Finite Group Databasebeta

← #46848205

Subgroups — #46848205

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#31774589C211yesmaximal, characteristic, complemented
2#31774538C211yesmaximal, characteristic
3#31774253C211yesmaximal, characteristic, complemented
4#31774225C211yesmaximal, characteristic, complemented
5#31774224C211yesmaximal, characteristic, complemented
6#31774200C211yesmaximal, characteristic, complemented
7#31774129C211yesmaximal, characteristic, complemented
8#31774121C211yesmaximal, characteristic, complemented
9#31774058C211yesmaximal, characteristic, complemented
10#31774047C211yesmaximal, characteristic, complemented
11#31773982C211yesmaximal, characteristic, complemented
12C8×C22⋊D271C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14C2×#22576082C211yesmaximal, characteristic, complemented
15#467×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C2×#16864641C2211yescharacteristic
19C2×#16864641C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic, complemented
21#22576155—21no—
22#22576151—21no—
23#22576151C2211yescharacteristic, complemented
24#22576149C2211yescharacteristic
25#22576145—21no—
26#22576144—21no—
27#22576136C2211yescharacteristic, complemented
28#22576135—21no—
29#22576135C2211yescharacteristic
30#22576130—21no—
31Q8×Dic271C2211yescharacteristic, complemented
32#22576122C2211yescharacteristic
33C4×C22⋊D271C2211yescharacteristic
34#22576118C2211yescharacteristic, complemented
35#22576116—21no—
36C2×C4⋊Dic271C2211yescharacteristic
37C22⋊C4×D271—21no—
38#22576091C2211yescharacteristic, complemented
39#22576091C2211yescharacteristic, complemented
40Dic271⋊C8C2211yescharacteristic, complemented
41#22576084C2211yescharacteristic, complemented
42#22576084C2211yescharacteristic, complemented
43#22576082C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C2×Q16×C271C2211yescharacteristic, complemented
53C22×C8×C271C2211yescharacteristic
54C22⋊Q8×C271C2211yescharacteristic
55C22⋊Q8×C271C2211yescharacteristic
56#107×C271C2211yescharacteristic, complemented
57#103×C271C2211yescharacteristic, complemented
58#103×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864655—22no—
61#16864655—41no—
62C2×D1084—22no—
63C2×D1084—21no—
64C2×D1084—21no—
65C2×D1084—21no—
66C2×D1084—41no—
67C2×D1084C2311yescharacteristic
68C2×D1084C2311yescharacteristic
69C2×C22⋊D271D411yescharacteristic
70C22×Dic271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271C2311yescharacteristic
74#16864641—22no—
75#16864641—41no—
76C2×C271⋊C8C2311yescharacteristic
77C2×C271⋊C8C2311yescharacteristic
78C8×D271—41no—
79C4⋊Dic271—21no—
80C4⋊Dic271—21no—
81C4⋊Dic271—21no—
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4D411yescharacteristic
87C22⋊Dic271D411yescharacteristic
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
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×Q8×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101Q16×C271—41no—
102C2×C8×C271—41no—
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C4⋊C4×C271—21no—
106C4⋊C4×C271—21no—
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C22⋊C4×C271—21no—
110C22⋊C4×C271—21no—
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—42no—
114C22×D271—42no—
115C22×D271—81no—
116C22×D271C2×D411yescharacteristic
117D1084—21no—
118D1084—21no—
119D1084—21no—
120D1084—42no—
121D1084—42no—
122D1084—81no—
123D1084#4112yes—
124C22⋊D271—22no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—42no—
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C2×Dic271C2×D411yescharacteristic
131C4×D271—21no—
132C4×D271—21no—
133C4×D271—42no—
134C4×D271—81no—
135C271⋊C8—21no—
136C271⋊C8—21no—
137C23×C271C2×D411yescharacteristic
138Q8×C271—21no—
139Q8×C271—21no—
140Q8×C271—21no—
141Q8×C271—21no—
142C2×C4×C271—21no—
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#13512yes—
155C22×C271C22×D411yescharacteristic
156C2×D271—21no—
157C2×D271—22no—
158C2×D271—21no—
159C2×D271—21no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C2×D271—84no—
166C4×C271—21no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271—41no—
170C4×C271—41no—
171C4×C271C2×#4111yescharacteristic
172C4×C271H(5,𝔽2)11yescharacteristic
