Finite Group Databasebeta

← #30648558

Subgroups — #30648558

Order 16256.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550506C211yesmaximal, characteristic, complemented
2#22550348C211yesmaximal, characteristic
3#22550179C211yesmaximal, characteristic, complemented
4#22550174C211yesmaximal, characteristic, complemented
5#22550173C211yesmaximal, characteristic, complemented
6#22550166C211yesmaximal, characteristic, complemented
7#22550165C211yesmaximal, characteristic, complemented
8#22550157C211yesmaximal, characteristic, complemented
9#22550154C211yesmaximal, characteristic, complemented
10#22549915C211yesmaximal, characteristic, complemented
11#22549913C211yesmaximal, characteristic, complemented
12#22549901C211yesmaximal, characteristic, complemented
13#22549879C211yesmaximal, characteristic
14#102×D127C211yesmaximal, characteristic
15#478×C127C211yesmaximal, characteristic, complemented
16C2×D4×D127C2211yescharacteristic
17C2×#14523231C2211yescharacteristic
18C2×C8×D127C2211yescharacteristic
19#16861209C2211yescharacteristic, complemented
20#16861208—21no—
21#16861208C2211yescharacteristic, complemented
22#16861200C2211yescharacteristic, complemented
23#16861199—21no—
24#16861189—21no—
25#16861187—21no—
26C4×D508C2211yescharacteristic
27D4×Dic127C2211yescharacteristic
28#16861171C2211yescharacteristic
29C4×C22⋊D127—21no—
30(C22⋊D127)⋊C4—21no—
31#16861167C2211yescharacteristic
32#16861167C2211yescharacteristic
33#16861166C2211yescharacteristic
34C4⋊C4×D127C2211yescharacteristic
35C2×(C2×D127)⋊C4—21no—
36C22⋊C4×D127—21no—
37#16861143C2211yescharacteristic, complemented
38#16861143C2211yescharacteristic, complemented
39#16861142C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41#16861140C2211yescharacteristic, complemented
42#16861140C2211yescharacteristic, complemented
43C4⋊(C127⋊C8)C2211yescharacteristic, complemented
44Dic127⋊C8C2211yescharacteristic, complemented
45Dic127⋊C8C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic
47#16861131C2211yescharacteristic
48#16861130C2211yescharacteristic
49#16861130C2211yescharacteristic
50#16861129C2211yescharacteristic
51#16861129C2211yescharacteristic
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×Dic254—21no—
61C2×Dic254—21no—
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64D4×D127—22no—
65D4×D127Q814yescomplemented
66C22×Dic127—21no—
67C2×C4×D127—21no—
68C2×C4×D127—21no—
69C2×C4×D127—21no—
70C2×C4×D127C2311yescharacteristic
71#14523231—22no—
72C2×C127⋊C8C2311yescharacteristic
73C2×C127⋊C8C2311yescharacteristic
74C8×D127—22no—
75C4⋊Dic127—21no—
76C4⋊Dic127—21no—
77C4⋊Dic127C2311yescharacteristic
78C4⋊Dic127C2311yescharacteristic
79C4⋊Dic127C2311yescharacteristic
80Dic127⋊C4—21no—
81Dic127⋊C4—21no—
82Dic127⋊C4—21no—
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)⋊C4D411yescharacteristic
91(C2×D127)⋊C4D411yescharacteristic
92C4×Dic127—21no—
93C4×Dic127—21no—
94C4×Dic127C2311yescharacteristic
95C2×D4×C127C2311yescharacteristic
96C22×C4×C127—21no—
97C2×C8×C127C2311yescharacteristic
98C2×C8×C127C2311yescharacteristic
99C4⋊C4×C127—21no—
100C4⋊C4×C127—21no—
