Finite Group Databasebeta

← #30650561

Subgroups — #30650561

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#22550561C211yesmaximal, characteristic, complemented
2#22550520C211yesmaximal, characteristic
3#22550229C211yesmaximal, characteristic, complemented
4#22550206C211yesmaximal, characteristic, complemented
5#22550198C211yesmaximal, characteristic, complemented
6#22550174C211yesmaximal, characteristic, complemented
7#22550117C211yesmaximal, characteristic, complemented
8#22550104C211yesmaximal, characteristic, complemented
9#22550097C211yesmaximal, characteristic, complemented
10#22550039C211yesmaximal, characteristic, complemented
11#22550027C211yesmaximal, characteristic, complemented
12#22549967C211yesmaximal, characteristic, complemented
13C8×C22⋊D127C211yesmaximal, characteristic, complemented
14#22549869C211yesmaximal, characteristic, complemented
15C2×#102×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic
17C2×Dic508C2211yescharacteristic, complemented
18C2×#14523236C2211yescharacteristic
19C2×#14523236C2211yescharacteristic
20C2×C8×D127C2211yescharacteristic, complemented
21#16861204—21no—
22#16861200C2211yescharacteristic, complemented
23#16861199—21no—
24#16861196C2211yescharacteristic
25#16861192—21no—
26#16861190—21no—
27#16861189C2211yescharacteristic, complemented
28#16861188C2211yescharacteristic
29#16861188C2211yescharacteristic
30#16861181—21no—
31#16861180C2211yescharacteristic
32#16861178—21no—
33#16861174C2211yescharacteristic, complemented
34D4×Dic127C2211yescharacteristic
35C4×C22⋊D127C2211yescharacteristic
36#16861168—21no—
37#16861167C2211yescharacteristic, complemented
38C2×C22⋊Dic127—21no—
39#16861143C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41#16861140C2211yescharacteristic, complemented
42Dic127⋊C8C2211yescharacteristic, complemented
43#16861134C2211yescharacteristic, complemented
44#16861134C2211yescharacteristic, complemented
45#16861132C2211yescharacteristic, complemented
46#16861132C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic
49C22⋊(C127⋊C8)C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C22×D4×C127C2211yescharacteristic
53C22×C8×C127C2211yescharacteristic
54C2×C4⋊C4×C127C2211yescharacteristic
55#102×C127C2211yescharacteristic
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59#14523247—22no—
60#14523247—41no—
61C2×Dic254C2311yescharacteristic
62C2×Dic254C2311yescharacteristic
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C22⋊D127D411yescharacteristic
66C22×Dic127—21no—
67C22×Dic127—21no—
68C2×C4×D127C2311yescharacteristic
69Dic508—41no—
70#14523236—22no—
71#14523236—41no—
72C2×C127⋊C8C2311yescharacteristic
73C2×C127⋊C8C2311yescharacteristic
74C8×D127—41no—
75C4⋊Dic127—21no—
76C4⋊Dic127—21no—
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79Dic127⋊C4—21no—
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
83Dic127⋊C4D411yescharacteristic
84C22⋊Dic127—21no—
85C22⋊Dic127—21no—
86C22⋊Dic127—21no—
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127D411yescharacteristic
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4D411yescharacteristic
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127C2311yescharacteristic
97C24×C127—21no—
98C2×D4×C127—22no—
99C2×D4×C127—21no—
100C2×D4×C127—41no—
101C2×D4×C127C2311yescharacteristic
102C2×D4×C127C2311yescharacteristic
103C22×C4×C127—21no—
104C22×C4×C127C2311yescharacteristic
105C2×C8×C127—41no—
106C2×C8×C127C2311yescharacteristic
107C2×C8×C127C2311yescharacteristic
108C4⋊C4×C127—21no—
109C4⋊C4×C127C2311yescharacteristic
110C4⋊C4×C127C2311yescharacteristic
111C22×D127C2×D411yescharacteristic
112Dic254—21no—
113Dic254—21no—
114Dic254—21no—
115Dic254—21no—
116C22⋊D127—22no—
117C22⋊D127—42no—
118C22⋊D127—81no—
119C2×Dic127—21no—
120C2×Dic127—21no—
121C2×Dic127—21no—
122C2×Dic127—21no—
123C2×Dic127—42no—
124C2×Dic127—81no—
125C2×Dic127C2×D411yescharacteristic
126C2×Dic127C2×D411yescharacteristic
127C2×Dic127C2×D411yescharacteristic
128C4×D127—21no—
129C4×D127—21no—
130C127⋊C8—21no—
131C127⋊C8—21no—
132C23×C127—21no—
133C23×C127—21no—
134C23×C127—42no—
135C23×C127—42no—
136C23×C127—81no—
137C23×C127C2×D411yescharacteristic
138D4×C127—21no—
139D4×C127—21no—
140D4×C127—21no—
141D4×C127—42no—
