋊(C3⋊(C3⋊D3))—22no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))—21no—
49C4⋊(C3⋊(C3⋊D3))—44no—
50C4⋊(C3⋊(C3⋊D3))D411yescharacteristic, complemented
51C4×C3⋊(C3⋊D3)—22no—
52C4×C3⋊(C3⋊D3)—21no—
53#6308D411yescharacteristic, complemented
54C33×C8—21no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56#4822—913no—
57C32×#41C2×D3113yes—
58#3832—613no—
59D4×C3⋊D3—626no—
60D4×C3⋊D3—613no—
61#3828—313no—
62#3828—613no—
63C2×C4⋊(C3⋊D3)—313no—
64C2×C4⋊(C3⋊D3)—313no—
65C2×C4⋊(C3⋊D3)—613no—
66C2×C4⋊(C3⋊D3)—626no—
67C32×M16C2×D3113yescomplemented
68C32×C42C2×D3113yescomplemented
69#3755—613no—
70#3753—613no—
71#3748—313no—
72#3745—613no—
73#3744—613no—
74C22×C33—21no—
75C22×C33C2×D411yescharacteristic
76C2×C3⋊(C3⋊D3)—22no—
77C2×C3⋊(C3⋊D3)—21no—
78C2×C3⋊(C3⋊D3)—42no—
79C2×C3⋊(C3⋊D3)—42no—
80C2×C3⋊(C3⋊D3)—41no—
81C2×C3⋊(C3⋊D3)—82no—
82C33×C4—22no—
83C33×C4—21no—
84C33×C4C2×D411yescharacteristic
85C33×C4C2×D411yescharacteristic
86C3⋊(C3⋊Dic3)—21no—
87#1009—913no—
88#949—913no—
89#908—913no—
90#874—913no—
91C3×#104C3⋊D3113yescomplemented
92#823—913no—
93#805—913no—
94C22×C3⋊D3—326no—
95C22×C3⋊D3—626no—
96C22×C3⋊D3—613no—
97C32×Q8D12113yescomplemented
98C32×D4—213no—
99C32×D4D12113yescomplemented
100C2×C32×C4—213no—
101C2×C32×C4—213no—
102C2×C32×C4C22×D3113yes—
103#638—626no—
104#638—613no—
105C4⋊(C3⋊D3)—313no—
106C4⋊(C3⋊D3)—626no—
107C4⋊(C3⋊D3)—613no—
108C4⋊(C3⋊D3)—613no—
109C4⋊(C3⋊D3)—613no—
110C4⋊(C3⋊D3)—613no—
111C4⋊(C3⋊D3)—1252no—
112C4×C3⋊D3—626no—
113C4×C3⋊D3—613no—
114#634—313no—
115C32×C8—213no—
116C3⋊(C3⋊C8)—613no—
117#453—271noSylow
118C2×C33—21no—
119C2×C33—41no—
120C2×C33#12011yescharacteristic
121C3⋊(C3⋊D3)—42no—
122C3⋊(C3⋊D3)—41no—
123C3⋊(C3⋊D3)—81no—
124C3×#41C2×C3⋊D3113yes—
125#239—1813no—
126D3×D4—1826no—
127D3×D4—1813no—
128#235—913no—
129#235—1813no—
130C2×D12—913no—
131C2×D12—913no—
132C2×D12—1813no—
133C2×D12—1826no—
134C3×M16C2×C3⋊D3113yescomplemented
135C3×C42C2×C3⋊D3113yescomplemented
136#215—1813no—
137#213—1813no—
138#208—913no—
139D24—1813no—
140#204—1813no—
141C22×C32—213no—
142C22×C32D3×D4113yes—
143C2×C3⋊D3—613no—
144C2×C3⋊D3—626no—
145C2×C3⋊D3—1226no—
146C2×C3⋊D3—1226no—
147C2×C3⋊D3—1213no—
148C2×C3⋊D3—2426no—
149C32×C4—226no—
150C32×C4—213no—
151C32×C4C2×D12113yes—
152C32×C4D3×D4113yes—
153C3⋊Dic3—613no—
154H(5,𝔽2)—271no—
155AGL(1,ℤ/8)—271no—
156AGL(1,ℤ/8)—271no—
157#127—271no—
158#104—271no—
159#104C3⋊(C3⋊D3)11yescharacteristic, complemented
160#100—271no—
161C33#45311yescharacteristic, complemented, Sylow
162C22×D3—926no—
163C22×D3—1826no—
164C22×D3—1813no—
165C3×Q8C4⋊(C3⋊D3)113yescomplemented
166C3×D4—213no—
167C3×D4C4⋊(C3⋊D3)113yescomplemented
168C2×C3×C4—213no—
169C2×C3×C4—213no—
170C2×C3×C4C22×C3⋊D3113yes—
171#67—1826no—
172#67—1813no—
173D12—913no—
174D12—1826no—
175D12—1813no—
176D12—1813no—
177D12—1813no—
178D12—1813no—
179D12—3652no—
180C4×D3—1826no—
181C4×D3—1813no—
182Dic6—913no—
183C3×C8—213no—
184C3⋊C8—1813no—
185C2×C32—213no—
186C2×C32—413no—
187C2×C32#882113yes—
