Finite Group Databasebeta

← #46848194

Subgroups — #46848194

Order 34688.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774581C211yesmaximal, characteristic, complemented
2#31774544C211yesmaximal, characteristic
3#31774250C211yesmaximal, characteristic, complemented
4#31774229C211yesmaximal, characteristic, complemented
5#31774218C211yesmaximal, characteristic, complemented
6#31774196C211yesmaximal, characteristic, complemented
7#31774137C211yesmaximal, characteristic, complemented
8#31774117C211yesmaximal, characteristic, complemented
9#31774057C211yesmaximal, characteristic, complemented
10#31774046C211yesmaximal, characteristic, complemented
11#31773982C211yesmaximal, characteristic, complemented
12C8×C22⋊D271C211yesmaximal, characteristic, complemented
13#31773894C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18C2×#16864644C2211yescharacteristic, complemented
19C2×#16864641C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21#22576153—21no—
22#22576151C2211yescharacteristic, complemented
23#22576150—21no—
24#22576147C2211yescharacteristic
25#22576142—21no—
26#22576142—21no—
27#22576140C2211yescharacteristic
28#22576136C2211yescharacteristic
29#22576131—21no—
30#22576128—21no—
31Q8×Dic271C2211yescharacteristic, complemented
32#22576122C2211yescharacteristic
33C4×C22⋊D271C2211yescharacteristic
34#22576119C2211yescharacteristic, complemented
35#22576118—21no—
36C4⋊C4×D271C2211yescharacteristic
37C22⋊C4×D271—21no—
38#22576093C2211yescharacteristic, complemented
39#22576091C2211yescharacteristic, complemented
40Dic271⋊C8C2211yescharacteristic, complemented
41#22576085C2211yescharacteristic, complemented
42#22576085C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic, complemented
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic, complemented
52C2×SD16×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic, complemented
54C22⋊Q8×C271C2211yescharacteristic, complemented
55#121×C271C2211yescharacteristic
56#106×C271C2211yescharacteristic, complemented
57#103×C271C2211yescharacteristic, complemented
58#102×C271C2211yescharacteristic
59C23×D271—21no—
60#16864654—22no—
61#16864654—41no—
62C2×Dic542C2311yescharacteristic
63C2×D1084C2311yescharacteristic
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271D411yescharacteristic
67D4×D271—22no—
68D4×D271—41no—
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271C2311yescharacteristic
73#16864644—41no—
74#16864641—22no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C8×D271—41no—
78C4⋊Dic271—21no—
79C4⋊Dic271—21no—
80C4⋊Dic271—21no—
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4D411yescharacteristic
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271D411yescharacteristic
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4D411yescharacteristic
95C4×Dic271—21no—
96C4×Dic271—21no—
97C4×Dic271C2311yescharacteristic
98C2×Q8×C271C2311yescharacteristic
99C2×D4×C271—21no—
100C2×D4×C271C2311yescharacteristic
101C22×C4×C271C2311yescharacteristic
102SD16×C271—41no—
103C2×C8×C271—41no—
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C4⋊C4×C271—21no—
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C22⋊C4×C271—21no—
110C22⋊C4×C271—21no—
111C22×D271—21no—
112C22×D271—42no—
113C22×D271—42no—
114C22×D271—81no—
115C22×D271C2×D411yescharacteristic
116Dic542—21no—
117Dic542—21no—
118D1084—21no—
119D1084#4112yes—
120C22⋊D271—22no—
121C22⋊D271—42no—
122C22⋊D271—42no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C4×D271—21no—
130C4×D271—21no—
131C4×D271—42no—
132C4×D271—42no—
133C4×D271—81no—
134C271⋊C8—21no—
135C271⋊C8—21no—
136C23×C271—21no—
137C23×C271C2×D411yescharacteristic
138Q8×C271—21no—
139Q8×C271—21no—
140D4×C271—21no—
141D4×C271—21no—
142D4×C271—81no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2411yescharacteristic
150C8×C271—21no—
151C8×C271—21no—
152C22×C271—21no—
153C22×C271—21no—
154C22×C271—21no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271C22×D411yescharacteristic
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—22no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—84no—
165C2×D271#13512yes—
166C4×C271—21no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271—41no—
170C4×C271C2×#4111yescharacteristic
171C4×C271H(5,𝔽2)11yescharacteristic
172Dic271—21no—
173Dic271—21no—
174Dic271—21no—
175Dic271—41no—
176Dic271—41no—
177Dic271—41no—
178C2×C271—41no—
179C2×C271—81no—
180C2×C271#54611yescharacteristic
181C2×C271C2×#13511yescharacteristic
182C2×C271#57711yescharacteristic
183D271—22no—
184D271—42no—
