Finite Group Databasebeta

← #46848325

Subgroups — #46848325

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#31774551C211yesmaximal, characteristic, complemented
2#31774256C211yesmaximal, characteristic, complemented
3#31774225C211yesmaximal, characteristic, complemented
4#31774213C211yesmaximal, characteristic, complemented
5#31774187C211yesmaximal, characteristic, complemented
6#31774148C211yesmaximal, characteristic, complemented
7#31774128C211yesmaximal, characteristic, complemented
8#31774109C211yesmaximal, characteristic, complemented
9#31774042C211yesmaximal, characteristic, complemented
10#31774031C211yesmaximal, characteristic, complemented
11#31774007C211yesmaximal, characteristic, complemented
12#31773969C211yesmaximal, characteristic, complemented
13#31773898C211yesmaximal, characteristic, complemented
14C2×#22576081C211yesmaximal, characteristic, complemented
15#546×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×#16864641C2211yescharacteristic, complemented
18C2×#16864639C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20C2×#16864638C2211yescharacteristic, complemented
21C22×C271⋊C8C2211yescharacteristic
22#22576160C2211yescharacteristic, complemented
23#22576144—21no—
24#22576144—21no—
25#22576140—21no—
26#22576134C2211yescharacteristic
27#22576134C2211yescharacteristic, complemented
28C4×D1084C2211yescharacteristic, complemented
29C2×C4⋊Dic271C2211yescharacteristic
30C2×C4⋊Dic271C2211yescharacteristic, complemented
31#22576091C2211yescharacteristic, complemented
32#22576091C2211yescharacteristic, complemented
33#22576091C2211yescharacteristic, complemented
34C4⋊(C271⋊C8)C2211yescharacteristic, complemented
35#22576084C2211yescharacteristic, complemented
36#22576084C2211yescharacteristic, complemented
37#22576084C2211yescharacteristic, complemented
38#22576081C2211yescharacteristic
39#22576081C2211yescharacteristic
40#22576081C2211yescharacteristic
41#22576081C2211yescharacteristic
42#22576081C2211yescharacteristic
43#22576081C2211yescharacteristic
44#22576080C2211yescharacteristic, complemented
45C22⋊(C271⋊C8)C2211yescharacteristic
46C22⋊(C271⋊C8)C2211yescharacteristic, complemented
47#22576071C2211yescharacteristic, complemented
48C2×#41×C271C2211yescharacteristic
49C22×D4×C271C2211yescharacteristic
50#124×C271C2211yescharacteristic
51#123×C271—21no—
52C22⋊Q8×C271C2211yescharacteristic
53#121×C271—21no—
54#121×C271C2211yescharacteristic
55#120×C271—21no—
56C4×D4×C271C2211yescharacteristic
57C4×D4×C271C2211yescharacteristic
58C2×C22⋊C4×C271—21no—
59#16864653—44no—
60C2×Dic542—21no—
61C2×D1084C2311yescharacteristic
62C2×C22⋊D271—21no—
63C2×C22⋊D271—21no—
64C22×Dic271—21no—
65C22×Dic271—21no—
66C2×C4×D271—21no—
67#16864641—41no—
68#16864639—22no—
69#16864639—41no—
70#16864638—41no—
71C2×C271⋊C8—41no—
72C2×C271⋊C8C2311yescharacteristic
73C2×C271⋊C8C2311yescharacteristic
74C2×C271⋊C8C2311yescharacteristic
75C2×C271⋊C8C2311yescharacteristic
76C4⋊Dic271—21no—
77C4⋊Dic271—21no—
78C4⋊Dic271—21no—
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82(C2×D271)⋊C4—21no—
83(C2×D271)⋊C4—21no—
84(C2×D271)⋊C4—21no—
85C24×C271—21no—
86#41×C271—22no—
87#41×C271—41no—
88C2×Q8×C271C2311yescharacteristic
89C2×D4×C271—22no—
90C2×D4×C271—21no—
91C2×D4×C271—21no—
92C2×D4×C271—21no—
93C2×D4×C271—41no—
94C2×D4×C271D411yescharacteristic
95C2×D4×C271C2311yescharacteristic
96C2×D4×C271C2311yescharacteristic