101C4⋊C4×C127C2311yescharacteristic
102C4⋊C4×C127C2311yescharacteristic
103C4⋊C4×C127C2311yescharacteristic
104C22⋊C4×C127—21no—
105C42×C127C2311yescharacteristic
106C22×D127—21no—
107C22×D127—24no—
108C22×D127—81no—
109C22×D127—81no—
110C22×D127C2×D411yescharacteristic
111Dic254—81no—
112Dic254—81no—
113D508—21no—
114D508C2×Q812yes—
115C22⋊D127—24no—
116C2×Dic127—21no—
117C2×Dic127—21no—
118C2×Dic127—21no—
119C2×Dic127—81no—
120C2×Dic127C2×D411yescharacteristic
121C2×Dic127C2×D411yescharacteristic
122C2×Dic127C2×D411yescharacteristic
123C4×D127—81no—
124C4×D127—81no—
125C4×D127—81no—
126C4×D127C2×Q812yes—
127C4×D127#4112yes—
128C127⋊C8—21no—
129C127⋊C8—21no—
130C23×C127—21no—
131D4×C127—21no—
132D4×C127C2×Q812yes—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—81no—
137C2×C4×C127C2×D411yescharacteristic
138C2×C4×C127C2×D411yescharacteristic
139C2×C4×C127C2411yescharacteristic
140C8×C127—21no—
141C8×C127—21no—
142C22×C127—22no—
143C22×C127—41no—
144C22×C127—41no—
145C22×C127C22×D411yescharacteristic
146C2×D127—21no—
147C2×D127—21no—
148C2×D127—22no—
149C2×D127—41no—
150C2×D127—41no—
151C2×D127—84no—
152C2×D127C22⋊Q812yes—
153C4×C127—21no—
154C4×C127—21no—
155C4×C127—41no—
156C4×C127—41no—
157C4×C127—41no—
158C4×C127C22×Q811yescharacteristic
159C4×C127C2×#4111yescharacteristic
160Dic127—21no—
161Dic127—21no—
162Dic127—41no—
163Dic127—41no—
164Dic127—41no—
165Dic127C22⋊Q812yes—
166C2×C127—42no—
167C2×C127C2×C22⋊Q811yescharacteristic
168C2×C127#57611yescharacteristic
169C2×C127#57711yescharacteristic
170D127—22no—
171D127—42no—
172#3081—1271nomaximal, Sylow
173C127#308111yescharacteristic, complemented, Sylow
174#544—1271no—
175#518—1271no—
176#478—1271no—
177#478—1271no—
178#478—1271no—
179#478D12711yescharacteristic, complemented
180#476—1271no—
181#476—1271no—
182#474—1271no—
183#474—1271no—
184#428—1271no—
185#427—1271no—
186#424—1271no—
187#418—1271no—
188C2×#102—1271no—
189C22×D4—1271no—
190C2×M16—1271no—
191C22×C8—1271no—
192C4⋊Q8—1271no—
193C4⋊Q8—1271no—
194C4⋊Q8—2541no—
195#125—1271no—
196#125—2541no—
197#125C2×D12711yescharacteristic, complemented
198C22⋊Q8—2541no—
199C22⋊Q8—2541no—
200C4×D4—1271no—
201C4×D4—1271no—
202C4×D4—1271no—
203C4×D4—2541no—
204C4×D4—2541no—
205C4×D4C2×D12711yescharacteristic
206#117—1271no—
207#117—1271no—
208#117—1271no—
209C2×C4⋊C4—1271no—
210C2×C22⋊C4—2541no—
211C2×C22⋊C4—2541no—
212#107—1271no—
213#107—1271no—
214#107—1271no—
215#107C2×D12711yescharacteristic, complemented
216#106—1271no—
217#106—1271no—
218#106—1271no—
219#106C2×D12711yescharacteristic, complemented
220C4⋊C8—1271no—
221C4⋊C8—1271no—
222C4⋊C8—1271no—
223C4⋊C8C2×D12711yescharacteristic, complemented
224#102—1271no—
225#102—1271no—
226#102—1271no—
227#102—1271no—
228#102—1271no—
229#102—1271no—
230#102C2×D12711yescharacteristic
231#102C2×D12711yescharacteristic
232C24—2541no—
233C2×Q8—2541no—
234C2×Q8—2541no—
235C2×D4—1274no—
236C2×D4—1271no—
237C2×D4—2542no—
238C2×D4—2541no—
239C2×D4C22×D12711yescharacteristic
240C22×C4—21no—
241C22×C4—1271no—
242C22×C4—2541no—
243C22×C4—2541no—
244C22×C4—2541no—
245C22×C4—2541no—
246M16—2542no—
