Finite Group Databasebeta

← #30650205

Subgroups — #30650205

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×#16861233C211yesmaximal, characteristic, complemented
2C2×D8×D127C211yesmaximal, characteristic, complemented
3#22550521C211yesmaximal, characteristic, complemented
4#22550511C211yesmaximal, characteristic, complemented
5#22550263C211yesmaximal, characteristic, complemented
6#22550128C211yesmaximal, characteristic, complemented
7#22550104C211yesmaximal, characteristic, complemented
8#22550092C211yesmaximal, characteristic, complemented
9#22550028C211yesmaximal, characteristic, complemented
10#22550018C211yesmaximal, characteristic, complemented
11#22550011C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549986C211yesmaximal, characteristic, complemented
14#22549946C211yesmaximal, characteristic, complemented
15#459×C127C211yesmaximal, characteristic, complemented
16C2×#14523247C2211yescharacteristic, complemented
17C2×D4×D127—21no—
18C2×D4×D127C2211yescharacteristic
19C2×D4×D127C2211yescharacteristic, complemented
20C2×D4×D127C2211yescharacteristic, complemented
21#16861233—24no—
22C2×#14523237C2211yescharacteristic, complemented
23C2×#14523236C2211yescharacteristic, complemented
24C2×#14523236C2211yescharacteristic, complemented
25C2×D1016C2211yescharacteristic
26C2×#14523233C2211yescharacteristic
27C2×#14523233C2211yescharacteristic, complemented
28C2×#14523233C2211yescharacteristic, complemented
29D8×D127—24no—
30C2×#14523231C2211yescharacteristic, complemented
31C2×C8×D127C2211yescharacteristic, complemented
32#16861197C2211yescharacteristic, complemented
33#16861193—21no—
34#16861189C2211yescharacteristic, complemented
35#16861184C2211yescharacteristic, complemented
36#16861183—21no—
37#16861180—21no—
38#16861180C2211yescharacteristic
39#16861180C2211yescharacteristic, complemented
40#16861179—21no—
41#16861177—21no—
42#16861177—21no—
43C4×D508C2211yescharacteristic
44C4×D508C2211yescharacteristic, complemented
45C2×(C2×D127)⋊C4—21no—
46#16861140C2211yescharacteristic, complemented
47#16861134C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic, complemented
49#16861130C2211yescharacteristic, complemented
50#16861129C2211yescharacteristic, complemented
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861120C2211yescharacteristic, complemented
54C2×D8×C127C2211yescharacteristic, complemented
55C2×D8×C127C2211yescharacteristic, complemented
56#127×C127C2211yescharacteristic, complemented
57C4×D4×C127C2211yescharacteristic, complemented
58C4⋊C8×C127C2211yescharacteristic, complemented
59#102×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63C23×D127—21no—
64#14523247—22no—
65#14523247—41no—
66C2×Dic254C2311yescharacteristic
67C2×D508—21no—
68C2×D508D411yescharacteristic
69C2×D508C2311yescharacteristic
70C2×D508C2311yescharacteristic
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76C2×C22⋊D127—21no—
77D4×D127—22no—
78D4×D127—22no—
79D4×D127—22no—
80D4×D127—41no—
81D4×D127—41no—
82D4×D127—41no—
83D4×D127—44no—
84C22×Dic127—21no—
85C2×C4×D127—21no—
86C2×C4×D127C2311yescharacteristic
87C2×C4×D127C2311yescharacteristic
88#14523237—22no—
89#14523236—22no—
90#14523236—41no—
91D1016—22no—
92#14523233—22no—
93#14523233—41no—
94#14523233D414yescomplemented
