Finite Group Databasebeta

← #30650568

Subgroups — #30650568

Order 16256.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550565C211yesmaximal, characteristic, complemented
2#22550523C211yesmaximal, characteristic
3#22550229C211yesmaximal, characteristic, complemented
4#22550208C211yesmaximal, characteristic, complemented
5#22550179C211yesmaximal, characteristic, complemented
6#22550118C211yesmaximal, characteristic, complemented
7#22550103C211yesmaximal, characteristic, complemented
8#22550097C211yesmaximal, characteristic, complemented
9#22550040C211yesmaximal, characteristic, complemented
10#22550028C211yesmaximal, characteristic, complemented
11#22549963C211yesmaximal, characteristic, complemented
12(C22⋊D127)⋊C8C211yesmaximal, characteristic, complemented
13#22549870C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#482×C127C211yesmaximal, characteristic, complemented
16C2×#14523248C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic
18C2×Dic508C2211yescharacteristic, complemented
19C2×#14523237C2211yescharacteristic
20C2×#14523236C2211yescharacteristic
21C22×C127⋊C8C2211yescharacteristic, complemented
22C2×C8×D127C2211yescharacteristic
23#16861202—21no—
24#16861201—21no—
25#16861200C2211yescharacteristic, complemented
26#16861196C2211yescharacteristic
27#16861191—21no—
28#16861191—21no—
29#16861189C2211yescharacteristic
30#16861189C2211yescharacteristic
31#16861188C2211yescharacteristic, complemented
32#16861180—21no—
33#16861177—21no—
34#16861174C2211yescharacteristic, complemented
35#16861171C2211yescharacteristic
36C4×C22⋊D127C2211yescharacteristic
37#16861166—21no—
38C4⋊C4×D127C2211yescharacteristic
39C22⋊C4×D127—21no—
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42#16861134C2211yescharacteristic, complemented
43#16861132C2211yescharacteristic, complemented
44#16861132C2211yescharacteristic, complemented
45#16861132C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861129C2211yescharacteristic
49#16861129C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic, complemented
52#121×C127C2211yescharacteristic
53#117×C127C2211yescharacteristic, complemented
54#107×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523248—22no—
61#14523248—41no—
62C2×Dic254C2311yescharacteristic
63C2×Dic254C2311yescharacteristic
64C2×D508C2311yescharacteristic
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127D411yescharacteristic
68D4×D127—22no—
69D4×D127—41no—
70C2×C4×D127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74Dic508—41no—
75#14523237—22no—
76#14523236—41no—
77C2×C127⋊C8—41no—
78C2×C127⋊C8C2311yescharacteristic
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—41no—
81C4⋊Dic127—21no—
82C4⋊Dic127—21no—
83C4⋊Dic127—21no—
84C4⋊Dic127C2311yescharacteristic
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4—21no—
89Dic127⋊C4D411yescharacteristic
90C22⋊Dic127—21no—
91C22⋊Dic127—21no—
92C22⋊Dic127—21no—
93C22⋊Dic127D411yescharacteristic
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4—21no—
96(C2×D127)⋊C4—21no—
97(C2×D127)⋊C4D411yescharacteristic
98C4×Dic127—21no—
99C4×Dic127C2311yescharacteristic
100C2×D4×C127—21no—
101C2×D4×C127C2311yescharacteristic
102C22×C4×C127C2311yescharacteristic
103C2×C8×C127C2311yescharacteristic
104C2×C8×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C4⋊C4×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C22⋊C4×C127—21no—
109C22⋊C4×C127—21no—
110C42×C127—21no—
111C22×D127—21no—
112C22×D127—42no—
113C22×D127—42no—
114C22×D127—81no—
115C22×D127C2×D411yescharacteristic
116Dic254—21no—
117Dic254—21no—
118Dic254—21no—
119Dic254—21no—
120D508—21no—
121D508#4112yes—
122C22⋊D127—22no—
123C22⋊D127—42no—
124C22⋊D127—42no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C2×Dic127C2×D411yescharacteristic
131C4×D127—21no—
132C4×D127—21no—
133C4×D127—42no—
134C4×D127—42no—
135C4×D127—81no—
136C127⋊C8—21no—
137C127⋊C8—21no—
138C23×C127—21no—
139C23×C127C2×D411yescharacteristic
140D4×C127—21no—
141D4×C127—21no—
142D4×C127—81no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127C22×D411yescharacteristic
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—22no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—84no—
165C2×D127#13512yes—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127—41no—
170C4×C127C2×#4111yescharacteristic
171C4×C127H(5,𝔽2)11yescharacteristic
172Dic127—21no—
173Dic127—21no—
174Dic127—21no—
175Dic127—41no—
176Dic127—41no—
177Dic127—41no—
178C2×C127—41no—
179C2×C127—81no—
180C2×C127#54611yescharacteristic
181C2×C127C2×#13511yescharacteristic
182C2×C127#57711yescharacteristic
183D127—22no—
