Finite Group Databasebeta

← #30650207

Subgroups — #30650207

Order 16256.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C211yesmaximal, characteristic, complemented
2C22×D1016C211yesmaximal, characteristic, complemented
3#22550517C211yesmaximal, characteristic, complemented
4#22550510C211yesmaximal, characteristic, complemented
5#22550262C211yesmaximal, characteristic, complemented
6#22550130C211yesmaximal, characteristic, complemented
7#22550094C211yesmaximal, characteristic, complemented
8#22550070C211yesmaximal, characteristic, complemented
9#22550026C211yesmaximal, characteristic, complemented
10#22550024C211yesmaximal, characteristic, complemented
11#22550011C211yesmaximal, characteristic, complemented
12#22550009C211yesmaximal, characteristic, complemented
13#22549987C211yesmaximal, characteristic, complemented
14#22549947C211yesmaximal, characteristic, complemented
15#465×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic, complemented
17C22×D508C2211yescharacteristic
18C22×D508C2211yescharacteristic, complemented
19C2×D4×D127—21no—
20C2×D4×D127C2211yescharacteristic, complemented
21#16861236—24no—
22C2×#14523235C2211yescharacteristic, complemented
23C2×#14523235C2211yescharacteristic, complemented
24C2×D1016—24no—
25C2×D1016C2211yescharacteristic
26C2×D1016C2211yescharacteristic
27C2×D1016C2211yescharacteristic, complemented
28C2×D1016C2211yescharacteristic, complemented
29C2×#14523233C2211yescharacteristic, complemented
30C2×#14523233C2211yescharacteristic, complemented
31C2×#14523231C2211yescharacteristic, complemented
32#16861205C2211yescharacteristic, complemented
33#16861198C2211yescharacteristic, complemented
34#16861194—21no—
35#16861184—21no—
36#16861184C2211yescharacteristic, complemented
37#16861184C2211yescharacteristic, complemented
38#16861182—21no—
39#16861179—21no—
40#16861179—21no—
41#16861177—21no—
42#16861171C2211yescharacteristic, complemented
43#16861171C2211yescharacteristic, complemented
44C4×C22⋊D127C2211yescharacteristic
45C22⋊C4×D127—21no—
46Dic127⋊C8C2211yescharacteristic, complemented
47#16861129C2211yescharacteristic, complemented
48#16861129C2211yescharacteristic, complemented
49#16861129C2211yescharacteristic, complemented
50#16861129C2211yescharacteristic, complemented
51C22⋊(C127⋊C8)C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861119C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C22×C8×C127C2211yescharacteristic, complemented
56C22⋊Q8×C127C2211yescharacteristic, complemented
57#121×C127C2211yescharacteristic
58#106×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63C23×D127—21no—
64#14523249—22no—
65#14523249—41no—
66C2×D508—22no—
67C2×D508—22no—
68C2×D508—21no—
69C2×D508—21no—
70C2×D508—21no—
71C2×D508—21no—
72C2×D508—21no—
73C2×D508—41no—
74C2×D508—41no—
75C2×D508C2311yescharacteristic
76C2×D508C2311yescharacteristic
77C2×D508C2311yescharacteristic
78C2×D508C2311yescharacteristic
79C2×C22⋊D127—21no—
80C2×C22⋊D127—21no—
81C2×C22⋊D127—21no—
82C2×C22⋊D127D411yescharacteristic
83D4×D127—22no—
84D4×D127—41no—
85D4×D127—44no—
86C2×C4×D127—21no—
87C2×C4×D127—21no—
88C2×C4×D127—21no—
89C2×C4×D127C2311yescharacteristic
90#14523235—22no—
91#14523235—41no—
92D1016—22no—
93D1016—22no—
94D1016—22no—
95D1016D414yescomplemented
96#14523233—22no—
97#14523233—41no—
98#14523231—22no—
99C2×C127⋊C8C2311yescharacteristic
100C2×C127⋊C8C2311yescharacteristic
101Dic127⋊C4D411yescharacteristic
102C22⋊Dic127D411yescharacteristic
103(C2×D127)⋊C4—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4—21no—
109(C2×D127)⋊C4D411yescharacteristic
110C4×Dic127C2311yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127C2311yescharacteristic
115SD16×C127—22no—
116C2×C8×C127—22no—
117C2×C8×C127C2311yescharacteristic
118C2×C8×C127C2311yescharacteristic
119C4⋊C4×C127—21no—
120C4⋊C4×C127C2311yescharacteristic
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22⋊C4×C127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—21no—
128C22×D127—42no—
