Finite Group Databasebeta

← #46849153

Subgroups — #46849153

Order 34688.

384 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774629C211yesmaximal, characteristic, complemented
2#31774562C211yesmaximal, characteristic, complemented
3#31774258C211yesmaximal, characteristic, complemented
4#31774242C211yesmaximal, characteristic, complemented
5#31774185C211yesmaximal, characteristic, complemented
6#31774184C211yesmaximal, characteristic, complemented
7#31774180C211yesmaximal, characteristic, complemented
8#31774115C211yesmaximal, characteristic, complemented
9#31774109C211yesmaximal, characteristic, complemented
10#31774091C211yesmaximal, characteristic
11#31774025C211yesmaximal, characteristic, complemented
12#31774010C211yesmaximal, characteristic, complemented
13#31774006C211yesmaximal, characteristic, complemented
14#31773982C211yesmaximal, characteristic, complemented
15#489×C271C211yesmaximal, characteristic, complemented
16C2×Q8×D271—21no—
17C2×#16864641C2211yescharacteristic
18C2×D2168C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20#22576159C2211yescharacteristic
21#22576159C2211yescharacteristic
22#22576156C2211yescharacteristic
23#22576149C2211yescharacteristic
24#22576146—21no—
25#22576144—21no—
26#22576142—21no—
27#22576138—21no—
28#22576138—21no—
29#22576136—21no—
30Q8×Dic271C2211yescharacteristic
31C4×Dic542C2211yescharacteristic
32C4×D1084C2211yescharacteristic
33C4×D1084C2211yescharacteristic
34D4×Dic271C2211yescharacteristic
35#22576122C2211yescharacteristic
36#22576122C2211yescharacteristic
37(C22⋊D271)⋊C4—21no—
38#22576092C2211yescharacteristic
39#22576091C2211yescharacteristic
40#22576091C2211yescharacteristic
41C4⋊(C271⋊C8)C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43Dic271⋊C8C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576080C2211yescharacteristic
49#22576080C2211yescharacteristic
50#22576070C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C4⋊Q8×C271C2211yescharacteristic
53#124×C271C2211yescharacteristic
54#103×C271C2211yescharacteristic
55#103×C271C2211yescharacteristic
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#97×C271C2211yescharacteristic
59C2×Dic542—21no—
60C2×Dic542—21no—
61C2×Dic542D411yescharacteristic
62Q8×D271—44no—
63C2×D1084C2311yescharacteristic
64C2×D1084C2311yescharacteristic
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C22×Dic271—21no—
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72#16864641—22no—
73D2168—22no—
74#16864639—22no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C4⋊Dic271—21no—
78C4⋊Dic271D411yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4D411yescharacteristic
84Dic271⋊C4D411yescharacteristic
85C22⋊Dic271—21no—
86(C2×D271)⋊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—
92C4×Dic271—21no—
93C4×Dic271C2311yescharacteristic
94C4×Dic271C2311yescharacteristic
95C2×Q8×C271—21no—
96C2×Q8×C271C2311yescharacteristic
97C2×D4×C271C2311yescharacteristic
98C2×C8×C271C2311yescharacteristic
99C2×C8×C271C2311yescharacteristic
100C4⋊C4×C271—21no—
101C4⋊C4×C271C2311yescharacteristic
102C4⋊C4×C271C2311yescharacteristic
103C22⋊C4×C271—21no—
104C22⋊C4×C271—21no—
105C42×C271C2311yescharacteristic
106C22×D271—21no—
107C22×D271—21no—
108Dic542—22no—
109Dic542—42no—
110Dic542—42no—
111D1084—21no—
112D1084—21no—
113D1084—21no—
114D1084#4112yes—
115C22⋊D271—42no—
116C22⋊D271—81no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—81no—
120C2×Dic271C2×D411yescharacteristic
121C2×Dic271C2×D411yescharacteristic
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C4×D271—42no—
125C4×D271—42no—
126C4×D271—42no—
127C4×D271—81no—
128C271⋊C8—21no—
129C271⋊C8—21no—
130C23×C271—21no—
131Q8×C271—21no—
132Q8×C271—21no—
133Q8×C271—42no—
134D4×C271—21no—
135D4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271C2×D411yescharacteristic
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2411yescharacteristic
142C8×C271—21no—
143C8×C271—21no—
144C22×C271—41no—
145C22×C271—41no—
146C22×C271—41no—
147C22×C271C22×D411yescharacteristic
148C2×D271—22no—
149C2×D271—41no—
150C2×D271—41no—
151C2×D271—41no—
152C2×D271—41no—
153C2×D271—41no—
154C4×C271—21no—
155C4×C271—21no—
156C4×C271—41no—
157C4×C271—41no—
158C4×C271—41no—
159C4×C271C2×#4111yescharacteristic
160C4×C2712−1+411yescharacteristic
161Dic271—21no—
162Dic271—21no—
163Dic271—21no—
164Dic271—41no—