142D4×C127#4112yes—
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—21no—
154C22×C127—22no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—41no—
160C22×C127—84no—
161C22×C127#13512yes—
162C22×C127C22×D411yescharacteristic
163C2×D127—21no—
164C2×D127—21no—
165C2×D127—21no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127C2×#4111yescharacteristic
170C4×C127H(5,𝔽2)11yescharacteristic
171Dic127—21no—
172Dic127—21no—
173Dic127—21no—
174Dic127—41no—
175Dic127—41no—
176Dic127—41no—
177Dic127—41no—
178C2×C127—22no—
179C2×C127—42no—
180C2×C127—81no—
181C2×C127#54611yescharacteristic
182C2×C127C2×#13511yescharacteristic
183C2×C127#57711yescharacteristic
184D127—41no—
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#416—1271no—
201C2×#102D12711yescharacteristic, complemented
202C2×#41—1271no—
203C22×D4C2×D12711yescharacteristic
204C2×Q16—1271no—
205C2×SD16—1271no—
206C2×SD16—1271no—
207C22×C8—1271no—
208C22×C8C2×D12711yescharacteristic
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—1271no—
225#117—2541no—
226C2×C4⋊C4C2×D12711yescharacteristic
227C2×C22⋊C4—2541no—
228#107—1271no—
229#106—1271no—
230#106—1271no—
231C4⋊C8—1271no—
232#103—1271no—
233#103—1271no—
234#103—1271no—
235#103—1271no—
236#102—1271no—
237#102—1271no—
238#102C2×D12711yescharacteristic
239#102C2×D12711yescharacteristic
240#102C2×D12711yescharacteristic
241#102C2×D12711yescharacteristic
242C22⋊C8—1271no—
243C22⋊C8—1271no—
244C4×C8—1271no—
245C24—21no—
246#41—5081no—
247#41—2542no—
248C2×Q8—1271no—
249C2×Q8—1271no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—1271no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4C22×D12711yescharacteristic
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—21no—
273C4⋊C4—1271no—
274C4⋊C4—1271no—
275C4⋊C4—1271no—
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—1271no—
285C22⋊C4—1271no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C22⋊C4—2541no—
294C42—1271no—
295C42—2541no—
296C42—2541no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—1271no—
303C23D4×D12711yescharacteristic
304Q8—2541no—
305Q8—2541no—
306Q8—2541no—
307Q8—2541no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—10161no—
313D4—5082no—
314D4—2542no—
315D4#1452324812yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—1271no—
322C2×C4—1271no—
323C2×C4—1271no—
324C2×C4—10161no—
325C2×C4—5082no—
326C2×C4—2541no—
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—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—84no—
347C22—2541no—
348C22—2541no—
349C22—2541no—
350C22#1686122812yes—
351C22C2×D4×D12711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—5081no—
356C4—5081no—
357C4—5081no—
358C4—5081no—
359C4—2541no—
360C4—2541no—
361C4—2541no—
362C4C2×#1452324811yescharacteristic
363C4#1686125411yescharacteristic
364C2—22no—
365C2—42no—
366C2—81no—
367C2—5081no—
368C2#2255051611yescharacteristic
369C2C2×#1686122811yescharacteristic
370C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC22×C127C22×D4
prime radical#22550520C2
hypercenterC2×#102D127
hypocenterC127#3296
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #22550561C2semidirect
3. #22550229C2semidirect
4. #22550206C2semidirect
5. #22550198C2semidirect
6. #22550174C2semidirect
7. #22550117C2semidirect
8. #22550104C2semidirect
9. #22550097C2semidirect
10. #22550039C2semidirect
11. #22550027C2semidirect
12. #22549967C2semidirect
13. C8×C22⋊D127C2semidirect
14. #22549869C2semidirect
15. C2×#102×C127C2semidirect
17. C2×Dic508C22semidirect
20. C2×C8×D127C22semidirect
22. #16861200C22semidirect
27. #16861189C22semidirect
33. #16861174C22semidirect
37. #16861167C22semidirect
39. #16861143C22semidirect
40. #16861140C22semidirect
41. #16861140C22semidirect
42. Dic127⋊C8C22semidirect
43. #16861134C22semidirect
44. #16861134C22semidirect
45. #16861132C22semidirect
46. #16861132C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. C8×Dic127C22semidirect
186. C127#3296semidirect
201. C2×#102D127semidirect

Maximal subgroups

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

subgroupindex
#225505612
#225505202
#225502292
#225502062
#225501982
#225501742
#225501172
#225501042
#225500972
#225500392
#225500272
#225499672
C8×C22⋊D1272
#225498692
C2×#102×C1272
#3296127