Finite Group Databasebeta

← #46846612

Subgroups — #46846612

Order 34688.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774507C211yesmaximal, characteristic, complemented
2#31774371C211yesmaximal, characteristic
3#31774238C211yesmaximal, characteristic, complemented
4#31774200C211yesmaximal, characteristic, complemented
5#31774196C211yesmaximal, characteristic, complemented
6#31774101C211yesmaximal, characteristic, complemented
7#31774101C211yesmaximal, characteristic, complemented
8#31774050C211yesmaximal, characteristic
9#31774047C211yesmaximal, characteristic
10#31774000C211yesmaximal, characteristic, complemented
11#31773996C211yesmaximal, characteristic, complemented
12#31773984C211yesmaximal, characteristic, complemented
13#31773982C211yesmaximal, characteristic, complemented
14#31773955C211yesmaximal, characteristic, complemented
15#485×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C2×#16864643C2211yescharacteristic
18C2×#16864643C2211yescharacteristic
19C2×#16864641C2211yescharacteristic
20C2×#16864641C2211yescharacteristic
21#22576158C2211yescharacteristic, complemented
22#22576153—21no—
23#22576153—21no—
24#22576151—21no—
25#22576151C2211yescharacteristic, complemented
26#22576150—21no—
27#22576139C2211yescharacteristic
28Q8×Dic271C2211yescharacteristic, complemented
29#22576125C2211yescharacteristic, complemented
30C4×D1084—21no—
31C4×D1084—21no—
32#22576122C2211yescharacteristic
33#22576122C2211yescharacteristic
34#22576122C2211yescharacteristic
35#22576122C2211yescharacteristic
36#22576119C2211yescharacteristic
37C22⋊C4×D271—21no—
38C22⋊C4×D271—21no—
39#22576093C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42Dic271⋊C8C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576083C2211yescharacteristic, complemented
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50#22576070C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C22⋊Q8×C271C2211yescharacteristic
53#117×C271C2211yescharacteristic
54#107×C271C2211yescharacteristic, complemented
55#106×C271C2211yescharacteristic, complemented
56#103×C271C2211yescharacteristic, complemented
57#103×C271C2211yescharacteristic, complemented
58C22⋊C8×C271C2211yescharacteristic
59C23×D271—21no—
60C2×Dic542C2311yescharacteristic
61C2×D1084—22no—
62C2×D1084—22no—
63C2×D1084—21no—
64C2×D1084C2311yescharacteristic
65C2×D1084C2311yescharacteristic
66C2×C4×D271—21no—
67C2×C4×D271—21no—
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70#16864643—22no—
71#16864643—22no—
72#16864641—22no—
73#16864641—22no—
74C2×C271⋊C8C2311yescharacteristic
75C2×C271⋊C8C2311yescharacteristic
76C4⋊Dic271—21no—
77C4⋊Dic271—21no—
78C4⋊Dic271D411yescharacteristic
79C4⋊Dic271D411yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271D411yescharacteristic
88C22⋊Dic271D411yescharacteristic
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96C4×Dic271C2311yescharacteristic
97C2×Q8×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C2×C8×C271C2311yescharacteristic
100C2×C8×C271C2311yescharacteristic
101C4⋊C4×C271—21no—
102C4⋊C4×C271C2311yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C42×C271—21no—
108C22×D271—21no—
109C22×D271—21no—
110C22×D271—24no—
111C22×D271—81no—
112C22×D271—81no—
113Dic542—21no—
114Dic542—21no—
115D1084—24no—
116D1084—21no—
117D1084—21no—
118D1084#4112yes—
119D1084#4112yes—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C4×D271—81no—
127C4×D271—81no—
128C4×D271—81no—
129C4×D271—81no—
130C271⋊C8—21no—
131C271⋊C8—21no—
132C23×C271C2×D411yescharacteristic
133Q8×C271—21no—
134Q8×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271C2×D411yescharacteristic
142C2×C4×C271C2411yescharacteristic
143C8×C271—21no—
144C8×C271—21no—
145C22×C271—21no—
146C22×C271—21no—
147C22×C271—21no—
148C22×C271C22×D411yescharacteristic
149C2×D271—22no—
150C2×D271—22no—
151C2×D271—41no—
152C2×D271—41no—
153C2×D271—41no—
154C2×D271—41no—
155C2×D271—84no—
156C4×C271—21no—
157C4×C271—41no—
158C4×C271—41no—
159C4×C271—41no—
160C4×C271—41no—
161C4×C271C2×#4111yescharacteristic
162C4×C271C2×#4111yescharacteristic
163Dic271—21no—
164Dic271—21no—
165Dic271—21no—
166Dic271—21no—
167Dic271—41no—
168Dic271—41no—
169C2×C271—41no—
170C2×C271#52511yescharacteristic
171C2×C271#57711yescharacteristic
172C2×C271#57711yescharacteristic
173D271—42no—
174D271—42no—
175C271#312311yescharacteristic, complemented, Sylow
176#3123—2711nomaximal, Sylow
177#541—2711no—
