Finite Group Databasebeta

← #13163840

Subgroups — #13163840

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
2C33×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#11723981C211yesmaximal, characteristic, complemented
4#11723973C211yesmaximal, characteristic, complemented
5#11722207C211yesmaximal, characteristic, complemented
6#11722195C211yesmaximal, characteristic, complemented
7#11722193C211yesmaximal, characteristic, complemented
8#10592897—313nomaximal
9C33×#41C2211yescharacteristic
10C2×C33×D4C2211yescharacteristic, complemented
11C2×#6312—22no—
12#89080—21no—
13D4×C3⋊(C3⋊D3)—22no—
14D4×C3⋊(C3⋊D3)—21no—
15#89076—21no—
16#89076C2211yescharacteristic
17C2×C4⋊(C3⋊(C3⋊D3))—21no—
18C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
19C33×SD16—21no—
20C33×D8—21no—
21C33×M16C2211yescharacteristic, complemented
22#88861—21no—
23#88860—21no—
24C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
25#88855C2211yescharacteristic, complemented
26#64366—313no—
27C32×AGL(1,ℤ/8)D3113yescomplemented
28#64149—313no—
29#64141—313no—
30#63664—313no—
31#63652—313no—
32#63650—313no—
33C23×C33—21no—
34C22×C3⋊(C3⋊D3)—22no—
35C22×C3⋊(C3⋊D3)D412yes—
36C33×Q8D411yescharacteristic, complemented
37C33×D4—21no—
38C33×D4—21no—
39C33×D4—21no—
40C33×D4D411yescharacteristic, complemented
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43#6312—22no—
44#6312—21no—
45#6312—44no—
46C2×C3⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊(C3⋊D3))—22no—
48C4⋊(C3⋊(C3⋊D3))—21no—
49C4⋊(C3⋊(C3⋊D3))—21no—
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#5198—913no—
57C32×#41C2×D3113yes—
58C2×C32×D4C2×D3113yescomplemented
59C2×#638—626no—
60#3832—613no—
61D4×C3⋊D3—626no—
62D4×C3⋊D3—613no—
63#3828—313no—
64#3828—613no—
65C2×C4⋊(C3⋊D3)—313no—
66C2×C4⋊(C3⋊D3)—613no—
67C32×SD16—213no—
68C32×D8—213no—
69C32×M16C2×D3113yescomplemented
70#3754—613no—
71#3753—613no—
72C4×C3⋊Dic3—313no—
73#3748—313no—
74C22×C33—21no—
75C22×C33—42no—
76C22×C33—41no—
77C22×C33C2×D411yescharacteristic
78C2×C3⋊(C3⋊D3)—22no—
79C2×C3⋊(C3⋊D3)—21no—
80C2×C3⋊(C3⋊D3)—42no—
81C2×C3⋊(C3⋊D3)—82no—
82C33×C4—21no—
83C33×C4C2×D411yescharacteristic
84C33×C4C2×D411yescharacteristic
85C3⋊(C3⋊Dic3)—22no—
86C3⋊(C3⋊Dic3)—21no—
87#1009—913no—
88C3×AGL(1,ℤ/8)C3⋊D3113yescomplemented
89#940—913no—
90#932—913no—
91#837—913no—
92#825—913no—
93#823—913no—
94C23×C32—213no—
95C22×C3⋊D3—326no—
96C22×C3⋊D3—626no—
97C32×Q8#67113yescomplemented
98C32×D4—213no—
99C32×D4—213no—
100C32×D4—213no—
101C32×D4#67113yescomplemented
102C2×C32×C4—213no—
103C2×C32×C4C22×D3113yes—
104#638—626no—
105#638—613no—
106#638—1252no—
107C2×C3⋊Dic3—613no—
108C4⋊(C3⋊D3)—313no—
109C4⋊(C3⋊D3)—626no—
110C4⋊(C3⋊D3)—613no—
111C4⋊(C3⋊D3)—613no—
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—81no—
121C2×C33#12011yescharacteristic
122C3⋊(C3⋊D3)—42no—
123C3⋊(C3⋊D3)—41no—
124C3×#41C2×C3⋊D3113yes—
125C2×C3×D4C2×C3⋊D3113yescomplemented
126C2×#67—1826no—
127#239—1813no—
128D3×D4—1826no—
129D3×D4—1813no—
130#235—913no—
131#235—1813no—
132C2×D12—913no—
133C2×D12—1813no—
134C3×SD16—213no—
135C3×D8—213no—
136C3×M16C2×C3⋊D3113yescomplemented
137#214—1813no—
138#213—1813no—
139C4×Dic3—913no—
140#208—913no—
141C22×C32—213no—
142C22×C32—426no—
143C22×C32—413no—
144C22×C32D3×D4113yes—
145C2×C3⋊D3—613no—
146C2×C3⋊D3—626no—
147C2×C3⋊D3—1226no—
148C2×C3⋊D3—2426no—
149C32×C4—213no—
150C32×C4D3×D4113yes—
151C32×C4C2×#67113yes—
152C3⋊Dic3—626no—
153C3⋊Dic3—613no—
154H(5,𝔽2)—271no—
155AGL(1,ℤ/8)—271no—