95#14523231—22no—
96C2×C127⋊C8C2311yescharacteristic
97C2×C127⋊C8C2311yescharacteristic
98C8×D127—22no—
99C4⋊Dic127—21no—
100C4⋊Dic127D411yescharacteristic
101C4⋊Dic127C2311yescharacteristic
102C22⋊Dic127—21no—
103C22⋊Dic127—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
110(C2×D127)⋊C4D411yescharacteristic
111C2×D4×C127—21no—
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115C2×D4×C127C2311yescharacteristic
116C22×C4×C127—21no—
117D8×C127—22no—
118D8×C127—22no—
119C2×C8×C127C2311yescharacteristic
120C2×C8×C127C2311yescharacteristic
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C42×C127C2311yescharacteristic
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—42no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127—42no—
130C22×D127—42no—
131C22×D127—42no—
132C22×D127—81no—
133C22×D127—81no—
134C22×D127—81no—
135C22×D127C2×D411yescharacteristic
136C22×D127C2×D411yescharacteristic
137Dic254—21no—
138Dic254—21no—
139D508—22no—
140D508—21no—
141D508—21no—
142D508—21no—
143D508—81no—
144D508C2×D412yes—
145C22⋊D127—42no—
146C22⋊D127—42no—
147C22⋊D127—42no—
148C22⋊D127—42no—
149C22⋊D127—42no—
150C22⋊D127—42no—
151C2×Dic127—21no—
152C2×Dic127—21no—
153C2×Dic127—42no—
154C2×Dic127C2×D411yescharacteristic
155C2×Dic127C2×D411yescharacteristic
156C4×D127—21no—
157C4×D127—21no—
158C4×D127—21no—
159C4×D127—21no—
160C4×D127—42no—
161C127⋊C8—21no—
162C127⋊C8C2×D412yescomplemented
163C23×C127—21no—
164C23×C127—21no—
165C23×C127—21no—
166D4×C127—21no—
167D4×C127—21no—
168D4×C127—21no—
169D4×C127—21no—
170D4×C127—21no—
171D4×C127—42no—
172D4×C127—81no—
173D4×C127C2×D412yescomplemented
174C2×C4×C127—21no—
175C2×C4×C127—81no—
176C2×C4×C127C2×D411yescharacteristic
177C2×C4×C127C2×D411yescharacteristic
178C2×C4×C127C2411yescharacteristic
179C8×C127—21no—
180C8×C127—21no—
181C22×C127—22no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127—41no—
188C22×C127—41no—
189C22×C127—41no—
190C22×C127C22×D411yescharacteristic
191C2×D127—21no—
192C2×D127—21no—
193C2×D127—21no—
194C2×D127—22no—
195C2×D127—21no—
196C2×D127—21no—
197C2×D127—41no—
198C2×D127—41no—
199C2×D127—41no—
200C2×D127—41no—
201C2×D127—41no—
202C2×D127—84no—
203C2×D127—84no—
204C2×D127—84no—
205C2×D127AGL(1,ℤ/8)12yescomplemented
206C4×C127—21no—
207C4×C127—21no—
208C4×C127—41no—
209C4×C127C22×D411yescharacteristic
210C4×C127H(5,𝔽2)11yescharacteristic
211Dic127—21no—
212Dic127—21no—
213Dic127—41no—
214Dic127—41no—
215C2×C127—42no—
216C2×C127—81no—
217C2×C127—81no—
218C2×C127D4211yescharacteristic
219C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C127#57611yescharacteristic
221D127—22no—
222D127—41no—
223D127—42no—
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#465—1271no—
234#463—1271no—
235#459D12711yescharacteristic, complemented
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×SD16—1271no—
251C2×D8—1271no—
252C2×D8—1271no—
253C2×D8—1271no—
254C2×D8—1271no—
255C2×D8—2544no—
256C2×D8C2×D12711yescharacteristic, complemented
257C2×D8C2×D12711yescharacteristic, complemented
258C2×M16—1271no—
259C22×C8—1271no—
260#127C2×D12711yescharacteristic, complemented
261#124—1271no—
262#123—2541no—
263C22⋊Q8—1271no—
264#121—1271no—
265#121—1271no—
266#121—1271no—
267#121—2541no—
268#121—2541no—