129C22×D127—42no—
130C22×D127—42no—
131C22×D127—42no—
132C22×D127—42no—
133C22×D127—42no—
134C22×D127—81no—
135C22×D127—81no—
136C22×D127—81no—
137C22×D127C2×D411yescharacteristic
138D508—21no—
139D508—21no—
140D508—21no—
141D508—21no—
142D508—21no—
143D508—21no—
144D508—42no—
145D508—42no—
146D508—42no—
147D508—42no—
148D508—81no—
149D508C2×D412yes—
150D508C2×D412yescomplemented
151C22⋊D127—22no—
152C22⋊D127—42no—
153C22⋊D127—42no—
154C22⋊D127—81no—
155C2×Dic127C2×D411yescharacteristic
156C2×Dic127C2×D411yescharacteristic
157C2×Dic127C2×D411yescharacteristic
158C4×D127—21no—
159C4×D127—21no—
160C4×D127—42no—
161C4×D127—42no—
162C4×D127—81no—
163C127⋊C8—21no—
164C127⋊C8—21no—
165C23×C127—21no—
166C23×C127C2×D411yescharacteristic
167Q8×C127—21no—
168Q8×C127—21no—
169D4×C127—21no—
170D4×C127—21no—
171D4×C127—42no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—21no—
177C2×C4×C127C2×D411yescharacteristic
178C2×C4×C127C2411yescharacteristic
179C8×C127—21no—
180C8×C127C2×D412yescomplemented
181C22×C127—21no—
182C22×C127—21no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127AGL(1,ℤ/8)12yescomplemented
187C22×C127C22×D411yescharacteristic
188C2×D127—21no—
189C2×D127—22no—
190C2×D127—22no—
191C2×D127—21no—
192C2×D127—21no—
193C2×D127—41no—
194C2×D127—41no—
195C2×D127—41no—
196C2×D127—41no—
197C2×D127—41no—
198C2×D127—41no—
199C2×D127—41no—
200C2×D127—41no—
201C2×D127—41no—
202C2×D127—41no—
203C2×D127—84no—
204C2×D127—84no—
205C2×D127—84no—
206C4×C127—21no—
207C4×C127—41no—
208C4×C127—41no—
209C4×C127—41no—
210C4×C127C22×D411yescharacteristic
211C4×C127H(5,𝔽2)11yescharacteristic
212Dic127—21no—
213Dic127—21no—
214Dic127—21no—
215C2×C127—22no—
216C2×C127—81no—
217C2×C127D4211yescharacteristic
218C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C127#57611yescharacteristic
220D127—41no—
221D127—42no—
222D127—42no—
223D127—81no—
224D127—81no—
225#3264—1271nomaximal, Sylow
226C127#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1271no—
228C22×D8—1271no—
229#546—1271no—
230D42—1271no—
231#496—1271no—
232#469—1271no—
233#465D12711yescharacteristic, complemented
234#463—1271no—
235#459—1271no—
236#450—1271no—
237#449—1271no—
238#447—1271no—
239#447—1271no—
240#442—1271no—
241#435—1271no—
242C2×#41—1271no—
243C22×D4—1271no—
244C22×D4—1271no—
245C22×D4—1271no—
246C22×D4—2541no—
247AGL(1,ℤ/8)—2544no—
248C2×SD16—1271no—
249C2×SD16—1271no—
250C2×SD16C2×D12711yescharacteristic, complemented
251C2×D8—1271no—
252C2×D8—1271no—
253C2×D8—1271no—
254C2×D8—1271no—
255C2×D8—1271no—
256C2×D8—1271no—
257C2×D8—2544no—
258C2×M16—1271no—
259C22×C8C2×D12711yescharacteristic, complemented
260#127—1271no—
261#124—1271no—
262#123—2541no—
263C22⋊Q8C2×D12711yescharacteristic, complemented
264#121—1271no—
265#121—1271no—
266#121—2541no—
267#121—2541no—
268#121C2×D12711yescharacteristic
269#120—2541no—
270#120—2541no—
271#120—2541no—
272C4×D4—1271no—
273C4×D4—1271no—
274C4×D4—1271no—
275C2×C22⋊C4—2541no—
276#106C2×D12711yescharacteristic, complemented
277C4⋊C8—1271no—
278#103C2×D12711yescharacteristic, complemented
279#102—1271no—
280#102—1271no—
281#102—1271no—
282#102—1271no—
283#102C2×D12711yescharacteristic, complemented
284C22⋊C8—1271no—
285C22⋊C8—1271no—
286#97—1271no—
287C24—2541no—
288C24—2541no—
289C24—2541no—
290#41—5081no—
291#41—2542no—
292C2×Q8C22×D12711yescharacteristic
293C2×D4—21no—
294C2×D4—1271no—
295C2×D4—1271no—
296C2×D4—1271no—
297C2×D4—1271no—
298C2×D4—1271no—
299C2×D4—5081no—
300C2×D4—5081no—
301C2×D4—5084no—
302C2×D4—5081no—
303C2×D4—2542no—
304C2×D4—2542no—
305C2×D4—2541no—
306C2×D4—2541no—
307C2×D4—2541no—
308C2×D4—2541no—
309C2×D4—2541no—
310C2×D4—2541no—
311C2×D4—2542no—
312C2×D4—2541no—
313C2×D4—2541no—
314C2×D4C22×D12711yescharacteristic
315C22×C4—1271no—
316C22×C4—2541no—
317C22×C4—2541no—
318C22×C4—2541no—
319C22×C4C22×D12711yescharacteristic
320SD16—22no—
321SD16—5081no—
322SD16—2542no—
323D8—1274no—
324D8—5081no—
325D8—2542no—
326D8—2542no—
327D8—2542no—
