Finite Group Databasebeta

← #30648975

Subgroups — #30648975

Order 16256.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550482C211yesmaximal, characteristic, complemented
2#22550348C211yesmaximal, characteristic, complemented
3#22550203C211yesmaximal, characteristic, complemented
4#22550202C211yesmaximal, characteristic, complemented
5#22550173C211yesmaximal, characteristic, complemented
6#22550072C211yesmaximal, characteristic, complemented
7#22550071C211yesmaximal, characteristic, complemented
8#22550011C211yesmaximal, characteristic, complemented
9#22550011C211yesmaximal, characteristic, complemented
10#22549987C211yesmaximal, characteristic, complemented
11#22549985C211yesmaximal, characteristic, complemented
12#22549958C211yesmaximal, characteristic, complemented
13#22549956C211yesmaximal, characteristic, complemented
14#22549936C211yesmaximal, characteristic, complemented
15#478×C127C211yesmaximal, characteristic, complemented
16C2×D4×D127C2211yescharacteristic
17C2×D1016C2211yescharacteristic
18C2×D1016C2211yescharacteristic
19C2×#14523233C2211yescharacteristic
20C2×#14523233C2211yescharacteristic
21#16861206C2211yescharacteristic, complemented
22#16861194—21no—
23#16861194—21no—
24#16861184—21no—
25#16861184—21no—
26#16861184C2211yescharacteristic
27#16861184C2211yescharacteristic
28C4×D508C2211yescharacteristic
29D4×Dic127C2211yescharacteristic
30#16861171C2211yescharacteristic
31#16861171C2211yescharacteristic, complemented
32#16861171C2211yescharacteristic, complemented
33C4×C22⋊D127—21no—
34(C22⋊D127)⋊C4—21no—
35#16861167C2211yescharacteristic
36#16861166C2211yescharacteristic
37C2×(C2×D127)⋊C4—21no—
38C22⋊C4×D127—21no—
39#16861142C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41C4⋊(C127⋊C8)C2211yescharacteristic, complemented
42#16861131C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic, complemented
44#16861129C2211yescharacteristic
45#16861129C2211yescharacteristic
46#16861129C2211yescharacteristic, complemented
47#16861129C2211yescharacteristic, complemented
48(C2×D127)⋊C8C2211yescharacteristic
49(C2×D127)⋊C8C2211yescharacteristic
50#16861119C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52#125×C127C2211yescharacteristic, complemented
53C4×D4×C127C2211yescharacteristic
54#107×C127C2211yescharacteristic, complemented
55#106×C127C2211yescharacteristic, complemented
56C4⋊C8×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic
58#102×C127C2211yescharacteristic
59C23×D127—21no—
60C2×D508—21no—
61C2×D508—21no—
62C2×D508—21no—
63C2×D508—21no—
64C2×D508C2311yescharacteristic
65C2×D508C2311yescharacteristic
66C2×D508C2311yescharacteristic
67C2×C22⋊D127—21no—
68D4×D127—22no—
69D4×D127—22no—
70C22×Dic127—21no—
71C2×C4×D127—21no—
72C2×C4×D127—21no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127C2311yescharacteristic
76D1016—22no—
77D1016—22no—
78#14523233—22no—
79#14523233—22no—
80C2×C127⋊C8C2311yescharacteristic
81C2×C127⋊C8C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4D411yescharacteristic
84Dic127⋊C4D411yescharacteristic
85C22⋊Dic127—21no—
86(C2×D127)⋊C4—21no—
87(C2×D127)⋊C4—21no—
88(C2×D127)⋊C4—21no—
89(C2×D127)⋊C4—21no—
90(C2×D127)⋊C4—21no—
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4D411yescharacteristic
94(C2×D127)⋊C4D411yescharacteristic
95C4×Dic127—21no—
96C4×Dic127C2311yescharacteristic
