Finite Group Databasebeta

← #46848175

Subgroups — #46848175

Order 34688.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774548C211yesmaximal, characteristic, complemented
2#31774531C211yesmaximal, characteristic, complemented
3#31774242C211yesmaximal, characteristic, complemented
4#31774212C211yesmaximal, characteristic, complemented
5#31774176C211yesmaximal, characteristic, complemented
6#31774134C211yesmaximal, characteristic, complemented
7#31774131C211yesmaximal, characteristic, complemented
8#31774091C211yesmaximal, characteristic, complemented
9#31774033C211yesmaximal, characteristic, complemented
10#31774032C211yesmaximal, characteristic, complemented
11#31773979C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13C2×#22576080C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#480×C271C211yesmaximal, characteristic, complemented
16C2×#16864654—21no—
17C22×D1084C2211yescharacteristic
18C2×D4×D271—21no—
19C2×D4×D271C2211yescharacteristic
20C2×D2168C2211yescharacteristic
21C2×D2168C2211yescharacteristic, complemented
22C2×#16864639C2211yescharacteristic, complemented
23C22×C271⋊C8C2211yescharacteristic, complemented
24C2×C8×D271C2211yescharacteristic, complemented
25#22576159C2211yescharacteristic, complemented
26#22576156C2211yescharacteristic
27#22576145—21no—
28#22576143—21no—
29#22576138—21no—
30#22576135C2211yescharacteristic
31#22576135C2211yescharacteristic, complemented
32#22576134C2211yescharacteristic, complemented
33#22576133—21no—
34#22576130—21no—
35#22576128—21no—
36#22576122C2211yescharacteristic
37#22576122C2211yescharacteristic, complemented
38C4×C22⋊D271C2211yescharacteristic
39C4⋊C4×D271C2211yescharacteristic, complemented
40#22576093C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42#22576082C2211yescharacteristic
43#22576082C2211yescharacteristic, complemented
44#22576081C2211yescharacteristic, complemented
45#22576080C2211yescharacteristic
46#22576080C2211yescharacteristic
47#22576080C2211yescharacteristic, complemented
48#22576080C2211yescharacteristic, complemented
49#22576080C2211yescharacteristic, complemented
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52#121×C271C2211yescharacteristic
53C2×C4⋊C4×C271C2211yescharacteristic, complemented
54#107×C271C2211yescharacteristic, complemented
55#107×C271C2211yescharacteristic, complemented
56#102×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C23×D271—21no—
61#16864654—44no—
62C2×Dic542—21no—
63C2×D1084—22no—
64C2×D1084—21no—
65C2×D1084—21no—
66C2×D1084—21no—
67C2×D1084—41no—
68C2×D1084C2311yescharacteristic
69C2×D1084C2311yescharacteristic
70C2×D1084C2311yescharacteristic
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271D411yescharacteristic
76D4×D271—22no—
77D4×D271—41no—
78D4×D271—44no—
79C2×C4×D271—21no—
80C2×C4×D271—21no—
81C2×C4×D271—21no—
82C2×C4×D271C2311yescharacteristic
83D2168—22no—
84D2168—41no—
85#16864639—41no—
86C2×C271⋊C8—41no—
87C2×C271⋊C8C2311yescharacteristic
88C2×C271⋊C8C2311yescharacteristic
89C8×D271—41no—
90C4⋊Dic271C2311yescharacteristic
91Dic271⋊C4—21no—
92Dic271⋊C4—21no—
93Dic271⋊C4D411yescharacteristic
94C22⋊Dic271D411yescharacteristic
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4D411yescharacteristic
101C4×Dic271C2311yescharacteristic
102C2×D4×C271—21no—
103C2×D4×C271C2311yescharacteristic
104C22×C4×C271—21no—
105C22×C4×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271—21no—
109C4⋊C4×C271C2311yescharacteristic
110C4⋊C4×C271C2311yescharacteristic
111C4⋊C4×C271C2311yescharacteristic
112C22⋊C4×C271—21no—
113C22×D271—21no—
114C22×D271—21no—
115C22×D271—21no—
116C22×D271—42no—
117C22×D271—42no—
