Finite Group Databasebeta

← #30649081

Subgroups — #30649081

Order 16256.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550503C211yesmaximal, characteristic, complemented
2#22550444C211yesmaximal, characteristic
3#22550241C211yesmaximal, characteristic, complemented
4#22550239C211yesmaximal, characteristic, complemented
5#22550237C211yesmaximal, characteristic, complemented
6#22550232C211yesmaximal, characteristic, complemented
7#22550230C211yesmaximal, characteristic, complemented
8#22550224C211yesmaximal, characteristic, complemented
9#22550221C211yesmaximal, characteristic, complemented
10#22549879C211yesmaximal, characteristic, complemented
11#22549876C211yesmaximal, characteristic, complemented
12#22549871C211yesmaximal, characteristic, complemented
13#102×D127C211yesmaximal, characteristic, complemented
14#22549833C211yesmaximal, characteristic, complemented
15#489×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17C2×D4×D127C2211yescharacteristic
18C2×#14523231C2211yescharacteristic
19C2×C8×D127C2211yescharacteristic
20#16861208C2211yescharacteristic, complemented
21#16861207—21no—
22#16861206C2211yescharacteristic
23#16861205C2211yescharacteristic
24#16861201C2211yescharacteristic, complemented
25#16861200C2211yescharacteristic, complemented
26#16861199—21no—
27#16861198C2211yescharacteristic
28#16861187—21no—
29#16861185—21no—
30#16861182—21no—
31#16861182—21no—
32#16861179—21no—
33#16861179—21no—
34#16861167C2211yescharacteristic
35#16861166C2211yescharacteristic
36C4⋊C4×D127C2211yescharacteristic
37#16861133C2211yescharacteristic, complemented
38#16861133C2211yescharacteristic, complemented
39#16861131C2211yescharacteristic
40#16861131C2211yescharacteristic
41#16861131C2211yescharacteristic, complemented
42#16861131C2211yescharacteristic, complemented
43#16861130C2211yescharacteristic
44#16861130C2211yescharacteristic
45#16861129C2211yescharacteristic
46#16861129C2211yescharacteristic
47#16861129C2211yescharacteristic, complemented
48#16861129C2211yescharacteristic, complemented
49#16861119C2211yescharacteristic, complemented
50C4×C127⋊C8C2211yescharacteristic, complemented
51C8×Dic127C2211yescharacteristic, complemented
52C4⋊Q8×C127C2211yescharacteristic, complemented
53#124×C127C2211yescharacteristic
54#103×C127C2211yescharacteristic, complemented
55#103×C127C2211yescharacteristic, complemented
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic
58#97×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523249—22no—
61#14523249—22no—
62C2×Dic254—21no—
63C2×D508—21no—
64C2×D508—21no—
65C2×D508—21no—
66C2×D508C2311yescharacteristic
67C2×D508C2311yescharacteristic
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71D4×D127—22no—
72D4×D127—22no—
73C2×C4×D127—21no—
74C2×C4×D127—21no—
75C2×C4×D127C2311yescharacteristic
76#14523231—22no—
77C2×C127⋊C8C2311yescharacteristic
78C2×C127⋊C8C2311yescharacteristic
79C8×D127—22no—
80C4⋊Dic127—21no—
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4—21no—
87Dic127⋊C4—21no—
88Dic127⋊C4D411yescharacteristic
89Dic127⋊C4D411yescharacteristic
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×Q8×C127—21no—
98C2×Q8×C127C2311yescharacteristic
99C2×D4×C127C2311yescharacteristic