184D127—42no—
185#3296—1271nomaximal, Sylow
186C127#329611yescharacteristic, complemented, Sylow
187#552—1271no—
188#546—1271no—
189#488—1271no—
190#483—1271no—
191#482D12711yescharacteristic, complemented
192#478—1271no—
193#467—1271no—
194#465—1271no—
195#464—1271no—
196#452—1271no—
197#450—1271no—
198C4×SD16—1271no—
199C8×D4—1271no—
200#416—1271no—
201C2×#102—1271no—
202C2×#41—1271no—
203C22×D4—1271no—
204C2×Q16—1271no—
205C2×SD16—1271no—
206C2×SD16—1271no—
207C22×C8—1271no—
208C22×C8—1271no—
209#126—2541no—
210#125—1271no—
211#125—2541no—
212#124—1271no—
213#123—2541no—
214#123—2541no—
215C22⋊Q8—1271no—
216C22⋊Q8—1271no—
217C22⋊Q8—1271no—
218#121—2541no—
219#121C2×D12711yescharacteristic
220#120—2541no—
221C4×Q8—1271no—
222C4×D4—1271no—
223C4×D4—1271no—
224#117—2541no—
225#117C2×D12711yescharacteristic, complemented
226C2×C4⋊C4—1271no—
227C2×C22⋊C4—2541no—
228#107C2×D12711yescharacteristic, complemented
229#106—1271no—
230#106C2×D12711yescharacteristic, complemented
231C4⋊C8—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102C2×D12711yescharacteristic
241#102C2×D12711yescharacteristic
242C22⋊C8—1271no—
243C22⋊C8C2×D12711yescharacteristic, complemented
244C4×C8—1271no—
245C24—2541no—
246#41—5081no—
247#41—2542no—
248C2×Q8—1271no—
249C2×Q8—1271no—
250C2×D4—21no—
251C2×D4—1271no—
252C2×D4—1271no—
253C2×D4—5081no—
254C2×D4—2541no—
255C2×D4—2542no—
256C2×D4—2541no—
257C2×D4C22×D12711yescharacteristic
258C22×C4—1271no—
259C22×C4—2541no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4C22×D12711yescharacteristic
263Q16—5081no—
264SD16—5081no—
265SD16—2542no—
266C2×C8—1271no—
267C2×C8—1271no—
268C2×C8—5081no—
269C2×C8—5081no—
270C2×C8C22×D12711yescharacteristic
271C2×C8C22×D12711yescharacteristic
272C4⋊C4—1271no—
273C4⋊C4—1271no—
274C4⋊C4—2541no—
275C4⋊C4—2541no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4C22×D12711yescharacteristic
282C4⋊C4C22×D12711yescharacteristic
283C4⋊C4C22×D12711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—1271no—
287C22⋊C4—1271no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C42—21no—
295C42—1271no—
296C42—2541no—
297C23—21no—
298C23—1271no—
299C23—10161no—
300C23—5082no—
301C23—5082no—
302C23—2541no—
303C23D4×D12711yescharacteristic
304Q8—2541no—
305Q8—2541no—
306Q8—2541no—
307Q8—2541no—
308D4—21no—
309D4—21no—
310D4—81no—
311D4—1272no—
312D4—5082no—
313D4—5082no—
314D4—2542no—
315D4—2541no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—1271no—
322C2×C4—1271no—
323C2×C4—1271no—
324C2×C4—10161no—
325C2×C4—5082no—
326C2×C4—5082no—
327C2×C4—2541no—
328C2×C4—2541no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4—2541no—
332C2×C4D4×D12711yescharacteristic
333C2×C4C23×D12711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2541no—
337C8—2541no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—1272no—
345C22—10164no—
346C22—5081no—
347C22—5081no—
348C22—2542no—
349C22—2541no—
350C22—2541no—
351C22C2×D4×D12711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—5081no—
357C4—5081no—
358C4—5081no—
359C4—2541no—
360C4—2541no—
361C4—2541no—
362C4#41×D12711yescharacteristic
363C4#1686125511yescharacteristic
364C2—41no—
365C2—81no—
366C2—5082no—
367C2—2542no—
368C2#2255051811yescharacteristic
369C2#135×D12711yescharacteristic
370C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#482×C127C2
socleC22×C127C22×D4
prime radical#22550523C2
hypercenter#482D127
hypocenterC127#3296
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #22550565C2semidirect
3. #22550229C2semidirect
4. #22550208C2semidirect
5. #22550179C2semidirect
6. #22550118C2semidirect
7. #22550103C2semidirect
8. #22550097C2semidirect
9. #22550040C2semidirect
10. #22550028C2semidirect
11. #22549963C2semidirect
12. (C22⋊D127)⋊C8C2semidirect
13. #22549870C2semidirect
14. #102×D127C2semidirect
15. #482×C127C2semidirect
18. C2×Dic508C22semidirect
21. C22×C127⋊C8C22semidirect
25. #16861200C22semidirect
31. #16861188C22semidirect
34. #16861174C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. #16861134C22semidirect
43. #16861132C22semidirect
44. #16861132C22semidirect
45. #16861132C22semidirect
51. C8×Dic127C22semidirect
53. #117×C127C22semidirect
54. #107×C127C22semidirect
55. #106×C127C22semidirect
58. C22⋊C8×C127C22semidirect
186. C127#3296semidirect
191. #482D127semidirect
225. #117C2×D127semidirect
228. #107C2×D127semidirect
230. #106C2×D127semidirect
243. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505652
#225505232
#225502292
#225502082
#225501792
#225501182
#225501032
#225500972
#225500402
#225500282
#225499632
(C22⋊D127)⋊C82
#225498702
#102×D1272
#482×C1272
#3296127