118C22×D271—42no—
119C22×D271—42no—
120C22×D271—81no—
121C22×D271—81no—
122C22×D271C2×D411yescharacteristic
123Dic542—42no—
124D1084—21no—
125D1084—21no—
126D1084—21no—
127D1084—21no—
128D1084—21no—
129D1084—42no—
130D1084—42no—
131D1084—42no—
132D1084#4112yescomplemented
133C22⋊D271—42no—
134C22⋊D271—42no—
135C22⋊D271—42no—
136C22⋊D271—81no—
137C22⋊D271D814yescomplemented
138C2×Dic271—21no—
139C2×Dic271C2×D411yescharacteristic
140C2×Dic271C2×D411yescharacteristic
141C2×Dic271C2×D411yescharacteristic
142C4×D271—21no—
143C4×D271—21no—
144C4×D271—42no—
145C4×D271—42no—
146C4×D271—42no—
147C271⋊C8—21no—
148C271⋊C8—21no—
149C23×C271—21no—
150C23×C271C2×D411yescharacteristic
151D4×C271—21no—
152D4×C271—21no—
153D4×C271—42no—
154C2×C4×C271—21no—
155C2×C4×C271—21no—
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—42no—
160C2×C4×C271C2×D411yescharacteristic
161C2×C4×C271C2411yescharacteristic
162C8×C271—21no—
163C8×C271—21no—
164C22×C271—21no—
165C22×C271—21no—
166C22×C271—41no—
167C22×C271—41no—
168C22×C271—41no—
169C22×C271C2×D812yes—
170C22×C271C22×D411yescharacteristic
171C2×D271—21no—
172C2×D271—21no—
173C2×D271—22no—
174C2×D271—41no—
175C2×D271—41no—
176C2×D271—41no—
177C2×D271—41no—
178C2×D271—41no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C2×D271—84no—
183C2×D271—84no—
184C2×D271C2×D812yes—
185C4×C271—21no—
186C4×C271—41no—
187C4×C271—41no—
188C4×C271—41no—
189C4×C271C2×#4111yescharacteristic
190C4×C271H(5,𝔽2)11yescharacteristic
191Dic271—21no—
192Dic271—21no—
193Dic271—41no—
194Dic271C2×D812yes—
195C2×C271—22no—
196C2×C271—81no—
197C2×C271#54611yescharacteristic
198C2×C271C22×D811yescharacteristic
199C2×C271#57711yescharacteristic
200D271—22no—
201D271—42no—
202D271—81no—
203D271—81no—
204C271#329411yescharacteristic, complemented, Sylow
205#3294—2711nomaximal, Sylow
206#547—2711no—
207D42—2711no—
208#486—2711no—
209#480—2711no—
210#480D27111yescharacteristic, complemented
211#474—2711no—
212#466—2711no—
213#466—2711no—
214#459—2711no—
215#447—2711no—
216#447—2711no—
217C4×D8—2711no—
218C8×D4—2711no—
219C2×#102—2711no—
220C2×#102—2711no—
221C2×#41—5421no—
222C22×D4—2711no—
223C22×D4—2711no—
224C22×D4—5421no—
225C2×D8—2711no—
226C2×D8—2711no—
227C2×D8—2711no—
228C22×C8—2711no—
229C22×C8—2711no—
230C4⋊Q8—2711no—
231#127—2711no—
232#123—5421no—
233#123—5421no—
234C22⋊Q8—5421no—
235#121—2711no—
236#121—2711no—
237#121—2711no—
238#121—5421no—
239#121C2×D27111yescharacteristic
240#120—5421no—
241#120—5421no—
242C4×D4—2711no—
243C4×D4—2711no—
244C4×D4—2711no—
245C2×C4⋊C4—2711no—
246C2×C4⋊C4C2×D27111yescharacteristic, complemented
247#107—2711no—
248#107C2×D27111yescharacteristic, complemented
249#107C2×D27111yescharacteristic, complemented
250C4⋊C8—2711no—
251#102—2711no—
252#102—2711no—
253#102—2711no—
254#102—2711no—
255#102—2711no—
256#102—2711no—
257#102—2711no—
258#102—2711no—
259#102C2×D27111yescharacteristic, complemented
260#102C2×D27111yescharacteristic, complemented
261C22⋊C8—2711no—
262C22⋊C8C2×D27111yescharacteristic, complemented
263C4×C8—2711no—
264C24—5421no—
265C24—5421no—
266#41—10844no—
267C2×Q8—5421no—
268C2×D4—21no—
269C2×D4—2711no—
270C2×D4—2711no—
271C2×D4—2711no—
272C2×D4—2711no—
273C2×D4—5422no—
274C2×D4—5421no—
275C2×D4—5421no—
276C2×D4—5421no—
277C2×D4—5421no—
278C2×D4—5421no—
279C2×D4—5422no—
280C2×D4—5421no—
281C2×D4—5421no—
282C2×D4—10841no—
283C2×D4—10841no—
284C2×D4—10844no—
285C2×D4C22×D27111yescharacteristic
286C22×C4—21no—
287C22×C4—2711no—
288C22×C4—5421no—
289C22×C4—5421no—
290C22×C4—5421no—
291C22×C4C22×D27111yescharacteristic
292D8—5422no—
293D8—10841no—
294D8—10841no—
295C2×C8—2711no—
296C2×C8—2711no—
