Finite Group Databasebeta

← #46848397

Subgroups — #46848397

Order 34688.

402 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#31774621C211yesmaximal, characteristic, complemented
2#31774542C211yesmaximal, characteristic
3#31774259C211yesmaximal, characteristic, complemented
4#31774241C211yesmaximal, characteristic, complemented
5#31774223C211yesmaximal, characteristic, complemented
6#31774156C211yesmaximal, characteristic, complemented
7#31774127C211yesmaximal, characteristic, complemented
8#31774120C211yesmaximal, characteristic, complemented
9#31774060C211yesmaximal, characteristic, complemented
10#31774049C211yesmaximal, characteristic, complemented
11#31773997C211yesmaximal, characteristic, complemented
12#31773967C211yesmaximal, characteristic, complemented
13#31773900C211yesmaximal, characteristic, complemented
14#31773890C211yesmaximal, characteristic, complemented
15#476×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18C2×Dic1084C2211yescharacteristic, complemented
19C2×#16864643C2211yescharacteristic
20C2×#16864642C2211yescharacteristic
21C2×#16864637C2211yescharacteristic
22C2×C8×D271C2211yescharacteristic, complemented
23#22576155—21no—
24#22576155—21no—
25#22576149—21no—
26#22576148C2211yescharacteristic
27#22576144—21no—
28#22576140C2211yescharacteristic
29#22576140C2211yescharacteristic
30#22576140C2211yescharacteristic, complemented
31#22576135C2211yescharacteristic
32#22576134—21no—
33#22576130—21no—
34Q8×Dic271—21no—
35C4×Dic542C2211yescharacteristic, complemented
36C4×D1084C2211yescharacteristic
37#22576118C2211yescharacteristic
38#22576118C2211yescharacteristic, complemented
39C2×(C2×D271)⋊C4—21no—
40#22576093C2211yescharacteristic, complemented
41#22576085C2211yescharacteristic, complemented
42#22576083C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576083C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51#22576071C2211yescharacteristic, complemented
52C4⋊Q8×C271C2211yescharacteristic, complemented
53C4×D4×C271C2211yescharacteristic
54#106×C271C2211yescharacteristic, complemented
55#106×C271C2211yescharacteristic, complemented
56C4⋊C8×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59C23×D271—21no—
60#16864653—22no—
61#16864653—41no—
62C2×Dic542C2311yescharacteristic
63C2×Dic542C2311yescharacteristic
64C2×D1084—21no—
65C2×D1084D411yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69D4×D271—22no—
70D4×D271—41no—
71C22×Dic271—21no—
72C2×C4×D271C2311yescharacteristic
73C2×C4×D271C2311yescharacteristic
74Dic1084—41no—
75#16864643—41no—
76#16864642—22no—
77#16864637—41no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C8×D271—41no—
81C4⋊Dic271—21no—
82C4⋊Dic271—21no—
83C4⋊Dic271—21no—
84C4⋊Dic271D411yescharacteristic
85C4⋊Dic271C2311yescharacteristic
86Dic271⋊C4—21no—
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)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4D411yescharacteristic
96(C2×D271)⋊C4D411yescharacteristic
97C4×Dic271—21no—
98C4×Dic271—21no—
99C4×Dic271—21no—
100C2×Q8×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C22×C4×C271—21no—
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C4⋊C4×C271—21no—
106C4⋊C4×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C22⋊C4×C271—21no—
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112C22×D271—42no—
113C22×D271—42no—
114C22×D271—81no—
115C22×D271C2×D411yescharacteristic
116C22×D271C2×D411yescharacteristic
117Dic542—21no—
118Dic542—21no—
119Dic542—21no—
120Dic542—21no—
121D1084—21no—
122D1084—21no—
123D1084—41no—
124D1084—81no—
125C22⋊D271—42no—
126C22⋊D271—42no—
127C2×Dic271—21no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—42no—
131C2×Dic271C2×D411yescharacteristic
132C2×Dic271C2×D411yescharacteristic
133C4×D271—21no—
134C4×D271—21no—
135C4×D271—21no—
136C4×D271—21no—
137C271⋊C8—21no—
138C271⋊C8—21no—
139C23×C271—21no—
140Q8×C271—42no—
141D4×C271—21no—
142D4×C271#4112yes—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—81no—
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2411yescharacteristic
150C8×C271—21no—
151C8×C271—21no—
152C22×C271—22no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271C22×D411yescharacteristic
156C2×D271—21no—
157C2×D271—21no—
158C2×D271—21no—
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—21no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C2×D271—84no—
166C4×C271—21no—
167C4×C271—21no—
168C4×C271—41no—
169C4×C271—41no—
170C4×C271—41no—
171C4×C271C2×#4111yescharacteristic
172C4×C271H(5,𝔽2)11yescharacteristic
173Dic271—21no—
174Dic271—21no—
175Dic271—41no—
176Dic271—41no—
177Dic271—41no—
178C2×C271—42no—
179C2×C271#54611yescharacteristic
