Finite Group Databasebeta

← #30650566

Subgroups — #30650566

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#22550514C211yesmaximal, characteristic
3#22550232C211yesmaximal, characteristic, complemented
4#22550205C211yesmaximal, characteristic, complemented
5#22550200C211yesmaximal, characteristic, complemented
6#22550178C211yesmaximal, characteristic, complemented
7#22550116C211yesmaximal, characteristic, complemented
8#22550105C211yesmaximal, characteristic, complemented
9#22550101C211yesmaximal, characteristic, complemented
10#22550038C211yesmaximal, characteristic, complemented
11#22550028C211yesmaximal, characteristic, complemented
12#22549967C211yesmaximal, characteristic, complemented
13C8×C22⋊D127C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#416×C127C211yesmaximal, characteristic, complemented
16C2×D4×D127C2211yescharacteristic
17C2×#14523238C2211yescharacteristic, complemented
18C2×#14523237C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic
21#16861202—21no—
22#16861201—21no—
23#16861200C2211yescharacteristic, complemented
24#16861198C2211yescharacteristic
25#16861192—21no—
26#16861191—21no—
27#16861189C2211yescharacteristic
28#16861188C2211yescharacteristic, complemented
29#16861185C2211yescharacteristic
30#16861183C2211yescharacteristic
31#16861181—21no—
32#16861177—21no—
33#16861174C2211yescharacteristic, complemented
34D4×Dic127C2211yescharacteristic
35C4×C22⋊D127C2211yescharacteristic
36#16861166—21no—
37C4⋊C4×D127C2211yescharacteristic
38C2×(C2×D127)⋊C4—21no—
39#16861142C2211yescharacteristic, complemented
40#16861141C2211yescharacteristic, complemented
41#16861140C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861133C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861129C2211yescharacteristic
48#16861129C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic
51C8×Dic127C2211yescharacteristic, complemented
52C2×#41×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic, complemented
54#117×C127C2211yescharacteristic, complemented
55#103×C127C2211yescharacteristic, complemented
56#103×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C23×D127—21no—
60C2×Dic254C2311yescharacteristic
61C2×D508C2311yescharacteristic
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64C2×C22⋊D127D411yescharacteristic
65D4×D127—22no—
66D4×D127—41no—
67C22×Dic127—21no—
68C2×C4×D127—21no—
69C2×C4×D127C2311yescharacteristic
70#14523238—41no—
71#14523237—41no—
72#14523236—22no—
73C2×C127⋊C8C2311yescharacteristic
74C2×C127⋊C8C2311yescharacteristic
75C8×D127—41no—
76C4⋊Dic127—21no—
77C4⋊Dic127—21no—
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4D411yescharacteristic
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88C22⋊Dic127D411yescharacteristic
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4—21no—
94(C2×D127)⋊C4D411yescharacteristic
95C4×Dic127—21no—
96C4×Dic127C2311yescharacteristic
97#41×C127—22no—
98#41×C127—41no—
99C2×Q8×C127C2311yescharacteristic
100C2×D4×C127—21no—
101C2×D4×C127C2311yescharacteristic
102C22×C4×C127—21no—
103C22×C4×C127C2311yescharacteristic
104C2×C8×C127—41no—
105C2×C8×C127C2311yescharacteristic
106C2×C8×C127C2311yescharacteristic
107C4⋊C4×C127C2311yescharacteristic
108C4⋊C4×C127C2311yescharacteristic
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—
118D508—21no—
119D508—21no—
120C22⋊D127—22no—
121C22⋊D127—42no—
122C22⋊D127—81no—
123C2×Dic127—21no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—81no—
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C4×D127—21no—
131C4×D127—21no—
132C4×D127—42no—
133C127⋊C8—21no—
134C127⋊C8—21no—
135C23×C127—21no—
136C23×C127C2×D411yescharacteristic
137Q8×C127—21no—
138Q8×C127—21no—
139D4×C127—21no—
140D4×C127—42no—
141D4×C127#4112yes—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—42no—
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2411yescharacteristic
150C8×C127—21no—
151C8×C127—21no—
152C22×C127—21no—
153C22×C127—22no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127C22×D411yescharacteristic
159C2×D127—21no—
160C2×D127—21no—
161C2×D127—41no—
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—42no—
