Finite Group Databasebeta

← #46848192

Subgroups — #46848192

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#31774535C211yesmaximal, characteristic
3#31774253C211yesmaximal, characteristic, complemented
4#31774226C211yesmaximal, characteristic, complemented
5#31774222C211yesmaximal, characteristic, complemented
6#31774195C211yesmaximal, characteristic, complemented
7#31774135C211yesmaximal, characteristic, complemented
8#31774128C211yesmaximal, characteristic, complemented
9#31774121C211yesmaximal, characteristic, complemented
10#31774046C211yesmaximal, characteristic, complemented
11SD16×Dic271C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13#31773893C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#452×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271C2211yescharacteristic
17C2×#16864641C2211yescharacteristic
18C22×C271⋊C8C2211yescharacteristic, complemented
19C2×C8×D271C2211yescharacteristic
20#22576153—21no—
21#22576151C2211yescharacteristic, complemented
22#22576150—21no—
23#22576149C2211yescharacteristic
24#22576143—21no—
25#22576142—21no—
26#22576136C2211yescharacteristic
27#22576136C2211yescharacteristic
28#22576134C2211yescharacteristic
29#22576132—21no—
30#22576128—21no—
31Q8×Dic271C2211yescharacteristic, complemented
32D4×Dic271C2211yescharacteristic
33C4×C22⋊D271C2211yescharacteristic
34#22576119C2211yescharacteristic, complemented
35#22576118—21no—
36C4⋊C4×D271C2211yescharacteristic
37C2×(C2×D271)⋊C4—21no—
38#22576094C2211yescharacteristic, complemented
39#22576092C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42#22576084C2211yescharacteristic, complemented
43#22576084C2211yescharacteristic, complemented
44#22576084C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic, complemented
52C2×#41×C271C2211yescharacteristic
53C2×Q16×C271C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic
55C22⋊Q8×C271C2211yescharacteristic, complemented
56#103×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C2×D1084C2311yescharacteristic
61C2×C22⋊D271—21no—
62C2×C22⋊D271—21no—
63C2×C22⋊D271D411yescharacteristic
64D4×D271—22no—
65D4×D271—41no—
66C22×Dic271—21no—
67C2×C4×D271—21no—
68C2×C4×D271C2311yescharacteristic
69#16864641—41no—
70C2×C271⋊C8—41no—
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C8×D271—41no—
74C4⋊Dic271—21no—
75C4⋊Dic271—21no—
76C4⋊Dic271C2311yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79Dic271⋊C4—21no—
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4D411yescharacteristic
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271D411yescharacteristic
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4D411yescharacteristic
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96#41×C271—22no—
97#41×C271—41no—
98C2×Q8×C271C2311yescharacteristic
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C22×C4×C271—21no—
103C22×C4×C271C2311yescharacteristic
104Q16×C271—41no—
105SD16×C271—22no—
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271—21no—
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22×D271—21no—
112C22×D271—42no—
113C22×D271—42no—
114C22×D271—81no—
115C22×D271C2×D411yescharacteristic
116D1084—21no—
117D1084—21no—
118C22⋊D271—22no—
119C22⋊D271—42no—
120C22⋊D271—81no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—81no—
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C4×D271—21no—
129C4×D271—21no—
130C4×D271—42no—
131C271⋊C8—21no—
132C271⋊C8—21no—
133C23×C271—21no—
134C23×C271C2×D411yescharacteristic
135Q8×C271—21no—
136Q8×C271—21no—
137Q8×C271—21no—
138Q8×C271—21no—
139D4×C271—21no—
140D4×C271—42no—
141D4×C271#4112yes—
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—22no—
154C22×C271—21no—
155C22×C271—21no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271C22×D411yescharacteristic
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—41no—
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—42no—
179C2×C271—41no—
180C2×C271#54611yescharacteristic
181C2×C271C2×#13511yescharacteristic
182C2×C271#57711yescharacteristic
183D271—22no—
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#467—2711no—
194#465—2711no—
195#464—2711no—
196#452D27111yescharacteristic, complemented
197#450—2711no—
198C4×SD16—2711no—
199C8×D4—2711no—
200#416—2711no—
201C2×#102—2711no—
202C2×#41C2×D27111yescharacteristic
203C22×D4—2711no—
204C2×Q16C2×D27111yescharacteristic, complemented
205C2×SD16—2711no—
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, complemented
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#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⋊C8C2×D27111yescharacteristic, complemented
244C4×C8—2711no—
245C24—5421no—
246#41—22no—
247#41—41no—
248C2×Q8C22×D27111yescharacteristic
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—21no—
259C22×C4—2711no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4C22×D27111yescharacteristic
263Q16—41no—
264SD16—22no—
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—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—2711no—
299C23—5421no—
300C23—10842no—
301C23—10842no—
302C23—21681no—
303C23D4×D27111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—42no—
310D4—5422no—
311D4—5421no—
312D4—5421no—
313D4—10842no—
314D4—21681no—
315D4#1686465312yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—2711no—
323C2×C4—2711no—
324C2×C4—2711no—
325C2×C4—5421no—
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—22no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—2712no—
345C22—5421no—
346C22—5421no—
347C22—10841no—
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—
362C4C2×#1686465311yescharacteristic
363C4#2257620511yescharacteristic
364C2—41no—
365C2—42no—
366C2—5422no—
367C2—21681no—
368C2#3177453411yescharacteristic
369C2#135×D27111yescharacteristic
370C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#452×C271C2
socleC22×C271C22×D4
prime radical#31774535C2
hypercenter#452D271
hypocenterC271#3296
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #31774581C2semidirect
3. #31774253C2semidirect
4. #31774226C2semidirect
5. #31774222C2semidirect
6. #31774195C2semidirect
7. #31774135C2semidirect
8. #31774128C2semidirect
9. #31774121C2semidirect
10. #31774046C2semidirect
11. SD16×Dic271C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. #31773893C2semidirect
14. #102×D271C2semidirect
15. #452×C271C2semidirect
18. C22×C271⋊C8C22semidirect
21. #22576151C22semidirect
31. Q8×Dic271C22semidirect
34. #22576119C22semidirect
38. #22576094C22semidirect
39. #22576092C22semidirect
40. #22576091C22semidirect
41. Dic271⋊C8C22semidirect
42. #22576084C22semidirect
43. #22576084C22semidirect
44. #22576084C22semidirect
51. C8×Dic271C22semidirect
53. C2×Q16×C271C22semidirect
55. C22⋊Q8×C271C22semidirect
56. #103×C271C22semidirect
58. C22⋊C8×C271C22semidirect
185. C271#3296semidirect
196. #452D271semidirect
204. C2×Q16C2×D271semidirect
217. C22⋊Q8C2×D271semidirect
235. #103C2×D271semidirect
243. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745812
#317745352
#317742532
#317742262
#317742222
#317741952
#317741352
#317741282
#317741212
#317740462
SD16×Dic2712
(C22⋊D271)⋊C82
#317738932
#102×D2712
#452×C2712
#3296271