Finite Group Databasebeta

← #46848318

Subgroups — #46848318

Order 34688.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774547C211yesmaximal, characteristic, complemented
2#31774535C211yesmaximal, characteristic, complemented
3#31774258C211yesmaximal, characteristic, complemented
4#31774226C211yesmaximal, characteristic, complemented
5#31774216C211yesmaximal, characteristic, complemented
6#31774185C211yesmaximal, characteristic, complemented
7#31774148C211yesmaximal, characteristic, complemented
8#31774123C211yesmaximal, characteristic, complemented
9#31774114C211yesmaximal, characteristic, complemented
10#31774029C211yesmaximal, characteristic, complemented
11#31774009C211yesmaximal, characteristic, complemented
12#31773971C211yesmaximal, characteristic, complemented
13#31773898C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×D4×D271C2211yescharacteristic
18C2×#16864639C2211yescharacteristic
19C2×#16864638C2211yescharacteristic, complemented
20C2×C8×D271C2211yescharacteristic
21#22576159C2211yescharacteristic, complemented
22#22576149C2211yescharacteristic
23#22576143—21no—
24#22576142—21no—
25#22576141—21no—
26#22576137—21no—
27#22576136C2211yescharacteristic
28#22576134C2211yescharacteristic
29#22576132—21no—
30#22576131C2211yescharacteristic
31#22576128—21no—
32D4×Dic271C2211yescharacteristic
33D4×Dic271C2211yescharacteristic, complemented
34C4×C22⋊D271C2211yescharacteristic
35C2×C4⋊Dic271C2211yescharacteristic, complemented
36C4⋊C4×D271C2211yescharacteristic
37C2×(C2×D271)⋊C4—21no—
38#22576092C2211yescharacteristic, complemented
39#22576092C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42#22576084C2211yescharacteristic, complemented
43#22576084C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic, complemented
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576070C2211yescharacteristic, complemented
52C2×#41×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic, complemented
54C2×D8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic, complemented
56#103×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864653—44no—
61C2×Dic542—21no—
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271D411yescharacteristic
67D4×D271—22no—
68D4×D271—41no—
69C22×Dic271—21no—
70C22×Dic271—21no—
71C22×Dic271—21no—
72C2×C4×D271—21no—
73C2×C4×D271C2311yescharacteristic
74#16864639—41no—
75#16864638—41no—
76C2×C271⋊C8C2311yescharacteristic
77C2×C271⋊C8C2311yescharacteristic
78C8×D271—41no—
79C4⋊Dic271—21no—
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4D411yescharacteristic
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271D411yescharacteristic
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4D411yescharacteristic
96C4×Dic271C2311yescharacteristic
97#41×C271—22no—
98#41×C271—41no—
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271—21no—
102C2×D4×C271C2311yescharacteristic
103C2×D4×C271C2311yescharacteristic
104C22×C4×C271—21no—
105C22×C4×C271C2311yescharacteristic
106SD16×C271—41no—
107D8×C271—22no—
108C2×C8×C271C2311yescharacteristic
109C2×C8×C271C2311yescharacteristic
110C4⋊C4×C271C2311yescharacteristic
111C22⋊C4×C271—21no—
112C22×D271—21no—
113C22×D271—42no—
114C22×D271—42no—
115C22×D271—81no—
116C22×D271C2×D411yescharacteristic
117Dic542—42no—
