Finite Group Databasebeta

← #46848288

Subgroups — #46848288

Order 34688.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774610C211yesmaximal, characteristic, complemented
2#31774546C211yesmaximal, characteristic, complemented
3#31774246C211yesmaximal, characteristic, complemented
4#31774238C211yesmaximal, characteristic, complemented
5#31774236C211yesmaximal, characteristic, complemented
6#31774185C211yesmaximal, characteristic, complemented
7#31774129C211yesmaximal, characteristic, complemented
8#31774126C211yesmaximal, characteristic, complemented
9#31774101C211yesmaximal, characteristic, complemented
10#31774043C211yesmaximal, characteristic, complemented
11#31774042C211yesmaximal, characteristic, complemented
12#31773977C211yesmaximal, characteristic, complemented
13C8×C22⋊D271C211yesmaximal, characteristic, complemented
14#31773890C211yesmaximal, characteristic, complemented
15#416×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×#16864643C2211yescharacteristic
18C2×#16864641C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21#22576159C2211yescharacteristic
22#22576158C2211yescharacteristic
23#22576148—21no—
24#22576146—21no—
25#22576143—21no—
26#22576141—21no—
27#22576139C2211yescharacteristic
28#22576136C2211yescharacteristic
29#22576135C2211yescharacteristic
30#22576134C2211yescharacteristic
31#22576133—21no—
32#22576132—21no—
33D4×Dic271C2211yescharacteristic
34#22576122C2211yescharacteristic
35C4×C22⋊D271C2211yescharacteristic
36#22576118C2211yescharacteristic
37#22576117—21no—
38C2×Dic271⋊C4—21no—
39#22576093C2211yescharacteristic
40#22576092C2211yescharacteristic
41#22576091C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576083C2211yescharacteristic
46#22576083C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54#117×C271C2211yescharacteristic
55#103×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864653—22no—
60#16864653—41no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63C2×D1084—21no—
64C2×D1084C2311yescharacteristic
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271D411yescharacteristic
68C22×Dic271—21no—
69C22×Dic271—21no—
70C2×C4×D271—21no—
71C2×C4×D271C2311yescharacteristic
72#16864643—41no—
73#16864641—41no—
74#16864639—22no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C8×D271—41no—
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4D411yescharacteristic
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271D411yescharacteristic
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)⋊C4D411yescharacteristic
95C4×Dic271—21no—
96C4×Dic271C2311yescharacteristic
97#41×C271—22no—
98#41×C271—41no—
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C22×C4×C271—21no—
103C22×C4×C271C2311yescharacteristic
104C2×C8×C271—41no—
105C2×C8×C271C2311yescharacteristic
106C2×C8×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C22⋊C4×C271—21no—
110C42×C271—21no—
111C22×D271—21no—
112C22×D271C2×D411yescharacteristic
113Dic542—21no—
114Dic542—21no—
115Dic542—81no—
116D1084—21no—
117D1084—21no—
118D1084—81no—
119C22⋊D271—22no—
120C22⋊D271—42no—
121C22⋊D271—81no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—42no—
126C2×Dic271—42no—
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—42no—
133C271⋊C8—21no—
134C271⋊C8—21no—
135C23×C271—21no—
136C23×C271C2×D411yescharacteristic
137Q8×C271—21no—
138Q8×C271—21no—
139D4×C271—21no—
140D4×C271—42no—
141D4×C271#4112yes—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—42no—
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2411yescharacteristic
150C8×C271—21no—
151C8×C271—21no—
152C22×C271—21no—
153C22×C271—22no—
154C22×C271—21no—
155C22×C271—21no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271C22×D411yescharacteristic
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—21no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C4×C271—21no—
166C4×C271—41no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271C2×#4111yescharacteristic
170C4×C271H(5,𝔽2)11yescharacteristic
171Dic271—21no—
172Dic271—21no—
173Dic271—41no—
174Dic271—41no—
175Dic271—41no—
176Dic271#13512yes—
177C2×C271—41no—
178C2×C271—42no—
179C2×C271#54611yescharacteristic