97C2×D4×C127C2311yescharacteristic
98C22×C4×C127—21no—
99C2×C8×C127C2311yescharacteristic
100C2×C8×C127C2311yescharacteristic
101C4⋊C4×C127—21no—
102C4⋊C4×C127—21no—
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C4⋊C4×C127C2311yescharacteristic
106C22⋊C4×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—24no—
110C22×D127—21no—
111C22×D127—21no—
112C22×D127—81no—
113C22×D127—81no—
114C22×D127C2×D411yescharacteristic
115D508—21no—
116D508—21no—
117D508—21no—
118D508—21no—
119D508—21no—
120D508—81no—
121D508—81no—
122D508—81no—
123D508—81no—
124D508#4112yes—
125C22⋊D127—24no—
126C2×Dic127—21no—
127C2×Dic127—81no—
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C2×Dic127C2×D411yescharacteristic
131C4×D127—21no—
132C4×D127—21no—
133C4×D127—81no—
134C4×D127—81no—
135C4×D127—81no—
136C4×D127—81no—
137C127⋊C8—21no—
138C127⋊C8—21no—
139C23×C127—21no—
140D4×C127—21no—
141D4×C127#4112yes—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127—81no—
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2411yescharacteristic
149C8×C127—21no—
150C8×C127—21no—
151C22×C127—22no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127C22×D411yescharacteristic
155C2×D127—21no—
156C2×D127—22no—
157C2×D127—21no—
158C2×D127—21no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—41no—
163C2×D127—41no—
164C2×D127—41no—
165C2×D127—41no—
166C2×D127—41no—
167C2×D127—84no—
168C4×C127—21no—
169C4×C127—21no—
170C4×C127—41no—
171C4×C127—41no—
172C4×C127—41no—
173C4×C127C2×#4111yescharacteristic
174C4×C127C2×#4111yescharacteristic
175Dic127—21no—
176Dic127—21no—
177Dic127—21no—
178Dic127—41no—
179C2×C127—42no—
180C2×C127#52511yescharacteristic
181C2×C127#57611yescharacteristic
182C2×C127#57611yescharacteristic
183D127—41no—
184D127—42no—
185D127—81no—
186D127—81no—
187#3122—1271nomaximal, Sylow
188C127#312211yescharacteristic, complemented, Sylow
189#540—1271no—
190#518—1271no—
191#482—1271no—
192#482—1271no—
193#478—1271no—
194#478D12711yescharacteristic, complemented
195#459—1271no—
196#459—1271no—
197#447—1271no—
198#447—1271no—
199#442—1271no—
200#442—1271no—
201C4×D8—1271no—
202C4×D8—1271no—
203#433—1271no—
204C22×D4—1271no—
205C2×D8—1271no—
206C2×D8—1271no—
207C2×D8—1271no—
208C2×D8—1271no—
209#127—1271no—
210#125C2×D12711yescharacteristic, complemented
211#123—2541no—
212#123—2541no—
213#121—1271no—
214#121—1271no—
215#121—2541no—
216#121—2541no—
217C4×D4—1271no—
218C4×D4—1271no—
219C4×D4—1271no—
220C4×D4—1271no—
221C4×D4—1271no—
222C4×D4—2541no—
223C4×D4—2541no—
224C4×D4C2×D12711yescharacteristic
225#117—1271no—
226#117—1271no—
227C2×C22⋊C4—2541no—
228C2×C22⋊C4—2541no—
229#107—1271no—
230#107C2×D12711yescharacteristic, complemented
231#106—1271no—
232#106C2×D12711yescharacteristic, complemented
233C4⋊C8—1271no—
234C4⋊C8C2×D12711yescharacteristic, complemented
235#102—1271no—
236#102—1271no—
237#102—1271no—
238#102—1271no—
239#102—1271no—
240#102—1271no—
241#102C2×D12711yescharacteristic
242#102C2×D12711yescharacteristic
243C22⋊C8—1271no—
244C22⋊C8—1271no—
245#97—1271no—
246C4×C8—1271no—
247C24—2541no—
248C2×D4—1271no—
249C2×D4—1271no—
250C2×D4—1271no—
251C2×D4—2541no—
252C2×D4—2541no—
253C2×D4—2541no—
254C2×D4—2542no—
255C2×D4—2542no—
256C2×D4—2541no—
257C2×D4—2541no—
258C2×D4C22×D12711yescharacteristic
259C22×C4—21no—
260C22×C4—1271no—
261C22×C4—2541no—
