Finite Group Databasebeta

← #46847966

Subgroups — #46847966

Order 34688.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22576188C212yesmaximal, complemented
2#31774552C211yesmaximal, characteristic, complemented
3D4×D1084C211yesmaximal, characteristic, complemented
4#31774270C211yesmaximal, characteristic, complemented
5#31774143C212yesmaximal, complemented
6#31774103C211yesmaximal, characteristic, complemented
7#31774093C211yesmaximal, characteristic, complemented
8#31774050C212yesmaximal, complemented
9#31774044C212yesmaximal, complemented
10C4×D2168C211yesmaximal, characteristic, complemented
11#436×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2212yescomplemented
13C22×D1084C2212yes—
14C2×D4×D271—21no—
15#22576188—28no—
16C2×#16864643C2212yes—
17C2×#16864643C2212yescomplemented
18C2×#16864643C2212yescomplemented
19C2×D2168C2211yescharacteristic
20C2×D2168C2212yescomplemented
21#22576160C2211yescharacteristic, complemented
22#22576157C2211yescharacteristic
23#22576142—22no—
24#22576139C2212yescomplemented
25#22576135—21no—
26#22576134C2212yes—
27#22576131—21no—
28#22576130—22no—
29C4×D1084C2211yescharacteristic
30C4×D1084C2211yescharacteristic, complemented
31C4⋊C4×D271—21no—
32#22576094C2211yescharacteristic, complemented
33#22576085C2212yescomplemented
34#22576082C2211yescharacteristic
35#22576082C2211yescharacteristic, complemented
36#22576082C2212yescomplemented
37C2×M16×C271C2212yescomplemented
38C4×D4×C271C2211yescharacteristic
39C4⋊C8×C271C2211yescharacteristic, complemented
40C22⋊C8×C271C2212yescomplemented
41C4×C8×C271C2211yescharacteristic, complemented
42C23×D271—22no—
43#16864654—24no—
44#16864654—42no—
45C2×Dic542C2312yes—
46C2×D1084—24no—
47C2×D1084—22no—
48C2×D1084—21no—
49C2×D1084—21no—
50C2×D1084—42no—
51C2×D1084C2312yes—
52C2×D1084C2311yescharacteristic
53C2×D1084C2311yescharacteristic
54C2×C22⋊D271—22no—
55C2×C22⋊D271—22no—
56D4×D271—44no—
57C2×C4×D271—21no—
58C2×C4×D271—21no—
59C2×C4×D271—22no—
60#16864643—24no—
61#16864643—24no—
62#16864643—42no—
63D2168—24no—
64D2168D414yescomplemented
65C4⋊Dic271—21no—
66C4⋊Dic271—21no—
67C4⋊Dic271C2311yescharacteristic
68C4⋊Dic271C2311yescharacteristic
69Dic271⋊C4—22no—
70C22⋊Dic271—22no—
71(C2×D271)⋊C4—21no—
72(C2×D271)⋊C4—22no—
73(C2×D271)⋊C4—21no—
74C2×D4×C271D411yescharacteristic
75C22×C4×C271C2312yes—
76M16×C271—24no—
77C2×C8×C271C2312yes—
78C2×C8×C271C2311yescharacteristic
79C2×C8×C271C2311yescharacteristic
80C4⋊C4×C271D411yescharacteristic
81C22⋊C4×C271D412yes—
82C42×C271C2311yescharacteristic
83C22×D271—21no—
84C22×D271—21no—
85C22×D271—22no—
86C22×D271—44no—
87C22×D271—44no—
88C22×D271—82no—
89Dic542—22no—
90Dic542—22no—
91D1084—22no—
92D1084—21no—
93D1084—21no—
94D1084—22no—
95D1084—44no—
96D1084—42no—
97D1084—81no—
98D1084C2×D412yes—
99D1084C2×D412yescomplemented
100C22⋊D271—44no—
101C22⋊D271—44no—
102C2×Dic271—21no—
103C2×Dic271—21no—
104C2×Dic271—22no—
105C4×D271—44no—
106C4×D271—42no—
107C4×D271—42no—
108C23×C271C2×D412yes—
109D4×C271—22no—
110C2×C4×C271—22no—
111C2×C4×C271—22no—
112C2×C4×C271C2×D411yescharacteristic
113C2×C4×C271C2×D412yes—
114C2×C4×C271C2×D411yescharacteristic
115C2×C4×C271C2411yescharacteristic
116C8×C271—22no—
117C8×C271—21no—
118C8×C271C2×D412yescomplemented
119C22×C271—22no—
120C22×C271—22no—
121C22×C271—22no—
122C22×C271C22×D411yescharacteristic
123C2×D271—22no—
124C2×D271—22no—
125C2×D271—41no—
126C2×D271—41no—
127C2×D271—41no—
128C2×D271—41no—
129C2×D271—42no—
130C2×D271—42no—
131C2×D271—42no—
132C2×D271—88no—
133C4×C271—22no—
134C4×C271—21no—
135C4×C271AGL(1,ℤ/8)12yescomplemented
136C4×C271C22×D411yescharacteristic
137C4×C271H(5,𝔽2)11yescharacteristic
138Dic271—42no—
139Dic271—41no—
140Dic271—41no—
141C2×C271—42no—
142C2×C271D4211yescharacteristic
143C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C271#57711yescharacteristic
145D271—42no—
146D271—42no—
147D271—82no—
148C271#327211yescharacteristic, complemented, Sylow
149#3272—2711nomaximal, Sylow
150C2×AGL(1,ℤ/8)—2712no—
151#547—2711no—
152D42—2711no—
153#492—2711no—
154#468—2712no—
155#461—2711no—
156#459—2711no—
157#450—2712no—
