Finite Group Databasebeta

← #46846713

Subgroups — #46846713

Order 34688.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774524C211yesmaximal, characteristic, complemented
2#31774460C211yesmaximal, characteristic
3#31774261C211yesmaximal, characteristic, complemented
4#31774260C211yesmaximal, characteristic, complemented
5#31774259C211yesmaximal, characteristic, complemented
6#31774255C211yesmaximal, characteristic, complemented
7#31774252C211yesmaximal, characteristic, complemented
8#31774250C211yesmaximal, characteristic, complemented
9#31774242C211yesmaximal, characteristic, complemented
10#31773900C211yesmaximal, characteristic, complemented
11#31773895C211yesmaximal, characteristic, complemented
12#31773890C211yesmaximal, characteristic, complemented
13#102×D271C211yesmaximal, characteristic, complemented
14#31773851C211yesmaximal, characteristic, complemented
15#486×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic
18C2×#16864637C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20#22576159C2211yescharacteristic, complemented
21#22576158—21no—
22#22576156C2211yescharacteristic
23#22576152C2211yescharacteristic, complemented
24#22576151C2211yescharacteristic, complemented
25#22576150—21no—
26#22576148C2211yescharacteristic
27#22576147C2211yescharacteristic
28#22576138—21no—
29#22576136—21no—
30#22576132—21no—
31#22576132—21no—
32#22576128—21no—
33#22576128—21no—
34#22576118C2211yescharacteristic
35#22576117C2211yescharacteristic
36C4⋊C4×D271C2211yescharacteristic
37#22576085C2211yescharacteristic, complemented
38#22576085C2211yescharacteristic, complemented
39#22576083C2211yescharacteristic, complemented
40#22576083C2211yescharacteristic, complemented
41#22576082C2211yescharacteristic
42#22576082C2211yescharacteristic
43#22576081C2211yescharacteristic
44#22576081C2211yescharacteristic
45#22576081C2211yescharacteristic, complemented
46#22576081C2211yescharacteristic, complemented
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49#22576071C2211yescharacteristic, complemented
50#22576070C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C4⋊Q8×C271C2211yescharacteristic, complemented
53#127×C271C2211yescharacteristic
54#102×C271C2211yescharacteristic
55#102×C271C2211yescharacteristic
56#102×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic, complemented
58C4×C8×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60#16864653—22no—
61#16864653—22no—
62C2×Dic542—21no—
63C2×Dic542C2311yescharacteristic
64C2×D1084C2311yescharacteristic
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69D4×D271—22no—
70D4×D271—22no—
71C22×Dic271—21no—
72C2×C4×D271—21no—
73C2×C4×D271C2311yescharacteristic
74#16864637—22no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C8×D271—22no—
78C4⋊Dic271—21no—
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4D411yescharacteristic
87Dic271⋊C4D411yescharacteristic
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4D411yescharacteristic
94(C2×D271)⋊C4D411yescharacteristic
95C4×Dic271—21no—
96C4×Dic271C2311yescharacteristic
97C2×Q8×C271—21no—
98C2×D4×C271—21no—
99C2×D4×C271—21no—
100C2×D4×C271C2311yescharacteristic
101C2×D4×C271C2311yescharacteristic
102C2×C8×C271C2311yescharacteristic
103C2×C8×C271C2311yescharacteristic
104C4⋊C4×C271—21no—
105C4⋊C4×C271C2311yescharacteristic
106C4⋊C4×C271C2311yescharacteristic
107C42×C271C2311yescharacteristic
108C22×D271—21no—
109C22×D271—42no—
110C22×D271—81no—
111C22×D271—81no—
112C22×D271C2×D411yescharacteristic
113Dic542—21no—
114Dic542—21no—
115Dic542—81no—
116D1084—21no—
117D1084—21no—
118C22⋊D271—42no—
119C22⋊D271—42no—
120C22⋊D271—81no—
121C22⋊D271—81no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—42no—
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C4×D271—81no—
130C4×D271#4112yes—
131C4×D271#4112yes—
132C271⋊C8—21no—
133C271⋊C8—21no—
134C23×C271—21no—
135C23×C271—21no—
136Q8×C271—81no—
137D4×C271—21no—
138D4×C271—21no—
139D4×C271—21no—
140D4×C271—21no—
141D4×C271—81no—
142D4×C271—81no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2411yescharacteristic
148C8×C271—21no—
149C8×C271—21no—
150C22×C271—41no—
151C22×C271—41no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271—41no—
156C22×C271C22×D411yescharacteristic
157C2×D271—21no—
158C2×D271—21no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271—84no—
163C2×D271#12412yes—
164C4×C271—21no—
165C4×C271—21no—
166C4×C271—41no—
167C4×C271—41no—
168C4×C271C2×#4111yescharacteristic
169C4×C271C2×#4111yescharacteristic
170Dic271—21no—
171Dic271—21no—
172Dic271—41no—
173Dic271—41no—
174Dic271—41no—
175Dic271#12412yes—
176C2×C271—81no—
177C2×C271—81no—
178C2×C271C2×#12411yescharacteristic
179C2×C271#57611yescharacteristic
180C2×C271#57711yescharacteristic
181D271—22no—
182D271—81no—
183C271#313611yescharacteristic, complemented, Sylow