262C22×C4—2541no—
263C22×C4—2541no—
264C22×C4—2541no—
265C22×C4—2541no—
266D8—2542no—
267D8—2542no—
268D8—2542no—
269D8—2542no—
270C2×C8—1271no—
271C2×C8—1271no—
272C2×C8C22×D12711yescharacteristic
273C2×C8C22×D12711yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—1271no—
277C4⋊C4—1271no—
278C4⋊C4—1271no—
279C4⋊C4C22×D12711yescharacteristic
280C4⋊C4C22×D12711yescharacteristic
281C4⋊C4C22×D12711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—1271no—
284C22⋊C4—1271no—
285C22⋊C4—2541no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C22⋊C4—2541no—
290C22⋊C4—2541no—
291C22⋊C4—2541no—
292C22⋊C4—2541no—
293C42—1271no—
294C42—2541no—
295C42C22×D12711yescharacteristic
296C23—21no—
297C23—1271no—
298C23—10161no—
299C23—10161no—
300C23—2544no—
301C23—2541no—
302C23—2541no—
303C23—2541no—
304D4—21no—
305D4—1272no—
306D4—10161no—
307D4—10161no—
308D4—10161no—
309D4—10161no—
310D4—2544no—
311D4—2541no—
312D4—2541no—
313D4—2541no—
314D4—2541no—
315D4—2541no—
316D4#1452324912yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—1271no—
322C2×C4—1271no—
323C2×C4—1271no—
324C2×C4—10161no—
325C2×C4—10161no—
326C2×C4—10161no—
327C2×C4—10161no—
328C2×C4—10161no—
329C2×C4—2541no—
330C2×C4—2541no—
331C2×C4—2541no—
332C2×C4D4×D12711yescharacteristic
333C2×C4D4×D12711yescharacteristic
334C2×C4C23×D12711yescharacteristic
335C8—21no—
336C8—21no—
337C8—2541no—
338C8—2541no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—10164no—
343C22—5081no—
344C22—5081no—
345C22—5081no—
346C22—5081no—
347C22—5081no—
348C22—5081no—
349C22—5081no—
350C22—5081no—
351C22—2541no—
352C22—2542no—
353C22—2541no—
354C22—2541no—
355C22C2×D4×D12711yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—5081no—
362C4—2541no—
363C4—2541no—
364C4—2541no—
365C4#41×D12711yescharacteristic
366C4C2×#1452324911yescharacteristic
367C2—42no—
368C2—10161no—
369C2—10161no—
370C2—5081no—
371C2—5082no—
372C2#2255038911yescharacteristic
373C2#2255067811yescharacteristic
374C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#478×C127C2
socleC22×C127C22×D4
hypercenter#478D127
hypocenterC127#3122
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #22550482C2semidirect
2. #22550348C2semidirect
3. #22550203C2semidirect
4. #22550202C2semidirect
5. #22550173C2semidirect
6. #22550072C2semidirect
7. #22550071C2semidirect
8. #22550011C2semidirect
9. #22550011C2semidirect
10. #22549987C2semidirect
11. #22549985C2semidirect
12. #22549958C2semidirect
13. #22549956C2semidirect
14. #22549936C2semidirect
15. #478×C127C2semidirect
21. #16861206C22semidirect
31. #16861171C22semidirect
32. #16861171C22semidirect
39. #16861142C22semidirect
40. #16861140C22semidirect
41. C4⋊(C127⋊C8)C22semidirect
42. #16861131C22semidirect
43. #16861131C22semidirect
46. #16861129C22semidirect
47. #16861129C22semidirect
50. #16861119C22semidirect
51. C8×Dic127C22semidirect
52. #125×C127C22semidirect
54. #107×C127C22semidirect
55. #106×C127C22semidirect
56. C4⋊C8×C127C22semidirect
188. C127#3122semidirect
194. #478D127semidirect
210. #125C2×D127semidirect
230. #107C2×D127semidirect
232. #106C2×D127semidirect
234. C4⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504822
#225503482
#225502032
#225502022
#225501732
#225500722
#225500712
#225500112
#225499872
#225499852
#225499582
#225499562
#225499362
#478×C1272
#3122127