Finite Group Databasebeta

← #46848197

Subgroups — #46848197

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#31774582C211yesmaximal, characteristic, complemented
2#31774538C211yesmaximal, characteristic
3#31774253C211yesmaximal, characteristic, complemented
4#31774223C211yesmaximal, characteristic, complemented
5#31774196C211yesmaximal, characteristic, complemented
6#31774136C211yesmaximal, characteristic, complemented
7#31774127C211yesmaximal, characteristic, complemented
8#31774122C211yesmaximal, characteristic, complemented
9#31774062C211yesmaximal, characteristic, complemented
10#31774050C211yesmaximal, characteristic, complemented
11#31773984C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14C2×#22576080C211yesmaximal, characteristic, complemented
15#483×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C2×#16864644C2211yescharacteristic, complemented
19C2×#16864643C2211yescharacteristic
20C2×#16864643C2211yescharacteristic
21C22×C271⋊C8C2211yescharacteristic
22C2×C8×D271C2211yescharacteristic, complemented
23#22576155—21no—
24#22576151C2211yescharacteristic, complemented
25#22576150—21no—
26#22576149C2211yescharacteristic
27#22576145—21no—
28#22576143—21no—
29#22576140C2211yescharacteristic, complemented
30#22576139C2211yescharacteristic
31#22576135—21no—
32#22576135C2211yescharacteristic
33#22576130—21no—
34#22576125C2211yescharacteristic, complemented
35#22576122C2211yescharacteristic
36C4×C22⋊D271C2211yescharacteristic
37#22576119—21no—
38#22576118C2211yescharacteristic, complemented
39C22⋊C4×D271—21no—
40#22576093C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42#22576084C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576080C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C22⋊Q8×C271C2211yescharacteristic
53C2×C4⋊C4×C271C2211yescharacteristic
54#106×C271C2211yescharacteristic, complemented
55#106×C271C2211yescharacteristic, complemented
56#103×C271C2211yescharacteristic, complemented
57#103×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic
59C23×D271—21no—
60#16864655—22no—
61#16864655—41no—
62C2×Dic542C2311yescharacteristic
63C2×D1084—22no—
64C2×D1084—21no—
65C2×D1084—21no—
66C2×D1084—21no—
67C2×D1084—41no—
68C2×D1084C2311yescharacteristic
69C2×D1084C2311yescharacteristic
70C2×C22⋊D271D411yescharacteristic
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271C2311yescharacteristic
74#16864644—41no—
75#16864643—22no—
76#16864643—41no—
77C2×C271⋊C8—41no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C8×D271—41no—
81C4⋊Dic271—21no—
82C4⋊Dic271—21no—
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4D411yescharacteristic
88C22⋊Dic271—21no—
89C22⋊Dic271D411yescharacteristic
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4D411yescharacteristic
97C4×Dic271—21no—
98C4×Dic271—21no—
99C4×Dic271C2311yescharacteristic
100C2×Q8×C271C2311yescharacteristic
101C22×C4×C271—21no—
102C22×C4×C271C2311yescharacteristic
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C4⋊C4×C271—21no—
106C4⋊C4×C271—21no—
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—42no—
114C22×D271—42no—
115C22×D271—81no—
116C22×D271C2×D411yescharacteristic
117Dic542—21no—
118Dic542—21no—
119D1084—21no—
120D1084—21no—
121D1084—21no—
122D1084—42no—
123D1084—42no—
124D1084—81no—
125D1084#4112yes—
126C22⋊D271—22no—
127C2×Dic271—21no—
128C2×Dic271—21no—
129C2×Dic271C2×D411yescharacteristic
130C2×Dic271C2×D411yescharacteristic
131C2×Dic271C2×D411yescharacteristic
132C4×D271—21no—
133C4×D271—21no—
134C4×D271—42no—
135C4×D271—81no—
136C271⋊C8—21no—
137C271⋊C8—21no—
138C23×C271C2×D411yescharacteristic
139Q8×C271—21no—
140Q8×C271—21no—
141C2×C4×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—42no—
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#483D27111yescharacteristic, complemented
191#482—2711no—
192#478—2711no—
193#467—2711no—
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×Q16—2711no—
205C2×SD16—2711no—
206C2×SD16—2711no—
207C22×C8—2711no—
208C22×C8—2711no—
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
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⋊C4C2×D27111yescharacteristic
227C2×C22⋊C4—5421no—
228#107—2711no—
229#106C2×D27111yescharacteristic, complemented
230#106C2×D27111yescharacteristic, complemented
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⋊C8C2×D27111yescharacteristic
244C4×C8—2711no—
245C24—5421no—
246#41—5422no—
247#41—10841no—
248C2×Q8—2711no—
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—21no—
259C22×C4—2711no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4C22×D27111yescharacteristic
263Q16—10841no—
264SD16—5422no—
265SD16—10841no—
266C2×C8—2711no—
267C2×C8—2711no—
268C2×C8—10841no—
269C2×C8—10841no—
270C2×C8C22×D27111yescharacteristic
271C2×C8C22×D27111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4C22×D27111yescharacteristic
282C4⋊C4C22×D27111yescharacteristic
283C4⋊C4C22×D27111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—2711no—
286C22⋊C4—2711no—
287C22⋊C4—5421no—
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—5421no—
307Q8—5421no—
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—42no—
323C2×C4—2711no—
324C2×C4—2711no—
325C2×C4—2711no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—5421no—
329C2×C4—5421no—
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#2257618112yes—
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#2257620611yescharacteristic
364C2—22no—
365C2—10841no—
366C2—10842no—
367C2—21681no—
368C2#3177453911yescharacteristic
369C2C2×#2257618111yescharacteristic
370C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#483×C271C2
socleC22×C271C22×D4
prime radical#31774538C2
hypercenter#483D271
hypocenterC271#3296
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #31774582C2semidirect
3. #31774253C2semidirect
4. #31774223C2semidirect
5. #31774196C2semidirect
6. #31774136C2semidirect
7. #31774127C2semidirect
8. #31774122C2semidirect
9. #31774062C2semidirect
10. #31774050C2semidirect
11. #31773984C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. #31773892C2semidirect
14. C2×#22576080C2semidirect
15. #483×C271C2semidirect
18. C2×#16864644C22semidirect
22. C2×C8×D271C22semidirect
24. #22576151C22semidirect
29. #22576140C22semidirect
34. #22576125C22semidirect
38. #22576118C22semidirect
40. #22576093C22semidirect
41. Dic271⋊C8C22semidirect
42. #22576084C22semidirect
43. #22576083C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. C8×Dic271C22semidirect
54. #106×C271C22semidirect
55. #106×C271C22semidirect
56. #103×C271C22semidirect
57. #103×C271C22semidirect
185. C271#3296semidirect
190. #483D271semidirect
229. #106C2×D271semidirect
230. #106C2×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
#317745822
#317745382
#317742532
#317742232
#317741962
#317741362
#317741272
#317741222
#317740622
#317740502
#317739842
(C22⋊D271)⋊C82
#317738922
C2×#225760802
#483×C2712
#3296271