297C2×C8—10841no—
298C2×C8—10841no—
299C2×C8C22×D27111yescharacteristic
300C2×C8C22×D27111yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—2711no—
303C4⋊C4—2711no—
304C4⋊C4—5421no—
305C4⋊C4—5421no—
306C4⋊C4C22×D27111yescharacteristic
307C4⋊C4C22×D27111yescharacteristic
308C4⋊C4C22×D27111yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—2711no—
311C22⋊C4—2711no—
312C22⋊C4—5421no—
313C22⋊C4—5421no—
314C22⋊C4—5421no—
315C22⋊C4—5421no—
316C22⋊C4—5421no—
317C42—2711no—
318C23—21no—
319C23—2711no—
320C23—5421no—
321C23—5421no—
322C23—5421no—
323C23—10842no—
324C23—10842no—
325C23—10842no—
326C23—10842no—
327C23—21681no—
328C23—21681no—
329C23D4×D27111yescharacteristic
330Q8—10842no—
331D4—21no—
332D4—21no—
333D4—42no—
334D4—2714no—
335D4—2712no—
336D4—5421no—
337D4—5421no—
338D4—5421no—
339D4—5421no—
340D4—5421no—
341D4—10842no—
342D4—10842no—
343D4—10842no—
344D4—10842no—
345D4—10842no—
346D4—10842no—
347D4—21681no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—2711no—
355C2×C4—2711no—
356C2×C4—2711no—
357C2×C4—5421no—
358C2×C4—5421no—
359C2×C4—5421no—
360C2×C4—10842no—
361C2×C4—10842no—
362C2×C4—10842no—
363C2×C4D4×D27111yescharacteristic
364C2×C4C23×D27111yescharacteristic
365C8—21no—
366C8—21no—
367C8—5421no—
368C8—5421no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—2712no—
375C22—5421no—
376C22—5421no—
377C22—5422no—
378C22—10841no—
379C22—10841no—
380C22—10841no—
381C22—10841no—
382C22—10841no—
383C22—10841no—
384C22—10841no—
385C22—10841no—
386C22—21684no—
387C22—21684no—
388C22D8×D27112yes—
389C22C2×D4×D27111yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—2712no—
395C4—5421no—
396C4—5421no—
397C4—10841no—
398C4#41×D27111yescharacteristic
399C4#2257620611yescharacteristic
400C2—22no—
401C2—81no—
402C2—5422no—
403C2—10842no—
404C2—21681no—
405C2—21681no—
406C2#3177453911yescharacteristic
407C2C2×D8×D27111yescharacteristic
408C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#480×C271C2
socleC22×C271C22×D4
hypercenter#480D271
hypocenterC271#3294
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #31774548C2semidirect
2. #31774531C2semidirect
3. #31774242C2semidirect
4. #31774212C2semidirect
5. #31774176C2semidirect
6. #31774134C2semidirect
7. #31774131C2semidirect
8. #31774091C2semidirect
9. #31774033C2semidirect
10. #31774032C2semidirect
11. #31773979C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. C2×#22576080C2semidirect
14. #102×D271C2semidirect
15. #480×C271C2semidirect
21. C2×D2168C22semidirect
22. C2×#16864639C22semidirect
23. C22×C271⋊C8C22semidirect
24. C2×C8×D271C22semidirect
25. #22576159C22semidirect
31. #22576135C22semidirect
32. #22576134C22semidirect
37. #22576122C22semidirect
39. C4⋊C4×D271C22semidirect
40. #22576093C22semidirect
41. Dic271⋊C8C22semidirect
43. #22576082C22semidirect
44. #22576081C22semidirect
47. #22576080C22semidirect
48. #22576080C22semidirect
49. #22576080C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. C8×Dic271C22semidirect
53. C2×C4⋊C4×C271C22semidirect
54. #107×C271C22semidirect
55. #107×C271C22semidirect
56. #102×C271C22semidirect
57. #102×C271C22semidirect
58. C22⋊C8×C271C22semidirect
132. D1084#41semidirect
137. C22⋊D271D8semidirect
204. C271#3294semidirect
210. #480D271semidirect
246. C2×C4⋊C4C2×D271semidirect
248. #107C2×D271semidirect
249. #107C2×D271semidirect
259. #102C2×D271semidirect
260. #102C2×D271semidirect
262. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745482
#317745312
#317742422
#317742122
#317741762
#317741342
#317741312
#317740912
#317740332
#317740322
#317739792
(C22⋊D271)⋊C82
C2×#225760802
#102×D2712
#480×C2712
#3294271