185C271#329611yescharacteristic, complemented, Sylow
186#3296—2711nomaximal, Sylow
187#552—2711no—
188#546—2711no—
189#488—2711no—
190#483—2711no—
191#482—2711no—
192#478—2711no—
193#467—2711no—
194#465D27111yescharacteristic, complemented
195#464—2711no—
196#452—2711no—
197#450—2711no—
198C4×SD16—2711no—
199C8×D4—2711no—
200#416—2711no—
201C2×#102—2711no—
202C2×#41—2711no—
203C22×D4—2711no—
204C2×Q16—2711no—
205C2×SD16—2711no—
206C2×SD16C2×D27111yescharacteristic
207C22×C8—2711no—
208C22×C8C2×D27111yescharacteristic, complemented
209#126—5421no—
210#125—2711no—
211#125—5421no—
212#124—2711no—
213#123—5421no—
214#123—5421no—
215C22⋊Q8—2711no—
216C22⋊Q8—2711no—
217C22⋊Q8C2×D27111yescharacteristic, complemented
218#121—5421no—
219#121C2×D27111yescharacteristic
220#120—5421no—
221C4×Q8—2711no—
222C4×D4—2711no—
223C4×D4—2711no—
224#117—2711no—
225#117—5421no—
226C2×C4⋊C4—2711no—
227C2×C22⋊C4—5421no—
228#107—2711no—
229#106—2711no—
230#106C2×D27111yescharacteristic, complemented
231C4⋊C8—2711no—
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103C2×D27111yescharacteristic, complemented
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241#102C2×D27111yescharacteristic
242C22⋊C8—2711no—
243C22⋊C8—2711no—
244C4×C8—2711no—
245C24—5421no—
246#41—5422no—
247#41—10841no—
248C2×Q8—2711no—
249C2×Q8C22×D27111yescharacteristic
250C2×D4—21no—
251C2×D4—2711no—
252C2×D4—2711no—
253C2×D4—5421no—
254C2×D4—5422no—
255C2×D4—5421no—
256C2×D4—10841no—
257C2×D4C22×D27111yescharacteristic
258C22×C4—2711no—
259C22×C4—5421no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4C22×D27111yescharacteristic
263Q16—10841no—
264SD16—41no—
265SD16—5422no—
266C2×C8—41no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8—10841no—
270C2×C8C22×D27111yescharacteristic
271C2×C8C22×D27111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4—5421no—
282C4⋊C4C22×D27111yescharacteristic
283C4⋊C4C22×D27111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—2711no—
287C22⋊C4—2711no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C42—2711no—
295C42—5421no—
296C42—5421no—
297C23—21no—
298C23—2711no—
299C23—5421no—
300C23—10842no—
301C23—10842no—
302C23—21681no—
303C23D4×D27111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5421no—
307Q8—5421no—
308D4—21no—
309D4—21no—
310D4—81no—
311D4—2712no—
312D4—5422no—
313D4—5421no—
314D4—10842no—
315D4—10842no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—2711no—
322C2×C4—2711no—
323C2×C4—2711no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—5421no—
329C2×C4—10842no—
330C2×C4—10842no—
331C2×C4—21681no—
332C2×C4D4×D27111yescharacteristic
333C2×C4C23×D27111yescharacteristic
334C8—21no—
335C8—21no—
336C8—5421no—
337C8—5421no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—2712no—
345C22—5421no—
346C22—5421no—
347C22—5422no—
348C22—10841no—
349C22—10841no—
350C22—21684no—
351C22C2×D4×D27111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—5421no—
357C4—5421no—
358C4—5421no—
359C4—10841no—
360C4—10841no—
361C4—10841no—
362C4#41×D27111yescharacteristic
363C4#2257620511yescharacteristic
364C2—41no—
365C2—81no—
366C2—5422no—
367C2—10842no—
368C2#3177453611yescharacteristic
369C2#135×D27111yescharacteristic
370C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
prime radical#31774544C2
hypercenter#465D271
hypocenterC271#3296
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #31774581C2semidirect
3. #31774250C2semidirect
4. #31774229C2semidirect
5. #31774218C2semidirect
6. #31774196C2semidirect
7. #31774137C2semidirect
8. #31774117C2semidirect
9. #31774057C2semidirect
10. #31774046C2semidirect
11. #31773982C2semidirect
12. C8×C22⋊D271C2semidirect
13. #31773894C2semidirect
14. #102×D271C2semidirect
15. #465×C271C2semidirect
18. C2×#16864644C22semidirect
22. #22576151C22semidirect
31. Q8×Dic271C22semidirect
34. #22576119C22semidirect
38. #22576093C22semidirect
39. #22576091C22semidirect
40. Dic271⋊C8C22semidirect
41. #22576085C22semidirect
42. #22576085C22semidirect
43. #22576083C22semidirect
49. C22⋊(C271⋊C8)C22semidirect
51. C8×Dic271C22semidirect
53. C22×C8×C271C22semidirect
54. C22⋊Q8×C271C22semidirect
56. #106×C271C22semidirect
57. #103×C271C22semidirect
185. C271#3296semidirect
194. #465D271semidirect
208. C22×C8C2×D271semidirect
217. C22⋊Q8C2×D271semidirect
230. #106C2×D271semidirect
235. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745812
#317745442
#317742502
#317742292
#317742182
#317741962
#317741372
#317741172
#317740572
#317740462
#317739822
C8×C22⋊D2712
#317738942
#102×D2712
#465×C2712
#3296271