158#449—2712no—
159C4×D8—2711no—
160#436D27111yescharacteristic, complemented
161C2×#41—2712no—
162C22×D4—2712no—
163C22×D4—5421no—
164AGL(1,ℤ/8)—5428no—
165C2×SD16—2712no—
166C2×SD16—2712no—
167C2×SD16—2712no—
168C2×D8—2711no—
169C2×D8—2712no—
170C2×M16C2×D27112yescomplemented
171C4⋊Q8—2711no—
172#127—2711no—
173#123—5422no—
174C22⋊Q8—2712no—
175#121—2712no—
176#121—5421no—
177#121—5421no—
178#120—5422no—
179C4×D4—2711no—
180C4×D4—2711no—
181C4×D4C2×D27111yescharacteristic
182C2×C4⋊C4—5421no—
183#107—2711no—
184C4⋊C8C2×D27111yescharacteristic, complemented
185#103—2712no—
186#102—2711no—
187#102—2711no—
188#102—2712no—
189C22⋊C8C2×D27112yescomplemented
190C4×C8C2×D27111yescharacteristic, complemented
191C24—5422no—
192#41—5424no—
193#41—10842no—
194C2×Q8—2712no—
195C2×D4—2711no—
196C2×D4—2711no—
197C2×D4—2712no—
198C2×D4—5424no—
199C2×D4—5422no—
200C2×D4—5422no—
201C2×D4—5422no—
202C2×D4—5421no—
203C2×D4—5421no—
204C2×D4—10842no—
205C2×D4—10844no—
206C2×D4D108411yescharacteristic
207C22×C4—5421no—
208C22×C4—5422no—
209C22×C4—5421no—
210C22×C4C22×D27112yes—
211SD16—5424no—
212SD16—5424no—
213SD16—10842no—
214D8—2714no—
215D8—5424no—
216M16—24no—
217C2×C8C22×D27111yescharacteristic
218C2×C8C22×D27112yes—
219C2×C8C22×D27111yescharacteristic
220C4⋊C4—2711no—
221C4⋊C4—2711no—
222C4⋊C4—5421no—
223C4⋊C4—5422no—
224C4⋊C4—5421no—
225C4⋊C4D108411yescharacteristic
226C22⋊C4—5422no—
227C22⋊C4—5421no—
228C22⋊C4—5421no—
229C22⋊C4—5422no—
230C22⋊C4D108412yes—
231C42C22×D27111yescharacteristic
232C23—5421no—
233C23—5421no—
234C23—5422no—
235C23—10844no—
236C23—10844no—
237C23—21682no—
238C23C2×D108412yes—
239Q8—5422no—
240Q8—5422no—
241D4—22no—
242D4—2712no—
243D4—2712no—
244D4—5422no—
245D4—5421no—
246D4—5421no—
247D4—5422no—
248D4—10844no—
249D4—10844no—
250D4—10844no—
251D4—10842no—
252D4—21681no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—5421no—
256C2×C4—5421no—
257C2×C4—5422no—
258C2×C4—10844no—
259C2×C4—10842no—
260C2×C4—10842no—
261C2×C4C2×D108411yescharacteristic
262C2×C4C2×D108412yes—
263C2×C4C2×D108411yescharacteristic
264C2×C4C23×D27111yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D27112yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—5422no—
272C22—5422no—
273C22—10841no—
274C22—10841no—
275C22—10841no—
276C22—10841no—
277C22—10842no—
278C22—10842no—
279C22—10842no—
280C22—21688no—
281C22C22×D108411yescharacteristic
282C4—22no—
283C4—21no—
284C4—10842no—
285C4—10841no—
286C4—10841no—
287C4#2257618812yescomplemented
288C4C2×D4×D27111yescharacteristic
289C4#2257620611yescharacteristic
290C2—42no—
291C2—10842no—
292C2—10842no—
293C2—21682no—
294C2D4×D108411yescharacteristic
295C2C2×#2257618811yescharacteristic
296C2#3177470911yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#436×C271C2
socleC22×C271C22×D4
hypercenter#436D271
hypocenterC271#3272
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. #31774552C2semidirect
3. D4×D1084C2semidirect
4. #31774270C2semidirect
5. #31774143C2semidirect
6. #31774103C2semidirect
7. #31774093C2semidirect
8. #31774050C2semidirect
9. #31774044C2semidirect
10. C4×D2168C2semidirect
11. #436×C271C2semidirect
12. C2×#16864654C22semidirect
17. C2×#16864643C22semidirect
18. C2×#16864643C22semidirect
20. C2×D2168C22semidirect
21. #22576160C22semidirect
24. #22576139C22semidirect
30. C4×D1084C22semidirect
32. #22576094C22semidirect
33. #22576085C22semidirect
35. #22576082C22semidirect
36. #22576082C22semidirect
37. C2×M16×C271C22semidirect
39. C4⋊C8×C271C22semidirect
40. C22⋊C8×C271C22semidirect
41. C4×C8×C271C22semidirect
64. D2168D4semidirect
99. D1084C2×D4semidirect
118. C8×C271C2×D4semidirect
135. C4×C271AGL(1,ℤ/8)semidirect
148. C271#3272semidirect
160. #436D271semidirect
170. C2×M16C2×D271semidirect
184. C4⋊C8C2×D271semidirect
189. C22⋊C8C2×D271semidirect
190. C4×C8C2×D271semidirect
267. C8D4×D271semidirect
287. C4#22576188semidirect

Maximal subgroups

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

subgroupindex
C2×#225761882
#317745522
D4×D10842
#317742702
#317741432
#317741032
#317740932
#317740502
#317740442
C4×D21682
#436×C2712
#3272271