97C22×C4×C271—21no—
98C22×C4×C271—21no—
99C22×C4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C4⋊C4×C271—21no—
102C4⋊C4×C271D411yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C22⋊C4×C271—21no—
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C22⋊C4×C271—21no—
108C22⋊C4×C271—21no—
109C22⋊C4×C271D411yescharacteristic
110C22⋊C4×C271D411yescharacteristic
111C42×C271C2311yescharacteristic
112C22×D271—21no—
113Dic542—42no—
114D1084—21no—
115D1084—21no—
116C22⋊D271—42no—
117C22⋊D271—42no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—42no—
122C2×Dic271—42no—
123C4×D271—42no—
124C271⋊C8—21no—
125C271⋊C8—21no—
126C271⋊C8—21no—
127C271⋊C8—21no—
128C23×C271—21no—
129C23×C271—21no—
130C23×C271—42no—
131C23×C271—42no—
132C23×C271—81no—
133C23×C271C2×D411yescharacteristic
134C23×C271C2×D411yescharacteristic
135Q8×C271—21no—
136Q8×C271—21no—
137D4×C271—22no—
138D4×C271—21no—
139D4×C271—21no—
140D4×C271—21no—
141D4×C271—42no—
142D4×C271—42no—
143D4×C271—81no—
144D4×C271#4112yescomplemented
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271—42no—
152C2×C4×C271—81no—
153C2×C4×C271C2×D411yescharacteristic
154C2×C4×C271C2×D411yescharacteristic
155C2×C4×C271C2×D411yescharacteristic
156C2×C4×C271C2×D411yescharacteristic
157C2×C4×C271C2411yescharacteristic
158C22×C271—21no—
159C22×C271—21no—
160C22×C271—21no—
161C22×C271—22no—
162C22×C271—21no—
163C22×C271—21no—
164C22×C271—41no—
165C22×C271—41no—
166C22×C271—41no—
167C22×C271—41no—
168C22×C271—41no—
169C22×C271—84no—
170C22×C271AGL(1,ℤ/8)12yes—
171C22×C271C22×D411yescharacteristic
172C2×D271—41no—
173C2×D271—41no—
174C2×D271—41no—
175C4×C271—21no—
176C4×C271—21no—
177C4×C271—21no—
178C4×C271—21no—
179C4×C271—41no—
180C4×C271—41no—
181C4×C271C2×#4111yescharacteristic
182C4×C271H(5,𝔽2)11yescharacteristic
183Dic271—41no—
184Dic271—41no—
185Dic271—41no—
186C2×C271—22no—
187C2×C271—41no—
188C2×C271—42no—
189C2×C271—81no—
190C2×C271#54611yescharacteristic
191C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C271#57711yescharacteristic
193D271—81no—
194C271#330611yescharacteristic, complemented, Sylow
195#3306—2711nomaximal, Sylow
196#547—2711no—
197#546D27111yescharacteristic, complemented
198#489—2711no—
199#483—2711no—
200#481—2711no—
201#476—2711no—
202#469—2711no—
203#465—2711no—
204#463—2711no—
205#449—2711no—
206#447—2711no—
207#442—2711no—
208#435—2711no—
209#417—2711no—
210C2×#102—2711no—
211C2×#41—5421no—
212C2×#41C2×D27111yescharacteristic
213C22×D4C2×D27111yescharacteristic
214C2×SD16—2711no—
215C2×D8—2711no—
216C2×D8—2711no—
217C2×M16—2711no—
218C22×C8—2711no—
219C4⋊Q8—2711no—
220#124C2×D27111yescharacteristic
221#123—21no—
222#123—5421no—
223#123—5421no—
224C22⋊Q8—5421no—
225C22⋊Q8C2×D27111yescharacteristic
226#121—21no—
227#121—2711no—
228#121—2711no—
229#121C2×D27111yescharacteristic
230#120—21no—
231C4×D4—2711no—
232C4×D4C2×D27111yescharacteristic
233C4×D4C2×D27111yescharacteristic
234C2×C4⋊C4—2711no—
235C2×C4⋊C4—2711no—
236C2×C22⋊C4—21no—
237#106—2711no—
238#106—2711no—
239#106—2711no—
240C4⋊C8—2711no—
241#103—2711no—
242#103—2711no—
243#103—2711no—
244#102—2711no—
245#102—2711no—
246#102—2711no—
247#102—2711no—
248#102—2711no—
249#102—2711no—
250#102—2711no—
251C22⋊C8—2711no—
252C22⋊C8—2711no—
253#97—2711no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—10844no—
258C2×Q8—5421no—
