Finite Group Databasebeta

← #46848203

Subgroups — #46848203

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#31774541C211yesmaximal, characteristic
3#31774250C211yesmaximal, characteristic, complemented
4#31774228C211yesmaximal, characteristic, complemented
5#31774220C211yesmaximal, characteristic, complemented
6#31774199C211yesmaximal, characteristic, complemented
7#31774136C211yesmaximal, characteristic, complemented
8#31774123C211yesmaximal, characteristic, complemented
9#31774117C211yesmaximal, characteristic, complemented
10#31774056C211yesmaximal, characteristic, complemented
11SD16×Dic271C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13#31773890C211yesmaximal, characteristic, complemented
14C2×#22576081C211yesmaximal, characteristic, complemented
15#450×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×#16864644C2211yescharacteristic, complemented
18C22×C271⋊C8C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic, complemented
20#22576155—21no—
21#22576151—21no—
22#22576151C2211yescharacteristic, complemented
23#22576147C2211yescharacteristic
24#22576144—21no—
25#22576141—21no—
26#22576139C2211yescharacteristic
27#22576136C2211yescharacteristic, complemented
28#22576132—21no—
29#22576131C2211yescharacteristic
30#22576129—21no—
31Q8×Dic271C2211yescharacteristic, complemented
32D4×Dic271C2211yescharacteristic
33C4×C22⋊D271C2211yescharacteristic
34#22576118C2211yescharacteristic, complemented
35#22576116—21no—
36C2×C4⋊Dic271C2211yescharacteristic
37C2×C22⋊Dic271—21no—
38#22576093C2211yescharacteristic, complemented
39#22576092C2211yescharacteristic, complemented
40#22576092C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42#22576085C2211yescharacteristic, complemented
43#22576085C2211yescharacteristic, complemented
44#22576084C2211yescharacteristic, complemented
45#22576083C2211yescharacteristic, complemented
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49#22576081C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C22×D4×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55C22⋊Q8×C271C2211yescharacteristic
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59#16864653—22no—
60#16864653—41no—
61C2×Dic542C2311yescharacteristic
62C2×C22⋊D271—21no—
63C2×C22⋊D271—21no—
64C2×C22⋊D271D411yescharacteristic
65C22×Dic271—21no—
66C22×Dic271—21no—
67C22×Dic271—21no—
68C2×C4×D271C2311yescharacteristic
69#16864644—41no—
70C2×C271⋊C8—41no—
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C8×D271—41no—
74C4⋊Dic271—21no—
75C4⋊Dic271—21no—
76C4⋊Dic271—21no—
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4D411yescharacteristic
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271D411yescharacteristic
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4D411yescharacteristic
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96C24×C271—21no—
97C2×Q8×C271C2311yescharacteristic
98C2×D4×C271—22no—
99C2×D4×C271—21no—
100C2×D4×C271—41no—
101C2×D4×C271C2311yescharacteristic
102C2×D4×C271C2311yescharacteristic
103C22×C4×C271C2311yescharacteristic
104SD16×C271—22no—
105SD16×C271—41no—
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271—21no—
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22×D271C2×D411yescharacteristic
112Dic542—21no—
113Dic542—21no—
114C22⋊D271—22no—
115C22⋊D271—42no—
116C22⋊D271—81no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—42no—
122C2×Dic271—42no—
123C2×Dic271—81no—
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C4×D271—21no—
128C4×D271—21no—
129C271⋊C8—21no—
130C271⋊C8—21no—
131C23×C271—21no—
132C23×C271—21no—
133C23×C271—42no—
134C23×C271—42no—
135C23×C271—81no—
136C23×C271C2×D411yescharacteristic
137Q8×C271—21no—
138Q8×C271—21no—
139D4×C271—21no—
140D4×C271—21no—
141D4×C271—21no—
142D4×C271—42no—
143D4×C271#4112yes—
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—22no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—84no—
161C22×C271#13512yes—
162C22×C271C22×D411yescharacteristic
163C2×D271—21no—
164C2×D271—21no—
165C2×D271—21no—
166C4×C271—21no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271C2×#4111yescharacteristic
