Finite Group Databasebeta

← #46848202

Subgroups — #46848202

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#31774586C211yesmaximal, characteristic, complemented
2#31774544C211yesmaximal, characteristic
3#31774250C211yesmaximal, characteristic, complemented
4#31774229C211yesmaximal, characteristic, complemented
5#31774200C211yesmaximal, characteristic, complemented
6#31774139C211yesmaximal, characteristic, complemented
7#31774124C211yesmaximal, characteristic, complemented
8#31774118C211yesmaximal, characteristic, complemented
9#31774061C211yesmaximal, characteristic, complemented
10#31774049C211yesmaximal, characteristic, complemented
11#31773984C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13#31773891C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#482×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18C2×Dic1084C2211yescharacteristic, complemented
19C2×#16864643C2211yescharacteristic
20C2×#16864642C2211yescharacteristic
21C22×C271⋊C8C2211yescharacteristic, complemented
22C2×C8×D271C2211yescharacteristic
23#22576153—21no—
24#22576152—21no—
25#22576151C2211yescharacteristic, complemented
26#22576147C2211yescharacteristic
27#22576142—21no—
28#22576142—21no—
29#22576140C2211yescharacteristic
30#22576140C2211yescharacteristic
31#22576139C2211yescharacteristic, complemented
32#22576131—21no—
33#22576128—21no—
34#22576125C2211yescharacteristic, complemented
35#22576122C2211yescharacteristic
36C4×C22⋊D271C2211yescharacteristic
37#22576117—21no—
38C4⋊C4×D271C2211yescharacteristic
39C22⋊C4×D271—21no—
40#22576091C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42#22576085C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576083C2211yescharacteristic, complemented
45#22576083C2211yescharacteristic, complemented
46#22576082C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic, complemented
52#121×C271C2211yescharacteristic
53#117×C271C2211yescharacteristic, complemented
54#107×C271C2211yescharacteristic, complemented
55#106×C271C2211yescharacteristic, complemented
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864654—22no—
61#16864654—41no—
62C2×Dic542C2311yescharacteristic
63C2×Dic542C2311yescharacteristic
64C2×D1084C2311yescharacteristic
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271D411yescharacteristic
68D4×D271—22no—
69D4×D271—41no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271C2311yescharacteristic
74Dic1084—41no—
75#16864643—22no—
76#16864642—41no—
77C2×C271⋊C8—41no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C8×D271—41no—
81C4⋊Dic271—21no—
82C4⋊Dic271—21no—
83C4⋊Dic271—21no—
84C4⋊Dic271C2311yescharacteristic
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊C4D411yescharacteristic
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92C22⋊Dic271—21no—
93C22⋊Dic271D411yescharacteristic
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97(C2×D271)⋊C4D411yescharacteristic
98C4×Dic271—21no—
99C4×Dic271C2311yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C22×C4×C271C2311yescharacteristic
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C4⋊C4×C271C2311yescharacteristic
106C4⋊C4×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C22⋊C4×C271—21no—
109C22⋊C4×C271—21no—
110C42×C271—21no—
111C22×D271—21no—
112C22×D271—42no—
113C22×D271—42no—
114C22×D271—81no—
115C22×D271C2×D411yescharacteristic
116Dic542—21no—
117Dic542—21no—
118Dic542—21no—
119Dic542—21no—
120D1084—21no—
121D1084#4112yes—
122C22⋊D271—22no—
123C22⋊D271—42no—
124C22⋊D271—42no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—21no—
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C2×Dic271C2×D411yescharacteristic
131C4×D271—21no—
132C4×D271—21no—
133C4×D271—42no—
134C4×D271—42no—
135C4×D271—81no—
136C271⋊C8—21no—
137C271⋊C8—21no—
138C23×C271—21no—
139C23×C271C2×D411yescharacteristic
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#482D27111yescharacteristic, complemented
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⋊Q8—2711no—
218#121—5421no—
219#121C2×D27111yescharacteristic
220#120—5421no—
221C4×Q8—2711no—
222C4×D4—2711no—
223C4×D4—2711no—
224#117—5421no—
225#117C2×D27111yescharacteristic, complemented
226C2×C4⋊C4—2711no—
227C2×C22⋊C4—5421no—
228#107C2×D27111yescharacteristic, complemented
229#106—2711no—
230#106C2×D27111yescharacteristic, complemented
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, complemented
244C4×C8—2711no—
245C24—5421no—
246#41—5422no—
247#41—10841no—
248C2×Q8—2711no—
249C2×Q8—2711no—
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—5422no—
265SD16—10841no—
266C2×C8—2711no—
267C2×C8—2711no—
268C2×C8—10841no—
269C2×C8—10841no—
270C2×C8C22×D27111yescharacteristic
271C2×C8C22×D27111yescharacteristic
272C4⋊C4—2711no—
273C4⋊C4—2711no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
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—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—21no—
295C42—2711no—
296C42—5421no—
297C23—21no—
298C23—2711no—
299C23—5421no—
300C23—10842no—
301C23—10842no—
302C23—21681no—
303C23D4×D27111yescharacteristic
304Q8—5421no—
305Q8—5421no—
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#2257620611yescharacteristic
364C2—41no—
365C2—81no—
366C2—5422no—
367C2—10842no—
368C2#3177453911yescharacteristic
369C2#135×D27111yescharacteristic
370C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#482×C271C2
socleC22×C271C22×D4
prime radical#31774544C2
hypercenter#482D271
hypocenterC271#3296
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #31774586C2semidirect
3. #31774250C2semidirect
4. #31774229C2semidirect
5. #31774200C2semidirect
6. #31774139C2semidirect
7. #31774124C2semidirect
8. #31774118C2semidirect
9. #31774061C2semidirect
10. #31774049C2semidirect
11. #31773984C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. #31773891C2semidirect
14. #102×D271C2semidirect
15. #482×C271C2semidirect
18. C2×Dic1084C22semidirect
21. C22×C271⋊C8C22semidirect
25. #22576151C22semidirect
31. #22576139C22semidirect
34. #22576125C22semidirect
40. #22576091C22semidirect
41. Dic271⋊C8C22semidirect
42. #22576085C22semidirect
43. #22576083C22semidirect
44. #22576083C22semidirect
45. #22576083C22semidirect
51. C8×Dic271C22semidirect
53. #117×C271C22semidirect
54. #107×C271C22semidirect
55. #106×C271C22semidirect
58. C22⋊C8×C271C22semidirect
185. C271#3296semidirect
191. #482D271semidirect
225. #117C2×D271semidirect
228. #107C2×D271semidirect
230. #106C2×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
#317745862
#317745442
#317742502
#317742292
#317742002
#317741392
#317741242
#317741182
#317740612
#317740492
#317739842
(C22⋊D271)⋊C82
#317738912
#102×D2712
#482×C2712
#3296271