179C2×C127—41no—
180C2×C127#54611yescharacteristic
181C2×C127C2×#13511yescharacteristic
182C2×C127#57711yescharacteristic
183D127—22no—
184D127—81no—
185#3296—1271nomaximal, Sylow
186C127#329611yescharacteristic, complemented, Sylow
187#552—1271no—
188#546—1271no—
189#488—1271no—
190#483—1271no—
191#482—1271no—
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#416D12711yescharacteristic, complemented
201C2×#102—1271no—
202C2×#41C2×D12711yescharacteristic
203C22×D4—1271no—
204C2×Q16—1271no—
205C2×SD16—1271no—
206C2×SD16—1271no—
207C22×C8—1271no—
208C22×C8C2×D12711yescharacteristic, complemented
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—1271no—
219#121—2541no—
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#107—1271no—
229#106—1271no—
230#106—1271no—
231C4⋊C8—1271no—
232#103—1271no—
233#103—1271no—
234#103C2×D12711yescharacteristic, complemented
235#103C2×D12711yescharacteristic, complemented
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102C2×D12711yescharacteristic
241#102C2×D12711yescharacteristic
242C22⋊C8—1271no—
243C22⋊C8—1271no—
244C4×C8—1271no—
245C24—2541no—
246#41—22no—
247#41—41no—
248C2×Q8—1271no—
249C2×Q8C22×D12711yescharacteristic
250C2×D4—21no—
251C2×D4—1271no—
252C2×D4—1271no—
253C2×D4—5081no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4—2542no—
257C2×D4C22×D12711yescharacteristic
258C22×C4—21no—
259C22×C4—1271no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4C22×D12711yescharacteristic
263Q16—5081no—
264SD16—5081no—
265SD16—2542no—
266C2×C8—41no—
267C2×C8—1271no—
268C2×C8—1271no—
269C2×C8—5081no—
270C2×C8C22×D12711yescharacteristic
271C2×C8C22×D12711yescharacteristic
272C4⋊C4—1271no—
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—2541no—
276C4⋊C4—2541no—
277C4⋊C4—2541no—
278C4⋊C4—2541no—
279C4⋊C4—2541no—
280C4⋊C4—2541no—
281C4⋊C4—2541no—
282C4⋊C4C22×D12711yescharacteristic
283C4⋊C4C22×D12711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—1271no—
286C22⋊C4—1271no—
287C22⋊C4—2541no—
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—21no—
305Q8—21no—
306Q8—2541no—
307Q8—2541no—
308D4—21no—
309D4—42no—
310D4—10161no—
311D4—5082no—
312D4—2542no—
313D4—2541no—
314D4—2541no—
315D4#1452324812yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—1271no—
323C2×C4—1271no—
324C2×C4—1271no—
325C2×C4—10161no—
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—22no—
342C22—41no—
343C22—41no—
344C22—1272no—
345C22—10164no—
346C22—5081no—
347C22—5081no—
348C22—5081no—
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—
362C4C2×#1452324811yescharacteristic
363C4#1686125411yescharacteristic
364C2—41no—
365C2—42no—
366C2—10161no—
367C2—2542no—
368C2#2255051611yescharacteristic
369C2#135×D12711yescharacteristic
370C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#416×C127C2
socleC22×C127C22×D4
prime radical#22550514C2
hypercenter#416D127
hypocenterC127#3296
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #22550565C2semidirect
3. #22550232C2semidirect
4. #22550205C2semidirect
5. #22550200C2semidirect
6. #22550178C2semidirect
7. #22550116C2semidirect
8. #22550105C2semidirect
9. #22550101C2semidirect
10. #22550038C2semidirect
11. #22550028C2semidirect
12. #22549967C2semidirect
13. C8×C22⋊D127C2semidirect
14. #102×D127C2semidirect
15. #416×C127C2semidirect
17. C2×#14523238C22semidirect
23. #16861200C22semidirect
28. #16861188C22semidirect
33. #16861174C22semidirect
39. #16861142C22semidirect
40. #16861141C22semidirect
41. #16861140C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861134C22semidirect
44. #16861133C22semidirect
49. C22⋊(C127⋊C8)C22semidirect
51. C8×Dic127C22semidirect
53. C22×C8×C127C22semidirect
54. #117×C127C22semidirect
55. #103×C127C22semidirect
56. #103×C127C22semidirect
186. C127#3296semidirect
200. #416D127semidirect
208. C22×C8C2×D127semidirect
225. #117C2×D127semidirect
234. #103C2×D127semidirect
235. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505652
#225505142
#225502322
#225502052
#225502002
#225501782
#225501162
#225501052
#225501012
#225500382
#225500282
#225499672
C8×C22⋊D1272
#102×D1272
#416×C1272
#3296127