247C2×C8—1271no—
248C2×C8—1271no—
249C2×C8—2542no—
250C2×C8C22×D12711yescharacteristic
251C2×C8C22×D12711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—1271no—
255C4⋊C4—1271no—
256C4⋊C4—1271no—
257C4⋊C4—1271no—
258C4⋊C4—1271no—
259C4⋊C4—2541no—
260C4⋊C4—2541no—
261C4⋊C4—2541no—
262C4⋊C4—2541no—
263C4⋊C4—2541no—
264C4⋊C4C22×D12711yescharacteristic
265C4⋊C4C22×D12711yescharacteristic
266C4⋊C4C22×D12711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—1271no—
269C22⋊C4—1271no—
270C22⋊C4—2541no—
271C22⋊C4—2541no—
272C22⋊C4—2541no—
273C22⋊C4—2541no—
274C22⋊C4—2541no—
275C42—1271no—
276C42—2541no—
277C42—2541no—
278C42C22×D12711yescharacteristic
279C23—21no—
280C23—1271no—
281C23—10161no—
282C23—10161no—
283C23—2544no—
284C23—2541no—
285Q8—10161no—
286Q8—10161no—
287D4—21no—
288D4—1272no—
289D4—2544no—
290D4—2541no—
291D4Q8×D12712yes—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—81no—
296C2×C4—1271no—
297C2×C4—1272no—
298C2×C4—1272no—
299C2×C4—1271no—
300C2×C4—1271no—
301C2×C4—10161no—
302C2×C4—10161no—
303C2×C4—10161no—
304C2×C4—10161no—
305C2×C4—2541no—
306C2×C4—2541no—
307C2×C4—2541no—
308C2×C4D4×D12711yescharacteristic
309C2×C4D4×D12711yescharacteristic
310C2×C4C23×D12711yescharacteristic
311C8—21no—
312C8—21no—
313C8—2541no—
314C8—2541no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—1272no—
319C22—10164no—
320C22—5081no—
321C22—5081no—
322C22—2542no—
323C22—2541no—
324C22—2541no—
325C22C2×D4×D12711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—1272no—
332C4—5081no—
333C4—5081no—
334C4—5081no—
335C4—2541no—
336C4—2541no—
337C4C2×Q8×D12711yescharacteristic
338C4#41×D12711yescharacteristic
339C2—42no—
340C2—5082no—
341C2—2542no—
342C2C22⋊Q8×D12711yescharacteristic
343C2#2255067811yescharacteristic
344C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#478×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×D127C22
hypercenter#478D127
hypocenterC127#3081
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. #22550506C2semidirect
3. #22550179C2semidirect
4. #22550174C2semidirect
5. #22550173C2semidirect
6. #22550166C2semidirect
7. #22550165C2semidirect
8. #22550157C2semidirect
9. #22550154C2semidirect
10. #22549915C2semidirect
11. #22549913C2semidirect
12. #22549901C2semidirect
15. #478×C127C2semidirect
19. #16861209C22semidirect
21. #16861208C22semidirect
22. #16861200C22semidirect
37. #16861143C22semidirect
38. #16861143C22semidirect
39. #16861142C22semidirect
40. #16861140C22semidirect
41. #16861140C22semidirect
42. #16861140C22semidirect
43. C4⋊(C127⋊C8)C22semidirect
44. Dic127⋊C8C22semidirect
45. Dic127⋊C8C22semidirect
52. #125×C127C22semidirect
54. #107×C127C22semidirect
55. #106×C127C22semidirect
56. C4⋊C8×C127C22semidirect
65. D4×D127Q8semidirect
173. C127#3081semidirect
179. #478D127semidirect
197. #125C2×D127semidirect
215. #107C2×D127semidirect
219. #106C2×D127semidirect
223. C4⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505062
#225503482
#225501792
#225501742
#225501732
#225501662
#225501652
#225501572
#225501542
#225499152
#225499132
#225499012
#225498792
#102×D1272
#478×C1272
#3081127