100C2×C8×C127C2311yescharacteristic
101C2×C8×C127C2311yescharacteristic
102C4⋊C4×C127—21no—
103C4⋊C4×C127C2311yescharacteristic
104C4⋊C4×C127C2311yescharacteristic
105C22⋊C4×C127—21no—
106C22⋊C4×C127—21no—
107C42×C127C2311yescharacteristic
108C22×D127—21no—
109C22×D127—21no—
110C22×D127—42no—
111C22×D127—81no—
112C22×D127—81no—
113C22×D127C2×D411yescharacteristic
114Dic254—81no—
115D508—21no—
116D508—21no—
117D508—21no—
118D508—21no—
119D508—42no—
120D508—81no—
121D508—81no—
122C22⋊D127—42no—
123C22⋊D127—81no—
124C22⋊D127—81no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127C2×D411yescharacteristic
128C2×Dic127C2×D411yescharacteristic
129C2×Dic127C2×D411yescharacteristic
130C4×D127—42no—
131C4×D127—81no—
132C4×D127#4112yes—
133C4×D127#4112yes—
134C127⋊C8—21no—
135C127⋊C8—21no—
136C23×C127—21no—
137Q8×C127—21no—
138Q8×C127—21no—
139Q8×C127—81no—
140D4×C127—21no—
141D4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—21no—
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C8×C127—21no—
149C8×C127—21no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127C22×D411yescharacteristic
154C2×D127—21no—
155C2×D127—21no—
156C2×D127—41no—
157C2×D127—41no—
158C2×D127—41no—
159C2×D127—41no—
160C2×D127—41no—
161C2×D127—41no—
162C2×D127—84no—
163C2×D127#12412yes—
164C4×C127—21no—
165C4×C127—21no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127C2×#4111yescharacteristic
170C4×C127C2×#4111yescharacteristic
171Dic127—21no—
172Dic127—21no—
173Dic127—41no—
174Dic127—41no—
175Dic127#12412yes—
176C2×C127—81no—
177C2×C127C2×#12411yescharacteristic
178C2×C127#57611yescharacteristic
179C2×C127#57711yescharacteristic
180D127—22no—
181D127—81no—
182D127—81no—
183#3136—1271nomaximal, Sylow
184C127#313611yescharacteristic, complemented, Sylow
185#544—1271no—
186#535—1271no—
187#490—1271no—
188#490—1271no—
189#489—1271no—
190#489D12711yescharacteristic, complemented
191#488—1271no—
192#488—1271no—
193#486—1271no—
194#486—1271no—
195#418—1271no—
196#417—1271no—
197#416—1271no—
198C2×#102—1271no—
199#405—1271no—
200C2×#41—1271no—
201C22×D4—1271no—
202C2×M16—1271no—
203C22×C8—1271no—
204C4⋊Q8—1271no—
205C4⋊Q8—2541no—
206C4⋊Q8C2×D12711yescharacteristic, complemented
207#127—1271no—
208#127—1271no—
209#125—1271no—
210#125—1271no—
211#125—2541no—
212#124—1271no—
213#124C2×D12711yescharacteristic
214C22⋊Q8—2541no—
215C22⋊Q8—2541no—
216#121—2541no—
217#121—2541no—
218#120—2541no—
219#120—2541no—
220#117—1271no—
221#117—1271no—
222C2×C4⋊C4—1271no—
223#103—1271no—
224#103—1271no—
225#103C2×D12711yescharacteristic, complemented
226#103C2×D12711yescharacteristic, complemented
227#102—1271no—
228#102—1271no—
229#102—1271no—
230#102—1271no—
231#102—1271no—
232#102—1271no—
233#102—1271no—
234#102—1271no—
235#102—1271no—
236#102—1271no—
237#102C2×D12711yescharacteristic
238#102C2×D12711yescharacteristic
239#97—1271no—
240#97C2×D12711yescharacteristic, complemented
241C4×C8—1271no—
242C4×C8—1271no—
243C24—2541no—
244#41—2542no—
245#41—2542no—
246C2×Q8—21no—
247C2×Q8—2541no—
248C2×Q8C22×D12711yescharacteristic
249C2×D4—1271no—
250C2×D4—1271no—
251C2×D4—2541no—
252C2×D4—2542no—
253C2×D4—2542no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4—2541no—