180C2×C271C2×#13511yescharacteristic
181C2×C271#57711yescharacteristic
182D271—41no—
183D271—81no—
184C271#330311yescharacteristic, complemented, Sylow
185#3303—2711nomaximal, Sylow
186#555—2711no—
187#547—2711no—
188#487—2711no—
189#485—2711no—
190#485—2711no—
191#476—2711no—
192#465—2711no—
193#465—2711no—
194#461—2711no—
195#449—2711no—
196#449—2711no—
197C4×D8—2711no—
198C8×D4—2711no—
199#416—2711no—
200#416D27111yescharacteristic, complemented
201C2×#41—2711no—
202C2×#41C2×D27111yescharacteristic
203C2×SD16—2711no—
204C2×SD16—2711no—
205C2×D8—2711no—
206C22×C8—2711no—
207C22×C8C2×D27111yescharacteristic
208C4⋊Q8—2711no—
209C4⋊Q8—2711no—
210#124—5421no—
211#124—5421no—
212#123—5421no—
213#123—5421no—
214C22⋊Q8—2711no—
215C22⋊Q8—2711no—
216#121—2711no—
217#121—2711no—
218#121—5421no—
219#121—5421no—
220C4×D4—2711no—
221C4×D4—2711no—
222C4×D4—2711no—
223#117—2711no—
224#117—5421no—
225#117C2×D27111yescharacteristic
226C2×C4⋊C4—5421no—
227#107—2711no—
228#106—2711no—
229#106—2711no—
230C4⋊C8—2711no—
231#103—2711no—
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103C2×D27111yescharacteristic
236#103C2×D27111yescharacteristic
237#102—2711no—
238#102—2711no—
239#102C2×D27111yescharacteristic
240#102C2×D27111yescharacteristic
241C22⋊C8—2711no—
242C22⋊C8—2711no—
243C4×C8—2711no—
244#41—22no—
245#41—41no—
246#41—5422no—
247#41—10841no—
248C2×Q8—2711no—
249C2×Q8—5421no—
250C2×Q8C22×D27111yescharacteristic
251C2×D4—21no—
252C2×D4—2711no—
253C2×D4—2711no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4—5421no—
257C2×D4C22×D27111yescharacteristic
258C22×C4—21no—
259C22×C4—2711no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4C22×D27111yescharacteristic
264SD16—10841no—
265SD16—10841no—
266D8—5422no—
267C2×C8—41no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8—10841no—
271C2×C8C22×D27111yescharacteristic
272C2×C8C22×D27111yescharacteristic
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4C22×D27111yescharacteristic
282C4⋊C4C22×D27111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—2711no—
285C22⋊C4—2711no—
286C22⋊C4—5421no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C42—21no—
294C42—2711no—
295C42—5421no—
296C23—21no—
297C23—2711no—
298C23—5421no—
299C23D4×D27111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—5421no—
303Q8—5421no—
304Q8—21681no—
305D4—21no—
306D4—42no—
307D4—5422no—
308D4—5421no—
309D4—5421no—
310D4—10842no—
311D4—21681no—
312D4—21681no—
313D4#1686465412yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—2711no—
321C2×C4—2711no—
322C2×C4—2711no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—10842no—
329C2×C4—10842no—
330C2×C4—10842no—
331C2×C4D4×D27111yescharacteristic
332C2×C4C23×D27111yescharacteristic
333C8—21no—
334C8—21no—
335C8—5421no—
336C8—5421no—
337C22—21no—
338C22—22no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—5421no—
344C22—5421no—
345C22—5421no—
346C22—10841no—
347C22—10841no—
348C22—10841no—
349C22C2×D4×D27111yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—2712no—
355C4—5421no—
356C4—5421no—
357C4—10841no—
358C4—10841no—
359C4—10841no—
360C4C2×#1686465411yescharacteristic
361C4#2257620511yescharacteristic
362C2—41no—
363C2—42no—
364C2—10841no—
365C2—21681no—
366C2#3177453711yescharacteristic
367C2#135×D27111yescharacteristic
368C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#416×C271C2
socleC22×C271C22×D4
hypercenter#416D271
hypocenterC271#3303
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #31774610C2semidirect
2. #31774546C2semidirect
3. #31774246C2semidirect
4. #31774238C2semidirect
5. #31774236C2semidirect
6. #31774185C2semidirect
7. #31774129C2semidirect
8. #31774126C2semidirect
9. #31774101C2semidirect
10. #31774043C2semidirect
11. #31774042C2semidirect
12. #31773977C2semidirect
13. C8×C22⋊D271C2semidirect
14. #31773890C2semidirect
15. #416×C271C2semidirect
184. C271#3303semidirect
200. #416D271semidirect

Maximal subgroups

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

subgroupindex
#317746102
#317745462
#317742462
#317742382
#317742362
#317741852
#317741292
#317741262
#317741012
#317740432
#317740422
#317739772
C8×C22⋊D2712
#317738902
#416×C2712
#3303271