156AGL(1,ℤ/8)C3⋊(C3⋊D3)11yescharacteristic, complemented
157#127—271no—
158#104—271no—
159#104—271no—
160#100—271no—
161C33#45311yescharacteristic, complemented, Sylow
162C23×C3—213no—
163C22×D3—926no—
164C22×D3—1826no—
165C3×Q8#638113yescomplemented
166C3×D4—213no—
167C3×D4—213no—
168C3×D4—213no—
169C3×D4#638113yescomplemented
170C2×C3×C4—213no—
171C2×C3×C4C22×C3⋊D3113yes—
172#67—1826no—
173#67—1813no—
174#67—3652no—
175C2×Dic3—1813no—
176D12—913no—
177D12—1826no—
178D12—1813no—
179D12—1813no—
180C4×D3—1826no—
181C4×D3—1813no—
182Dic6—913no—
183C3×C8—213no—
184C3⋊C8—1813no—
185C2×C32—213no—
186C2×C32—413no—
187C2×C32—813no—
188C2×C32#937113yes—
189C3⋊D3—1226no—
190C3⋊D3—1213no—
191#41—271no—
192#41—541no—
193#41—541no—
194#41C2×C3⋊(C3⋊D3)11yescharacteristic
195C2×D4—271no—
196C2×D4—541no—
197C2×D4—542no—
198C2×D4—541no—
199C2×D4—542no—
200C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
201SD16—21no—
202SD16—541no—
203D8—21no—
204D8—541no—
205M16—271no—
206M16C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
207C42—271no—
208C22×C3—213no—
209C22×C3—426no—
210C22×C3—413no—
211C22×C3D4×C3⋊D3113yes—
212C2×D3—1813no—
213C2×D3—1826no—
214C2×D3—3626no—
215C2×D3—7226no—
216C3×C4—213no—
217C3×C4D4×C3⋊D3113yes—
218C3×C4C2×#638113yes—
219Dic3—1826no—
220Dic3—1813no—
221C32#5198113yescomplemented
222C23—21no—
223C23—272no—
224C23—542no—
225Q8—271no—
226Q8#631211yescharacteristic, complemented
227D4—21no—
228D4—21no—
229D4—21no—
230D4—271no—
231D4—542no—
232D4—542no—
233D4—541no—
234D4—541no—
235D4—541no—
236D4—1084no—
237D4#631211yescharacteristic, complemented
238C2×C4—21no—
239C2×C4—542no—
240C2×C4—541no—
241C2×C4—541no—
242C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
243C8—21no—
244C8—541no—
245C2×C3—213no—
246C2×C3—413no—
247C2×C3—813no—
248C2×C3#64146113yes—
249D3—3626no—
250D3—3613no—
251C22—21no—
252C22—42no—
253C22—41no—
254C22—541no—
255C22—542no—
256C22—1082no—
257C22—2162no—
258C22D4×C3⋊(C3⋊D3)11yescharacteristic
259C4—21no—
260C4—542no—
261C4—541no—
262C4D4×C3⋊(C3⋊D3)11yescharacteristic
263C4C2×#631211yescharacteristic
264C3#10592897113yescomplemented
265C2—21no—
266C2—41no—
267C2—81no—
268C2—1082no—
269C2—1081no—
270C2#1172397811yescharacteristic
subgroupG/H
centerC2#11723978
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×AGL(1,ℤ/8)C2
socleC2×C33#120
hypercenterAGL(1,ℤ/8)C3⋊(C3⋊D3)
hypocenterC33#453
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #11724684C2semidirect
2. C33×AGL(1,ℤ/8)C2semidirect
3. #11723981C2semidirect
4. #11723973C2semidirect
5. #11722207C2semidirect
6. #11722195C2semidirect
7. #11722193C2semidirect
10. C2×C33×D4C22semidirect
21. C33×M16C22semidirect
24. C4×C3⋊(C3⋊Dic3)C22semidirect
25. #88855C22semidirect
27. C32×AGL(1,ℤ/8)D3semidirect
36. C33×Q8D4semidirect
40. C33×D4D4semidirect
50. C4⋊(C3⋊(C3⋊D3))D4semidirect
53. #6308D4semidirect
58. C2×C32×D4C2×D3semidirect
69. C32×M16C2×D3semidirect
88. C3×AGL(1,ℤ/8)C3⋊D3semidirect
97. C32×Q8#67semidirect
101. C32×D4#67semidirect
125. C2×C3×D4C2×C3⋊D3semidirect
136. C3×M16C2×C3⋊D3semidirect
156. AGL(1,ℤ/8)C3⋊(C3⋊D3)semidirect
161. C33#453semidirect
165. C3×Q8#638semidirect
169. C3×D4#638semidirect
200. C2×D4C2×C3⋊(C3⋊D3)semidirect
206. M16C2×C3⋊(C3⋊D3)semidirect
221. C32#5198semidirect
226. Q8#6312semidirect
237. D4#6312semidirect
264. C3#10592897semidirect

Maximal subgroups

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

subgroupindex
#117246842
C33×AGL(1,ℤ/8)2
#117239812
#117239732
#117222072
#117221952
#117221932
#105928973