328D8—2542no—
329M16—2542no—
330C2×C8—22no—
331C2×C8—1271no—
332C2×C8—1271no—
333C2×C8C22×D12711yescharacteristic
334C2×C8C22×D12711yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—1271no—
337C4⋊C4C22×D12711yescharacteristic
338C4⋊C4C22×D12711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—1271no—
342C22⋊C4—1271no—
343C22⋊C4—2541no—
344C22⋊C4—2541no—
345C22⋊C4—2541no—
346C22⋊C4—2541no—
347C22⋊C4—2541no—
348C22⋊C4—2541no—
349C42—1271no—
350C23—21no—
351C23—1271no—
352C23—10161no—
353C23—10161no—
354C23—10161no—
355C23—5082no—
356C23—5082no—
357C23—5082no—
358C23—5082no—
359C23—5082no—
360C23—5082no—
361C23—2541no—
362C23—2541no—
363C23—2541no—
364C23—2541no—
365C23D4×D12711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—42no—
371D4—1272no—
372D4—1272no—
373D4—10161no—
374D4—10161no—
375D4—5082no—
376D4—5082no—
377D4—5082no—
378D4—5082no—
379D4—5082no—
380D4—5082no—
381D4—2542no—
382D4—2541no—
383D4—2541no—
384D4—2541no—
385D4—2541no—
386D4—2541no—
387D4—2541no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—1271no—
394C2×C4—1271no—
395C2×C4—1271no—
396C2×C4—10161no—
397C2×C4—5082no—
398C2×C4—5082no—
399C2×C4—2541no—
400C2×C4—2541no—
401C2×C4D4×D12711yescharacteristic
402C2×C4C23×D12711yescharacteristic
403C8—21no—
404C8—2541no—
405C8—2541no—
406C8D4×D12712yescomplemented
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—10164no—
413C22—10164no—
414C22—10164no—
415C22—5081no—
416C22—5081no—
417C22—5081no—
418C22—5081no—
419C22—5081no—
420C22—5081no—
421C22—5081no—
422C22—5081no—
423C22—5081no—
424C22—5081no—
425C22—2541no—
426C22—2541no—
427C22—2542no—
428C22—2541no—
429C22—2542no—
430C22#1686123612yescomplemented
431C22C2×D4×D12711yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—2541no—
437C4—2541no—
438C4—2541no—
439C4C2×D4×D12711yescharacteristic
440C4#1686125511yescharacteristic
441C2—22no—
442C2—81no—
443C2—10161no—
444C2—10161no—
445C2—5082no—
446C2—5081no—
447C2—5082no—
448C2#2255051011yescharacteristic
449C2C2×#1686123611yescharacteristic
450C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#465×C127C2
socleC22×C127C22×D4
hypercenter#465D127
hypocenterC127#3264
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C22×D1016C2semidirect
3. #22550517C2semidirect
4. #22550510C2semidirect
5. #22550262C2semidirect
6. #22550130C2semidirect
7. #22550094C2semidirect
8. #22550070C2semidirect
9. #22550026C2semidirect
10. #22550024C2semidirect
11. #22550011C2semidirect
12. #22550009C2semidirect
13. #22549987C2semidirect
14. #22549947C2semidirect
15. #465×C127C2semidirect
16. C2×#14523249C22semidirect
18. C22×D508C22semidirect
20. C2×D4×D127C22semidirect
22. C2×#14523235C22semidirect
23. C2×#14523235C22semidirect
27. C2×D1016C22semidirect
28. C2×D1016C22semidirect
29. C2×#14523233C22semidirect
30. C2×#14523233C22semidirect
31. C2×#14523231C22semidirect
32. #16861205C22semidirect
33. #16861198C22semidirect
36. #16861184C22semidirect
37. #16861184C22semidirect
42. #16861171C22semidirect
43. #16861171C22semidirect
46. Dic127⋊C8C22semidirect
47. #16861129C22semidirect
48. #16861129C22semidirect
49. #16861129C22semidirect
50. #16861129C22semidirect
51. C22⋊(C127⋊C8)C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861119C22semidirect
54. C2×SD16×C127C22semidirect
55. C22×C8×C127C22semidirect
56. C22⋊Q8×C127C22semidirect
58. #106×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
95. D1016D4semidirect
150. D508C2×D4semidirect
180. C8×C127C2×D4semidirect
186. C22×C127AGL(1,ℤ/8)semidirect
226. C127#3264semidirect
233. #465D127semidirect
250. C2×SD16C2×D127semidirect
259. C22×C8C2×D127semidirect
263. C22⋊Q8C2×D127semidirect
276. #106C2×D127semidirect
278. #103C2×D127semidirect
283. #102C2×D127semidirect
406. C8D4×D127semidirect
430. C22#16861236semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C22×D10162
#225505172
#225505102
#225502622
#225501302
#225500942
#225500702
#225500262
#225500242
#225500112
#225500092
#225499872
#225499472
#465×C1272
#3264127