184#3136—2711nomaximal, Sylow
185#544—2711no—
186#535—2711no—
187#490—2711no—
188#490—2711no—
189#489—2711no—
190#489—2711no—
191#488—2711no—
192#488—2711no—
193#486—2711no—
194#486D27111yescharacteristic, complemented
195#418—2711no—
196#417—2711no—
197#416—2711no—
198C2×#102—2711no—
199#405—2711no—
200C2×#41—2711no—
201C22×D4—2711no—
202C2×M16—2711no—
203C22×C8—2711no—
204C4⋊Q8—2711no—
205C4⋊Q8—5421no—
206C4⋊Q8C2×D27111yescharacteristic, complemented
207#127—2711no—
208#127C2×D27111yescharacteristic
209#125—2711no—
210#125—2711no—
211#125—5421no—
212#124—2711no—
213#124—2711no—
214C22⋊Q8—5421no—
215C22⋊Q8—5421no—
216#121—5421no—
217#121—5421no—
218#120—5421no—
219#120—5421no—
220#117—2711no—
221#117—2711no—
222C2×C4⋊C4—2711no—
223#103—2711no—
224#103—2711no—
225#103—2711no—
226#103—2711no—
227#102—2711no—
228#102—2711no—
229#102—2711no—
230#102—2711no—
231#102—2711no—
232#102—2711no—
233#102—2711no—
234#102—2711no—
235#102C2×D27111yescharacteristic
236#102C2×D27111yescharacteristic
237#102C2×D27111yescharacteristic, complemented
238#102C2×D27111yescharacteristic, complemented
239#97—2711no—
240#97—2711no—
241C4×C8—2711no—
242C4×C8C2×D27111yescharacteristic, complemented
243C24—5421no—
244#41—5422no—
245#41—5422no—
246C2×Q8—21no—
247C2×Q8—2711no—
248C2×Q8—5421no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—2711no—
252C2×D4—5422no—
253C2×D4—5422no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4—5421no—
257C2×D4—5421no—
258C2×D4C22×D27111yescharacteristic
259C2×D4C22×D27111yescharacteristic
260C22×C4—2711no—
261C22×C4—5421no—
262C22×C4—5421no—
263M16—5422no—
264C2×C8—2711no—
265C2×C8—2711no—
266C2×C8—5422no—
267C2×C8C22×D27111yescharacteristic
268C2×C8C22×D27111yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—2711no—
271C4⋊C4—2711no—
272C4⋊C4—2711no—
273C4⋊C4—2711no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
276C4⋊C4—5421no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4C22×D27111yescharacteristic
281C4⋊C4C22×D27111yescharacteristic
282C22⋊C4—2711no—
283C22⋊C4—2711no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C22⋊C4—5421no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C42—2711no—
290C42—5421no—
291C42C22×D27111yescharacteristic
292C23—21no—
293C23—21no—
294C23—2711no—
295C23—5421no—
296C23—10842no—
297C23—21681no—
298C23—21681no—
299Q8—81no—
300Q8—5421no—
301Q8—5421no—
302Q8—21681no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—81no—
309D4—5421no—
310D4—5421no—
311D4—10842no—
312D4—10842no—
313D4—21681no—
314D4—21681no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—2711no—
318C2×C4—2712no—
319C2×C4—2712no—
320C2×C4—2711no—
321C2×C4—2711no—
322C2×C4—5421no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—10842no—
326C2×C4—21681no—
327C2×C4D4×D27111yescharacteristic
328C2×C4D4×D27111yescharacteristic
329C2×C4C23×D27111yescharacteristic
330C8—21no—
331C8—21no—
332C8—5421no—
333C8—5421no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—2712no—
341C22—5421no—
342C22—5421no—
343C22—10841no—
344C22—10841no—
345C22—10841no—
346C22—21684no—
347C22C2×D4×D27111yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—2712no—
353C4—5421no—
354C4—5421no—
355C4—10841no—
356C4—10841no—
357C4—10841no—
358C4#41×D27111yescharacteristic
359C4#41×D27111yescharacteristic
360C2—81no—
361C2—81no—
362C2—5422no—
363C2—21681no—
364C2#124×D27111yescharacteristic
365C2#3177469811yescharacteristic
366C2#3177470311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#486×C271C2
socleC22×C271C22×D4
prime radical#31774460C2
hypercenter#486D271
hypocenterC271#3136
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #31774524C2semidirect
3. #31774261C2semidirect
4. #31774260C2semidirect
5. #31774259C2semidirect
6. #31774255C2semidirect
7. #31774252C2semidirect
8. #31774250C2semidirect
9. #31774242C2semidirect
10. #31773900C2semidirect
11. #31773895C2semidirect
12. #31773890C2semidirect
13. #102×D271C2semidirect
14. #31773851C2semidirect
15. #486×C271C2semidirect
20. #22576159C22semidirect
23. #22576152C22semidirect
24. #22576151C22semidirect
37. #22576085C22semidirect
38. #22576085C22semidirect
39. #22576083C22semidirect
40. #22576083C22semidirect
45. #22576081C22semidirect
46. #22576081C22semidirect
49. #22576071C22semidirect
50. #22576070C22semidirect
51. C8×Dic271C22semidirect
52. C4⋊Q8×C271C22semidirect
56. #102×C271C22semidirect
57. #102×C271C22semidirect
58. C4×C8×C271C22semidirect
183. C271#3136semidirect
194. #486D271semidirect
206. C4⋊Q8C2×D271semidirect
237. #102C2×D271semidirect
238. #102C2×D271semidirect
242. C4×C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745242
#317744602
#317742612
#317742602
#317742592
#317742552
#317742522
#317742502
#317742422
#317739002
#317738952
#317738902
#102×D2712
#317738512
#486×C2712
#3136271