178#518—2711no—
179#485—2711no—
180#485D27111yescharacteristic, complemented
181#478—2711no—
182#478—2711no—
183#461—2711no—
184#461—2711no—
185#450—2711no—
186#450—2711no—
187#440—2711no—
188#440—2711no—
189C4×SD16—2711no—
190C4×SD16—2711no—
191#433—2711no—
192C22×D4—2711no—
193C2×SD16—2711no—
194C2×SD16—2711no—
195C2×SD16—2711no—
196C2×SD16—2711no—
197C4⋊Q8—2711no—
198#126—5421no—
199#126—5421no—
200#125—2711no—
201#125—5421no—
202#125—5421no—
203C22⋊Q8—2711no—
204C22⋊Q8C2×D27111yescharacteristic
205C4×Q8—2711no—
206C4×Q8—2711no—
207C4×D4—2711no—
208C4×D4—2711no—
209C4×D4—2711no—
210C4×D4—2711no—
211C4×D4—5421no—
212C4×D4—5421no—
213#117—2711no—
214#117C2×D27111yescharacteristic
215C2×C22⋊C4—5421no—
216C2×C22⋊C4—5421no—
217#107—2711no—
218#107C2×D27111yescharacteristic, complemented
219#106—2711no—
220#106C2×D27111yescharacteristic, complemented
221C4⋊C8—2711no—
222C4⋊C8—2711no—
223#103—2711no—
224#103—2711no—
225#103C2×D27111yescharacteristic, complemented
226#103C2×D27111yescharacteristic, complemented
227#102—2711no—
228#102—2711no—
229#102—2711no—
230#102—2711no—
231C22⋊C8—2711no—
232C22⋊C8C2×D27111yescharacteristic
233#97—2711no—
234C4×C8—2711no—
235C24—5421no—
236C2×Q8—2711no—
237C2×Q8C22×D27111yescharacteristic
238C2×D4—2711no—
239C2×D4—2711no—
240C2×D4—5422no—
241C2×D4—5422no—
242C2×D4—5421no—
243C22×C4—5421no—
244C22×C4—5421no—
245C22×C4—5421no—
246C22×C4—5421no—
247C22×C4C22×D27111yescharacteristic
248SD16—5422no—
249SD16—5422no—
250SD16—5422no—
251SD16—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—5421no—
261C4⋊C4—5421no—
262C4⋊C4—5421no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4—5421no—
266C4⋊C4—5421no—
267C4⋊C4C22×D27111yescharacteristic
268C4⋊C4C22×D27111yescharacteristic
269C4⋊C4C22×D27111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—2711no—
273C22⋊C4—2711no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C22⋊C4—5421no—
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C42—21no—
280C42—2711no—
281C42—2711no—
282C42—5421no—
283C42—5421no—
284C23—5424no—
285C23—5421no—
286C23—5421no—
287C23—21681no—
288C23—21681no—
289C23D4×D27111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—5421no—
293Q8—5421no—
294D4—2712no—
295D4—2712no—
296D4—5424no—
297D4—5421no—
298D4—5421no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—2711no—
306C2×C4—2711no—
307C2×C4—2711no—
308C2×C4—2711no—
309C2×C4—5421no—
310C2×C4—5421no—
311C2×C4—21681no—
312C2×C4—21681no—
313C2×C4—21681no—
314C2×C4—21681no—
315C2×C4D4×D27111yescharacteristic
316C2×C4C23×D27111yescharacteristic
317C8—21no—
318C8—21no—
319C8—5421no—
320C8—5421no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—5422no—
325C22—5422no—
326C22—10841no—
327C22—10841no—
328C22—10841no—
329C22—10841no—
330C22—21684no—
331C22C2×D4×D27111yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—5421no—
338C4—5421no—
339C4—5421no—
340C4—5421no—
341C4—10841no—
342C4—10841no—
343C4#41×D27111yescharacteristic
344C4#41×D27111yescharacteristic
345C2—41no—
346C2—10842no—
347C2—10842no—
348C2#3177440511yescharacteristic
349C2#3177470411yescharacteristic
350C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#485×C271C2
socleC22×C271C22×D4
prime radicalC22×D1084C22
hypercenter#485D271
hypocenterC271#3123
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #31774507C2semidirect
3. #31774238C2semidirect
4. #31774200C2semidirect
5. #31774196C2semidirect
6. #31774101C2semidirect
7. #31774101C2semidirect
10. #31774000C2semidirect
11. #31773996C2semidirect
12. #31773984C2semidirect
13. #31773982C2semidirect
14. #31773955C2semidirect
15. #485×C271C2semidirect
21. #22576158C22semidirect
25. #22576151C22semidirect
28. Q8×Dic271C22semidirect
29. #22576125C22semidirect
39. #22576093C22semidirect
40. #22576091C22semidirect
41. Dic271⋊C8C22semidirect
42. Dic271⋊C8C22semidirect
43. #22576083C22semidirect
44. #22576083C22semidirect
50. #22576070C22semidirect
51. C8×Dic271C22semidirect
54. #107×C271C22semidirect
55. #106×C271C22semidirect
56. #103×C271C22semidirect
57. #103×C271C22semidirect
175. C271#3123semidirect
180. #485D271semidirect
218. #107C2×D271semidirect
220. #106C2×D271semidirect
225. #103C2×D271semidirect
226. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745072
#317743712
#317742382
#317742002
#317741962
#317741012
#317740502
#317740472
#317740002
#317739962
#317739842
#317739822
#317739552
#485×C2712
#3123271