259C2×Q8C22×D27111yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—2711no—
266C2×D4—5421no—
267C2×D4—5421no—
268C2×D4C22⋊D27111yescharacteristic
269C2×D4C22×D27111yescharacteristic
270C2×D4C22×D27111yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—5421no—
274C22×C4—5421no—
275C22×C4—5421no—
276C22×C4C22×D27111yescharacteristic
277C22×C4C22×D27111yescharacteristic
278SD16—10841no—
279D8—5422no—
280D8—10841no—
281M16—10841no—
282C2×C8—2711no—
283C2×C8—2711no—
284C2×C8—2711no—
285C2×C8—2711no—
286C2×C8—10841no—
287C4⋊C4—21no—
288C4⋊C4—2711no—
289C4⋊C4—2711no—
290C4⋊C4—2711no—
291C4⋊C4—5421no—
292C4⋊C4—5421no—
293C4⋊C4—5421no—
294C4⋊C4C22⋊D27111yescharacteristic
295C4⋊C4C22×D27111yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—5421no—
302C22⋊C4—5421no—
303C22⋊C4—5421no—
304C22⋊C4C22⋊D27111yescharacteristic
305C22⋊C4C22⋊D27111yescharacteristic
306C42C22×D27111yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—5421no—
313C23C2×C22⋊D27111yescharacteristic
314C23C2×C22⋊D27111yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—10842no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—5421no—
326D4—5421no—
327D4—10842no—
328D4—10842no—
329D4#1686465312yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—5421no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—10842no—
342C2×C4—10842no—
343C2×C4—10842no—
344C2×C4C2×C22⋊D27111yescharacteristic
345C2×C4C2×C22⋊D27111yescharacteristic
346C2×C4C2×C22⋊D27111yescharacteristic
347C2×C4C2×C22⋊D27111yescharacteristic
348C2×C4C23×D27111yescharacteristic
349C8—5421no—
350C8—5421no—
351C8—5421no—
352C8—5421no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—10841no—
366C22—10841no—
367C22—10841no—
368C22#2257618512yes—
369C22C22×C22⋊D27111yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—10841no—
377C4—10841no—
378C4—10841no—
379C4C2×#1686465311yescharacteristic
380C4#2257620511yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—21681no—
386C2#3177454111yescharacteristic
387C2C2×#2257618511yescharacteristic
388C2#3177470811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#546×C271C2
socleC22×C271C22×D4
hypercenter#546D271
hypocenterC271#3306
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #31774551C2semidirect
2. #31774256C2semidirect
3. #31774225C2semidirect
4. #31774213C2semidirect
5. #31774187C2semidirect
6. #31774148C2semidirect
7. #31774128C2semidirect
8. #31774109C2semidirect
9. #31774042C2semidirect
10. #31774031C2semidirect
11. #31774007C2semidirect
12. #31773969C2semidirect
13. #31773898C2semidirect
14. C2×#22576081C2semidirect
15. #546×C271C2semidirect
17. C2×#16864641C22semidirect
20. C2×#16864638C22semidirect
22. #22576160C22semidirect
27. #22576134C22semidirect
28. C4×D1084C22semidirect
30. C2×C4⋊Dic271C22semidirect
31. #22576091C22semidirect
32. #22576091C22semidirect
33. #22576091C22semidirect
34. C4⋊(C271⋊C8)C22semidirect
35. #22576084C22semidirect
36. #22576084C22semidirect
37. #22576084C22semidirect
44. #22576080C22semidirect
46. C22⋊(C271⋊C8)C22semidirect
47. #22576071C22semidirect
144. D4×C271#41semidirect
194. C271#3306semidirect
197. #546D271semidirect
329. D4#16864653semidirect

Maximal subgroups

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

subgroupindex
#317745512
#317742562
#317742252
#317742132
#317741872
#317741482
#317741282
#317741092
#317740422
#317740312
#317740072
#317739692
#317738982
C2×#225760812
#546×C2712
#3306271