188C3⋊D3—1226no—
189C3⋊D3—1213no—
190C3⋊D3—2413no—
191#41—271no—
192#41—541no—
193#41—541no—
194#41C2×C3⋊(C3⋊D3)11yescharacteristic
195C2×D4—271no—
196C2×D4—271no—
197C2×D4—541no—
198C2×D4—542no—
199C2×D4—542no—
200C2×D4—541no—
201SD16—541no—
202SD16—541no—
203D8—541no—
204D8—541no—
205M16—271no—
206M16C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
207C42C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
208C22×C3—213no—
209C22×C3D4×C3⋊D3113yes—
210C2×D3—1813no—
211C2×D3—1826no—
212C2×D3—3626no—
213C2×D3—3626no—
214C2×D3—3613no—
215C2×D3—7226no—
216C3×C4—226no—
217C3×C4—213no—
218C3×C4C2×C4⋊(C3⋊D3)113yes—
219C3×C4D4×C3⋊D3113yes—
220Dic3—1813no—
221C32#4822113yescomplemented
222C23—272no—
223C23—542no—
224C23—541no—
225Q8—271no—
226Q8C4⋊(C3⋊(C3⋊D3))11yescharacteristic, complemented
227D4—21no—
228D4—271no—
229D4—542no—
230D4—542no—
231D4—541no—
232D4—541no—
233D4—541no—
234D4—541no—
235D4—541no—
236D4—1084no—
237D4C4⋊(C3⋊(C3⋊D3))11yescharacteristic, complemented
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—542no—
241C2×C4—541no—
242C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
243C8—21no—
244C8—541no—
245C2×C3—213no—
246C2×C3—413no—
247C2×C3#64091113yes—
248D3—3626no—
249D3—3613no—
250D3—7213no—
251C22—21no—
252C22—541no—
253C22—542no—
254C22—1082no—
255C22—1082no—
256C22—1081no—
257C22—2162no—
258C22D4×C3⋊(C3⋊D3)11yescharacteristic
259C4—22no—
260C4—21no—
261C4—541no—
262C4C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic
263C4D4×C3⋊(C3⋊D3)11yescharacteristic
264C3#10592521113yescomplemented
265C2—21no—
266C2—41no—
267C2—1082no—
268C2—1081no—
269C2—2161no—
270C2#1172392311yescharacteristic
subgroupG/H
centerC2#11723923
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#104C2
socleC2×C33#120
hypercenter#104C3⋊(C3⋊D3)
hypocenterC33#453
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
1. #11724684C2semidirect
2. #11723990C2semidirect
3. #11723949C2semidirect
4. #11723915C2semidirect
5. C33×#104C2semidirect
6. #11722193C2semidirect
7. #11722175C2semidirect
18. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C33×M16C22semidirect
20. C33×C42C22semidirect
23. #88855C22semidirect
30. C32×#104D3semidirect
36. C33×Q8D4semidirect
38. C33×D4D4semidirect
50. C4⋊(C3⋊(C3⋊D3))D4semidirect
53. #6308D4semidirect
67. C32×M16C2×D3semidirect
68. C32×C42C2×D3semidirect
91. C3×#104C3⋊D3semidirect
97. C32×Q8D12semidirect
99. C32×D4D12semidirect
134. C3×M16C2×C3⋊D3semidirect
135. C3×C42C2×C3⋊D3semidirect
159. #104C3⋊(C3⋊D3)semidirect
161. C33#453semidirect
165. C3×Q8C4⋊(C3⋊D3)semidirect
167. C3×D4C4⋊(C3⋊D3)semidirect
206. M16C2×C3⋊(C3⋊D3)semidirect
207. C42C2×C3⋊(C3⋊D3)semidirect
221. C32#4822semidirect
226. Q8C4⋊(C3⋊(C3⋊D3))semidirect
237. D4C4⋊(C3⋊(C3⋊D3))semidirect
264. C3#10592521semidirect

Maximal subgroups

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

subgroupindex
#117246842
#117239902
#117239492
#117239152
C33×#1042
#117221932
#117221752
#105925213