173Dic271—21no—
174Dic271—21no—
175Dic271—21no—
176Dic271—41no—
177Dic271—41no—
178C2×C271—22no—
179C2×C271#54611yescharacteristic
180C2×C271C2×#13511yescharacteristic
181C2×C271#57711yescharacteristic
182D271—41no—
183D271—42no—
184D271—81no—
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#467D27111yescharacteristic, complemented
194#465—2711no—
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×Q16C2×D27111yescharacteristic, complemented
205C2×SD16—2711no—
206C2×SD16—2711no—
207C22×C8—2711no—
208C22×C8C2×D27111yescharacteristic
209#126—5421no—
210#125—2711no—
211#125—5421no—
212#124—2711no—
213#123—5421no—
214#123—5421no—
215C22⋊Q8—2711no—
216C22⋊Q8C2×D27111yescharacteristic
217C22⋊Q8C2×D27111yescharacteristic
218#121—2711no—
219#121—5421no—
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#107C2×D27111yescharacteristic, complemented
229#106—2711no—
230#106—2711no—
231C4⋊C8—2711no—
232#103—2711no—
233#103—2711no—
234#103C2×D27111yescharacteristic, complemented
235#103C2×D27111yescharacteristic, complemented
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241#102—2711no—
242C22⋊C8—2711no—
243C22⋊C8—2711no—
244C4×C8—2711no—
245C24—5421no—
246#41—5422no—
247#41—10841no—
248C2×Q8C22×D27111yescharacteristic
249C2×Q8C22×D27111yescharacteristic
250C2×D4—2711no—
251C2×D4—2711no—
252C2×D4—2711no—
253C2×D4—5422no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4—5421no—
257C2×D4—10841no—
258C22×C4—2711no—
259C22×C4—5421no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4C22×D27111yescharacteristic
263Q16—41no—
264SD16—5422no—
265SD16—10841no—
266C2×C8—41no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8—10841no—
270C2×C8C22×D27111yescharacteristic
271C2×C8C22×D27111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—2711no—
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—2711no—
298C23—5421no—
299C23—5421no—
300C23—10842no—
301C23—10842no—
302C23—21681no—
303C23D4×D27111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—2712no—
309D4—5422no—
310D4—5421no—
311D4—5421no—
312D4—5421no—
313D4—10842no—
314D4—10842no—
315D4—21681no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—2711no—
323C2×C4—2711no—
324C2×C4—2711no—
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—5421no—
341C22—5422no—
342C22—5421no—
343C22—5421no—
344C22—10841no—
345C22—10841no—
346C22—10841no—
347C22—10841no—
348C22—10841no—
349C22—21684no—
350C22#2257618312yes—
351C22C2×D4×D27111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5421no—
358C4—5421no—
359C4—5421no—
360C4—10841no—
361C4—10841no—
362C4#41×D27111yescharacteristic
363C4#2257620511yescharacteristic
364C2—22no—
365C2—10841no—
366C2—10842no—
367C2—21681no—
368C2#3177453611yescharacteristic
369C2C2×#2257618311yescharacteristic
370C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#467×C271C2
socleC22×C271C22×D4
prime radical#31774538C2
hypercenter#467D271
hypocenterC271#3296
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #31774589C2semidirect
3. #31774253C2semidirect
4. #31774225C2semidirect
5. #31774224C2semidirect
6. #31774200C2semidirect
7. #31774129C2semidirect
8. #31774121C2semidirect
9. #31774058C2semidirect
10. #31774047C2semidirect
11. #31773982C2semidirect
12. C8×C22⋊D271C2semidirect
13. #31773892C2semidirect
14. C2×#22576082C2semidirect
15. #467×C271C2semidirect
20. C2×C8×D271C22semidirect
23. #22576151C22semidirect
27. #22576136C22semidirect
31. Q8×Dic271C22semidirect
34. #22576118C22semidirect
38. #22576091C22semidirect
39. #22576091C22semidirect
40. Dic271⋊C8C22semidirect
41. #22576084C22semidirect
42. #22576084C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. C8×Dic271C22semidirect
52. C2×Q16×C271C22semidirect
56. #107×C271C22semidirect
57. #103×C271C22semidirect
58. #103×C271C22semidirect
185. C271#3296semidirect
193. #467D271semidirect
204. C2×Q16C2×D271semidirect
228. #107C2×D271semidirect
234. #103C2×D271semidirect
235. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745892
#317745382
#317742532
#317742252
#317742242
#317742002
#317741292
#317741212
#317740582
#317740472
#317739822
C8×C22⋊D2712
#317738922
C2×#225760822
#467×C2712
#3296271