118D1084—21no—
119D1084—21no—
120C22⋊D271—22no—
121C22⋊D271—42no—
122C22⋊D271—42no—
123C22⋊D271—81no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—42no—
128C2×Dic271—42no—
129C2×Dic271—81no—
130C2×Dic271C2×D411yescharacteristic
131C2×Dic271C2×D411yescharacteristic
132C2×Dic271C2×D411yescharacteristic
133C4×D271—21no—
134C4×D271—21no—
135C4×D271—42no—
136C271⋊C8—21no—
137C271⋊C8—21no—
138C23×C271—21no—
139C23×C271—21no—
140C23×C271C2×D411yescharacteristic
141Q8×C271—21no—
142Q8×C271—21no—
143D4×C271—21no—
144D4×C271—21no—
145D4×C271—21no—
146D4×C271—42no—
147D4×C271—42no—
148D4×C271#4112yescomplemented
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271—42no—
154C2×C4×C271C2×D411yescharacteristic
155C2×C4×C271C2411yescharacteristic
156C8×C271—21no—
157C8×C271—21no—
158C22×C271—21no—
159C22×C271—22no—
160C22×C271—21no—
161C22×C271—21no—
162C22×C271—41no—
163C22×C271—41no—
164C22×C271—41no—
165C22×C271—41no—
166C22×C271—41no—
167C22×C271C22×D411yescharacteristic
168C2×D271—21no—
169C2×D271—21no—
170C2×D271—41no—
171C2×D271—41no—
172C2×D271—41no—
173C2×D271—84no—
174C2×D271AGL(1,ℤ/8)12yes—
175C4×C271—21no—
176C4×C271—41no—
177C4×C271—41no—
178C4×C271C2×#4111yescharacteristic
179C4×C271H(5,𝔽2)11yescharacteristic
180Dic271—21no—
181Dic271—21no—
182Dic271—21no—
183Dic271—41no—
184Dic271—41no—
185Dic271—41no—
186C2×C271—42no—
187C2×C271—41no—
188C2×C271—81no—
189C2×C271#54611yescharacteristic
190C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C271#57711yescharacteristic
192D271—22no—
193D271—81no—
194C271#330611yescharacteristic, complemented, Sylow
195#3306—2711nomaximal, Sylow
196#547—2711no—
197#546—2711no—
198#489—2711no—
199#483—2711no—
200#481—2711no—
201#476—2711no—
202#469—2711no—
203#465—2711no—
204#463—2711no—
205#449D27111yescharacteristic, complemented
206#447—2711no—
207#442—2711no—
208#435—2711no—
209#417—2711no—
210C2×#102—2711no—
211C2×#41—5421no—
212C2×#41C2×D27111yescharacteristic
213C22×D4—2711no—
214C2×SD16C2×D27111yescharacteristic, complemented
215C2×D8—2711no—
216C2×D8C2×D27111yescharacteristic
217C2×M16—2711no—
218C22×C8—2711no—
219C4⋊Q8—2711no—
220#124—2711no—
221#123—5421no—
222#123—5421no—
223#123—5421no—
224C22⋊Q8—2711no—
225C22⋊Q8—5421no—
226#121—2711no—
227#121—2711no—
228#121—5421no—
229#121C2×D27111yescharacteristic, complemented
230#120—5421no—
231C4×D4—2711no—
232C4×D4—2711no—
233C4×D4—2711no—
234C2×C4⋊C4—2711no—
235C2×C4⋊C4—2711no—
236C2×C22⋊C4—5421no—
237#106—2711no—
238#106—2711no—
239#106—2711no—
240C4⋊C8—2711no—
241#103—2711no—
242#103—2711no—
243#103C2×D27111yescharacteristic, complemented
244#102—2711no—
245#102—2711no—
246#102—2711no—
247#102—2711no—
248#102—2711no—
249#102—2711no—
250#102C2×D27111yescharacteristic
251C22⋊C8—2711no—
252C22⋊C8C2×D27111yescharacteristic, complemented
253#97—2711no—
254C24—5421no—
255#41—22no—
256#41—41no—
257#41—10844no—
258C2×Q8—5421no—
259C2×Q8C22×D27111yescharacteristic
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—2711no—
263C2×D4—2711no—
264C2×D4—5421no—
265C2×D4—5422no—
266C2×D4—5421no—
267C2×D4—5421no—
268C2×D4—10841no—
269C2×D4C22×D27111yescharacteristic
270C2×D4C22×D27111yescharacteristic
271C22×C4—21no—
272C22×C4—2711no—
273C22×C4—5421no—
274C22×C4—5421no—
275C22×C4—5421no—
276C22×C4—5421no—