257C2×D4—2541no—
258C2×D4—2541no—
259C2×D4C22×D12711yescharacteristic
260C22×C4—1271no—
261C22×C4—2541no—
262C22×C4—2541no—
263M16—2542no—
264C2×C8—1271no—
265C2×C8—1271no—
266C2×C8—2542no—
267C2×C8C22×D12711yescharacteristic
268C2×C8C22×D12711yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—1271no—
271C4⋊C4—1271no—
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⋊C4C22×D12711yescharacteristic
281C4⋊C4C22×D12711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—1271no—
285C22⋊C4—1271no—
286C22⋊C4—2541no—
287C22⋊C4—2541no—
288C22⋊C4—2541no—
289C42—1271no—
290C42—2541no—
291C42C22×D12711yescharacteristic
292C23—21no—
293C23—1271no—
294C23—10161no—
295C23—10161no—
296C23—5082no—
297C23—2541no—
298C23—2541no—
299Q8—21no—
300Q8—21no—
301Q8—81no—
302Q8—10161no—
303D4—21no—
304D4—21no—
305D4—10161no—
306D4—10161no—
307D4—10161no—
308D4—10161no—
309D4—5082no—
310D4—5082no—
311D4—2541no—
312D4—2541no—
313D4—2541no—
314D4—2541no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—1271no—
319C2×C4—1272no—
320C2×C4—1272no—
321C2×C4—1271no—
322C2×C4—1271no—
323C2×C4—10161no—
324C2×C4—5082no—
325C2×C4—2541no—
326C2×C4—2541no—
327C2×C4D4×D12711yescharacteristic
328C2×C4D4×D12711yescharacteristic
329C2×C4C23×D12711yescharacteristic
330C8—21no—
331C8—21no—
332C8—2541no—
333C8—2541no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—1272no—
338C22—10164no—
339C22—5081no—
340C22—5081no—
341C22—5081no—
342C22—5081no—
343C22—5081no—
344C22—5081no—
345C22—2541no—
346C22—2541no—
347C22C2×D4×D12711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—1272no—
354C4—5081no—
355C4—5081no—
356C4—2541no—
357C4—2541no—
358C4#41×D12711yescharacteristic
359C4#41×D12711yescharacteristic
360C2—81no—
361C2—10161no—
362C2—10161no—
363C2—2542no—
364C2#124×D12711yescharacteristic
365C2#2255068011yescharacteristic
366C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#489×C127C2
socleC22×C127C22×D4
prime radical#22550444C2
hypercenter#489D127
hypocenterC127#3136
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #22550503C2semidirect
3. #22550241C2semidirect
4. #22550239C2semidirect
5. #22550237C2semidirect
6. #22550232C2semidirect
7. #22550230C2semidirect
8. #22550224C2semidirect
9. #22550221C2semidirect
10. #22549879C2semidirect
11. #22549876C2semidirect
12. #22549871C2semidirect
13. #102×D127C2semidirect
14. #22549833C2semidirect
15. #489×C127C2semidirect
20. #16861208C22semidirect
24. #16861201C22semidirect
25. #16861200C22semidirect
37. #16861133C22semidirect
38. #16861133C22semidirect
41. #16861131C22semidirect
42. #16861131C22semidirect
47. #16861129C22semidirect
48. #16861129C22semidirect
49. #16861119C22semidirect
50. C4×C127⋊C8C22semidirect
51. C8×Dic127C22semidirect
52. C4⋊Q8×C127C22semidirect
54. #103×C127C22semidirect
55. #103×C127C22semidirect
58. #97×C127C22semidirect
184. C127#3136semidirect
190. #489D127semidirect
206. C4⋊Q8C2×D127semidirect
225. #103C2×D127semidirect
226. #103C2×D127semidirect
240. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505032
#225504442
#225502412
#225502392
#225502372
#225502322
#225502302
#225502242
#225502212
#225498792
#225498762
#225498712
#102×D1272
#225498332
#489×C1272
#3136127