170C4×C271H(5,𝔽2)11yescharacteristic
171Dic271—21no—
172Dic271—21no—
173Dic271—21no—
174Dic271—41no—
175Dic271—41no—
176Dic271—41no—
177Dic271—41no—
178C2×C271—22no—
179C2×C271—42no—
180C2×C271—81no—
181C2×C271#54611yescharacteristic
182C2×C271C2×#13511yescharacteristic
183C2×C271#57711yescharacteristic
184D271—41no—
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#465—2711no—
195#464—2711no—
196#452—2711no—
197#450D27111yescharacteristic, complemented
198C4×SD16—2711no—
199C8×D4—2711no—
200#416—2711no—
201C2×#102—2711no—
202C2×#41—2711no—
203C22×D4C2×D27111yescharacteristic
204C2×Q16—2711no—
205C2×SD16C2×D27111yescharacteristic
206C2×SD16C2×D27111yescharacteristic
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⋊C4—2711no—
227C2×C22⋊C4—5421no—
228#107—2711no—
229#106—2711no—
230#106—2711no—
231C4⋊C8—2711no—
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103—2711no—
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102C2×D27111yescharacteristic
241#102C2×D27111yescharacteristic
242C22⋊C8—2711no—
243C22⋊C8C2×D27111yescharacteristic
244C4×C8—2711no—
245C24—21no—
246#41—5422no—
247#41—10841no—
248C2×Q8—2711no—
249C2×Q8C22×D27111yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—2711no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4C22×D27111yescharacteristic
257C2×D4C22×D27111yescharacteristic
258C22×C4—2711no—
259C22×C4—5421no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4C22×D27111yescharacteristic
263Q16—10841no—
264SD16—22no—
265SD16—41no—
266C2×C8—2711no—
267C2×C8—2711no—
268C2×C8—10841no—
269C2×C8—10841no—
270C2×C8C22×D27111yescharacteristic
271C2×C8C22×D27111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2711no—
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⋊C4—5421no—
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—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—2711no—
303C23D4×D27111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5421no—
307Q8—5421no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—5422no—
313D4—10842no—
314D4—21681no—
315D4#1686465312yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—2711no—
321C2×C4—2711no—
322C2×C4—2711no—
323C2×C4—5421no—
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—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—5421no—
348C22—5421no—
349C22—5421no—
350C22#2257618112yes—
351C22C2×D4×D27111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—5421no—
356C4—5421no—
357C4—5421no—
358C4—10841no—
359C4—10841no—
360C4—10841no—
361C4—10841no—
362C4C2×#1686465311yescharacteristic
363C4#2257620511yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—10841no—
368C2#3177453411yescharacteristic
369C2C2×#2257618111yescharacteristic
370C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#450×C271C2
socleC22×C271C22×D4
prime radical#31774541C2
hypercenter#450D271
hypocenterC271#3296
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #31774589C2semidirect
3. #31774250C2semidirect
4. #31774228C2semidirect
5. #31774220C2semidirect
6. #31774199C2semidirect
7. #31774136C2semidirect
8. #31774123C2semidirect
9. #31774117C2semidirect
10. #31774056C2semidirect
11. SD16×Dic271C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. #31773890C2semidirect
14. C2×#22576081C2semidirect
15. #450×C271C2semidirect
17. C2×#16864644C22semidirect
19. C2×C8×D271C22semidirect
22. #22576151C22semidirect
27. #22576136C22semidirect
31. Q8×Dic271C22semidirect
34. #22576118C22semidirect
38. #22576093C22semidirect
39. #22576092C22semidirect
40. #22576092C22semidirect
41. Dic271⋊C8C22semidirect
42. #22576085C22semidirect
43. #22576085C22semidirect
44. #22576084C22semidirect
45. #22576083C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. C8×Dic271C22semidirect
185. C271#3296semidirect
197. #450D271semidirect

Maximal subgroups

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

subgroupindex
#317745892
#317745412
#317742502
#317742282
#317742202
#317741992
#317741362
#317741232
#317741172
#317740562
SD16×Dic2712
(C22⋊D271)⋊C82
#317738902
C2×#225760812
#450×C2712
#3296271