180C2×C271#57511yescharacteristic
181C2×C271#57711yescharacteristic
182D271—41no—
183D271—41no—
184D271—81no—
185C271#331111yescharacteristic, complemented, Sylow
186#3311—2711nomaximal, Sylow
187#556—2711no—
188#546—2711no—
189#489—2711no—
190#485—2711no—
191#482—2711no—
192#476D27111yescharacteristic, complemented
193#470—2711no—
194#465—2711no—
195#464—2711no—
196#452—2711no—
197#450—2711no—
198#440—2711no—
199#435—2711no—
200#418—2711no—
201#416—2711no—
202C2×#41—2711no—
203C22×D4—2711no—
204C2×Q16—2711no—
205C2×SD16—2711no—
206C2×SD16—2711no—
207C2×M16—2711no—
208C22×C8—2711no—
209C4⋊Q8C2×D27111yescharacteristic, complemented
210#126—5421no—
211#126—5421no—
212#124—2711no—
213#124—5421no—
214#123—5421no—
215C22⋊Q8—2711no—
216C22⋊Q8—2711no—
217C22⋊Q8—2711no—
218#121—2711no—
219#121—5421no—
220#120—5421no—
221C4×Q8—2711no—
222C4×Q8—5421no—
223C4×D4—2711no—
224C4×D4C2×D27111yescharacteristic
225#117—2711no—
226#117—2711no—
227C2×C22⋊C4—5421no—
228#107—2711no—
229#106C2×D27111yescharacteristic, complemented
230#106C2×D27111yescharacteristic, complemented
231C4⋊C8C2×D27111yescharacteristic, complemented
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⋊C8—2711no—
244#97—2711no—
245C24—5421no—
246#41—5422no—
247#41—10841no—
248C2×Q8—21no—
249C2×Q8—2711no—
250C2×Q8—2711no—
251C2×D4—2711no—
252C2×D4—2711no—
253C2×D4—5421no—
254C2×D4—5421no—
255C2×D4—5422no—
256C2×D4—5421no—
257C2×D4—10841no—
258C2×D4C22×D27111yescharacteristic
259C22×C4—21no—
260C22×C4—2711no—
261C22×C4—2711no—
262C22×C4—5421no—
263Q16—10841no—
264SD16—5422no—
265SD16—10841no—
266M16—10841no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8—10841no—
270C2×C8C22×D27111yescharacteristic
271C2×C8C22×D27111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—5421no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4C22×D27111yescharacteristic
280C4⋊C4C22×D27111yescharacteristic
281C4⋊C4C22×D27111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—2711no—
284C22⋊C4—2711no—
285C22⋊C4—5421no—
286C22⋊C4—5421no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C42—5421no—
294C42—5421no—
295C42—5421no—
296C42C22×D27111yescharacteristic
297C23—21no—
298C23—2711no—
299C23—2711no—
300C23—5421no—
301C23—10842no—
302C23—10842no—
303C23—21681no—
304Q8—42no—
305Q8—5421no—
306Q8—5421no—
307Q8—5421no—
308Q8—5421no—
309D4—21no—
310D4—5421no—
311D4—5421no—
312D4—10841no—
313D4—10842no—
314D4—10842no—
315D4—21681no—
316D4#1686465512yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
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—5421no—
330C2×C4—10842no—
331C2×C4D4×D27111yescharacteristic
332C2×C4D4×D27111yescharacteristic
333C2×C4C23×D27111yescharacteristic
334C8—21no—
335C8—21no—
336C8—5421no—
337C8—5421no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—5421no—
342C22—5421no—
343C22—5421no—
344C22—5421no—
345C22—5421no—
346C22—5421no—
347C22—10841no—
348C22—10841no—
349C22—10841no—
350C22—21684no—
351C22C2×D4×D27111yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5421no—
358C4—5421no—
359C4—10841no—
360C4—10841no—
361C4—10841no—
362C4C2×#1686465511yescharacteristic
363C4#2257620611yescharacteristic
364C2—42no—
365C2—10841no—
366C2—10841no—
367C2—21681no—
368C2#3177453811yescharacteristic
369C2#3177469311yescharacteristic
370C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#476×C271C2
socleC22×C271C22×D4
prime radical#31774542C2
hypercenter#476D271
hypocenterC271#3311
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #31774621C2semidirect
3. #31774259C2semidirect
4. #31774241C2semidirect
5. #31774223C2semidirect
6. #31774156C2semidirect
7. #31774127C2semidirect
8. #31774120C2semidirect
9. #31774060C2semidirect
10. #31774049C2semidirect
11. #31773997C2semidirect
12. #31773967C2semidirect
13. #31773900C2semidirect
14. #31773890C2semidirect
15. #476×C271C2semidirect
18. C2×Dic1084C22semidirect
22. C2×C8×D271C22semidirect
30. #22576140C22semidirect
35. C4×Dic542C22semidirect
38. #22576118C22semidirect
40. #22576093C22semidirect
41. #22576085C22semidirect
42. #22576083C22semidirect
43. #22576083C22semidirect
44. #22576083C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. #22576071C22semidirect
52. C4⋊Q8×C271C22semidirect
54. #106×C271C22semidirect
55. #106×C271C22semidirect
56. C4⋊C8×C271C22semidirect
185. C271#3311semidirect
192. #476D271semidirect
209. C4⋊Q8C2×D271semidirect
229. #106C2×D271semidirect
230. #106C2×D271semidirect
231. C4⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746212
#317745422
#317742592
#317742412
#317742232
#317741562
#317741272
#317741202
#317740602
#317740492
#317739972
#317739672
#317739002
#317738902
#476×C2712
#3311271