165Dic271—41no—
166Dic271AGL(1,ℤ/8)12yes—
167C2×C271—81no—
168C2×C271#54811yescharacteristic
169C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
170C2×C271#57711yescharacteristic
171D271—42no—
172D271—81no—
173C271#337311yescharacteristic, complemented, Sylow
174#3373—2711nomaximal, Sylow
175#559—2711no—
176D4×Q8—2711no—
177#489—2711no—
178#489D27111yescharacteristic, complemented
179#486—2711no—
180#476—2711no—
181#476—2711no—
182#475—2711no—
183#463—2711no—
184#463—2711no—
185#459—2711no—
186#446—2711no—
187#442—2711no—
188#442—2711no—
189C4×SD16—2711no—
190C22×Q8—5421no—
191C2×SD16—2711no—
192C2×D8—2711no—
193C2×D8—2711no—
194C4⋊Q8—2711no—
195C4⋊Q8—2711no—
196C4⋊Q8C2×D27111yescharacteristic
197#127—2711no—
198#124—2711no—
199#124—5421no—
200#124C2×D27111yescharacteristic
201#123—5421no—
202#123—5421no—
203C22⋊Q8—5421no—
204C22⋊Q8—5421no—
205C22⋊Q8—5421no—
206C4×Q8—2711no—
207C4×Q8—2711no—
208C4×D4—2711no—
209C4×D4—2711no—
210C4×D4—2711no—
211C4×D4—2711no—
212C4×D4—2711no—
213C4×D4—5421no—
214#106—2711no—
215#106—2711no—
216#106—2711no—
217C4⋊C8—2711no—
218C4⋊C8—2711no—
219C4⋊C8—2711no—
220#103—2711no—
221#103C2×D27111yescharacteristic
222#103C2×D27111yescharacteristic
223#102—2711no—
224#102—2711no—
225#102—2711no—
226#102—2711no—
227#102—2711no—
228#102C2×D27111yescharacteristic
229#102C2×D27111yescharacteristic
230#97—2711no—
231#97C2×D27111yescharacteristic
232C4×C8—2711no—
233C2×Q8—21no—
234C2×Q8—2711no—
235C2×Q8—5421no—
236C2×Q8—5421no—
237C2×Q8—10844no—
238C2×Q8C22×D27111yescharacteristic
239C2×D4—2711no—
240C2×D4—2711no—
241C2×D4—5421no—
242C2×D4—5421no—
243C2×D4C22×D27111yescharacteristic
244C22×C4—5421no—
245C22×C4—5421no—
246C22×C4—5421no—
247C22×C4—5421no—
248C22×C4—5421no—
249SD16—5422no—
250D8—5422no—
251D8—5422no—
252C2×C8—2711no—
253C2×C8—2711no—
254C2×C8C22×D27111yescharacteristic
255C2×C8C22×D27111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—2711no—
258C4⋊C4—2711no—
259C4⋊C4—2711no—
260C4⋊C4—2711no—
261C4⋊C4—2711no—
262C4⋊C4—5421no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4C22×D27111yescharacteristic
266C4⋊C4C22×D27111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—5421no—
270C22⋊C4—5421no—
271C22⋊C4—5421no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C42—2711no—
277C42—2711no—
278C42—5421no—
279C42C22×D27111yescharacteristic
280C23—21no—
281C23—5421no—
282C23—5421no—
283Q8—21no—
284Q8—21no—
285Q8—42no—
286Q8—5422no—
287Q8—10842no—
288Q8—10842no—
289D4—21no—
290D4—21no—
291D4—2712no—
292D4—5421no—
293D4—5421no—
294D4—5421no—
295D4—10842no—
296D4—21681no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—2711no—
301C2×C4—2711no—
302C2×C4—2711no—
303C2×C4—2711no—
304C2×C4—5421no—
305C2×C4—5421no—
306C2×C4—10842no—
307C2×C4—10842no—
308C2×C4—10842no—
309C2×C4—21681no—
310C2×C4—21681no—
311C2×C4D4×D27111yescharacteristic
312C2×C4D4×D27111yescharacteristic
313C2×C4C23×D27111yescharacteristic
314C8—21no—
315C8—21no—
316C8—5421no—
317C8—5421no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—5422no—
322C22—10841no—
323C22—10841no—
324C22—10841no—
325C22—10841no—
326C22—10841no—
327C22C2×D4×D27111yescharacteristic
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—2712no—
334C4—5421no—
335C4—5421no—
336C4—5421no—
337C4—10841no—
338C4—10841no—
339C4#41×D27111yescharacteristic
340C4#2257620711yescharacteristic
341C2—81no—
342C2—10842no—
343C2—21681no—
344C2#3177455511yescharacteristic
345C2AGL(1,ℤ/8)×D27111yescharacteristic
346C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#489×C271C2
socleC22×C271C22×D4
prime radical#31774091C2
hypercenter#489D271
hypocenterC271#3373
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #31774629C2semidirect
2. #31774562C2semidirect
3. #31774258C2semidirect
4. #31774242C2semidirect
5. #31774185C2semidirect
6. #31774184C2semidirect
7. #31774180C2semidirect
8. #31774115C2semidirect
9. #31774109C2semidirect
11. #31774025C2semidirect
12. #31774010C2semidirect
13. #31774006C2semidirect
14. #31773982C2semidirect
15. #489×C271C2semidirect
173. C271#3373semidirect
178. #489D271semidirect

Maximal subgroups

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

subgroupindex
#317746292
#317745622
#317742582
#317742422
#317741852
#317741842
#317741802
#317741152
#317741092
#317740912
#317740252
#317740102
#317740062
#317739822
#489×C2712
#3373271