277C22×C4C22×D27111yescharacteristic
278SD16—41no—
279D8—22no—
280D8—10841no—
281M16—10841no—
282C2×C8—2711no—
283C2×C8—2711no—
284C2×C8—10841no—
285C2×C8C22×D27111yescharacteristic
286C2×C8C22×D27111yescharacteristic
287C4⋊C4—2711no—
288C4⋊C4—2711no—
289C4⋊C4—2711no—
290C4⋊C4—2711no—
291C4⋊C4—5421no—
292C4⋊C4—5421no—
293C4⋊C4—5421no—
294C4⋊C4—5421no—
295C4⋊C4C22×D27111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—2711no—
298C22⋊C4—2711no—
299C22⋊C4—5421no—
300C22⋊C4—5421no—
301C22⋊C4—5421no—
302C22⋊C4—5421no—
303C22⋊C4—5421no—
304C22⋊C4—5421no—
305C22⋊C4—5421no—
306C42—2711no—
307C23—21no—
308C23—21no—
309C23—2711no—
310C23—5421no—
311C23—10842no—
312C23—10842no—
313C23—21681no—
314C23D4×D27111yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—10842no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—5422no—
324D4—5421no—
325D4—5421no—
326D4—10842no—
327D4—10842no—
328D4—21681no—
329D4#1686465312yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—2711no—
336C2×C4—2711no—
337C2×C4—2711no—
338C2×C4—5421no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—5421no—
342C2×C4—5421no—
343C2×C4—10842no—
344C2×C4—10842no—
345C2×C4—10842no—
346C2×C4—21681no—
347C2×C4D4×D27111yescharacteristic
348C2×C4C23×D27111yescharacteristic
349C8—21no—
350C8—21no—
351C8—5421no—
352C8—5421no—
353C22—21no—
354C22—22no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—2712no—
363C22—5421no—
364C22—5421no—
365C22—10841no—
366C22—10841no—
367C22—10841no—
368C22—21684no—
369C22C2×D4×D27111yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—5421no—
374C4—5421no—
375C4—5421no—
376C4—10841no—
377C4—10841no—
378C4—10841no—
379C4C2×#1686465311yescharacteristic
380C4#2257620511yescharacteristic
381C2—41no—
382C2—42no—
383C2—81no—
384C2—5422no—
385C2—21681no—
386C2#3177453411yescharacteristic
387C2AGL(1,ℤ/8)×D27111yescharacteristic
388C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3306
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31774547C2semidirect
2. #31774535C2semidirect
3. #31774258C2semidirect
4. #31774226C2semidirect
5. #31774216C2semidirect
6. #31774185C2semidirect
7. #31774148C2semidirect
8. #31774123C2semidirect
9. #31774114C2semidirect
10. #31774029C2semidirect
11. #31774009C2semidirect
12. #31773971C2semidirect
13. #31773898C2semidirect
14. #102×D271C2semidirect
15. #449×C271C2semidirect
19. C2×#16864638C22semidirect
21. #22576159C22semidirect
33. D4×Dic271C22semidirect
35. C2×C4⋊Dic271C22semidirect
38. #22576092C22semidirect
39. #22576092C22semidirect
40. #22576091C22semidirect
41. Dic271⋊C8C22semidirect
42. #22576084C22semidirect
43. #22576084C22semidirect
48. #22576081C22semidirect
51. #22576070C22semidirect
53. C2×SD16×C271C22semidirect
55. #121×C271C22semidirect
56. #103×C271C22semidirect
58. C22⋊C8×C271C22semidirect
148. D4×C271#41semidirect
194. C271#3306semidirect
205. #449D271semidirect
214. C2×SD16C2×D271semidirect
229. #121C2×D271semidirect
243. #103C2×D271semidirect
252. C22⋊C8C2×D271semidirect
329. D4#16864653semidirect

Maximal subgroups

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

subgroupindex
#317745472
#317745352
#317742582
#317742262
#317742162
#317741852
#317741482
#317741232
#317741142
#317740292
#317740092
#317739712
#317738982
#102×D2712
#449×C2712
#3306271