269#120—2541no—
270#120—2541no—
271#120—2541no—
272C4×D4—1271no—
273C4×D4—1271no—
274C4×D4C2×D12711yescharacteristic, complemented
275C2×C22⋊C4—2541no—
276#106—1271no—
277C4⋊C8C2×D12711yescharacteristic, complemented
278#103—1271no—
279#102—1271no—
280#102—1271no—
281#102—1271no—
282#102C2×D12711yescharacteristic, complemented
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×Q8—1271no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1271no—
296C2×D4—1271no—
297C2×D4—1271no—
298C2×D4—5084no—
299C2×D4—5081no—
300C2×D4—5081no—
301C2×D4—5081no—
302C2×D4—2541no—
303C2×D4—2541no—
304C2×D4—2541no—
305C2×D4—2541no—
306C2×D4—2542no—
307C2×D4—2542no—
308C2×D4—2541no—
309C2×D4—2541no—
310C2×D4—2541no—
311C2×D4—2542no—
312C2×D4C22×D12711yescharacteristic
313C2×D4C22×D12711yescharacteristic
314C2×D4C22×D12711yescharacteristic
315C22×C4—21no—
316C22×C4—1271no—
317C22×C4—1271no—
318C22×C4—2541no—
319C22×C4—2541no—
320SD16—5081no—
321SD16—2542no—
322SD16—2542no—
323D8—22no—
324D8—22no—
325D8—1274no—
326D8—5081no—
327D8—2542no—
328D8—2542no—
329M16—2542no—
330C2×C8—1271no—
331C2×C8—1271no—
332C2×C8—2542no—
333C2×C8C22×D12711yescharacteristic
334C2×C8C22×D12711yescharacteristic
335C4⋊C4—1271no—
336C4⋊C4—1271no—
337C4⋊C4—2541no—
338C4⋊C4C22×D12711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—1271no—
341C22⋊C4—1271no—
342C22⋊C4—2541no—
343C22⋊C4—2541no—
344C22⋊C4—2541no—
345C22⋊C4—2541no—
346C22⋊C4—2541no—
347C22⋊C4—2541no—
348C22⋊C4—2541no—
349C42C22×D12711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—1271no—
354C23—1271no—
355C23—10161no—
356C23—10161no—
357C23—10161no—
358C23—5082no—
359C23—5082no—
360C23—5082no—
361C23—5082no—
362C23—5082no—
363C23—5082no—
364C23—2541no—
365C23—2541no—
366Q8—2541no—
367Q8—2541no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—1272no—
376D4—10161no—
377D4—5082no—
378D4—5082no—
379D4—5082no—
380D4—5082no—
381D4—5082no—
382D4—5082no—
383D4—2542no—
384D4—2541no—
385D4—2541no—
386D4—2541no—
387D4D4×D12712yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—1271no—
391C2×C4—1271no—
392C2×C4—5082no—
393C2×C4—5082no—
394C2×C4—2541no—
395C2×C4—2541no—
396C2×C4—2541no—
397C2×C4—2541no—
398C2×C4—2541no—
399C2×C4—2541no—
400C2×C4D4×D12711yescharacteristic
401C2×C4D4×D12711yescharacteristic
402C2×C4C23×D12711yescharacteristic
403C8—21no—
404C8—21no—
405C8—1272no—
406C8—2541no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—1272no—
417C22—10164no—
418C22—10164no—
419C22—10164no—
420C22—5081no—
421C22—5081no—
422C22—5081no—
423C22—5081no—
424C22—5081no—
425C22—2541no—
426C22—2541no—
427C22—2541no—
428C22—2542no—
429C22—2541no—
430C22—2541no—
431C22C2×D4×D12711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—5081no—
436C4—5081no—
437C4—2541no—
438C4—2541no—
439C4C2×D4×D12711yescharacteristic
440C4#1686125511yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—10161no—
445C2—5082no—
446C2—5081no—
447C2—2542no—
448C2#2255051011yescharacteristic
449C2AGL(1,ℤ/8)×D12711yescharacteristic
450C2#2255067711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#459×C127C2
socleC22×C127C22×D4
hypercenter#459D127
hypocenterC127#3264
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. C2×D8×D127C2semidirect
3. #22550521C2semidirect
4. #22550511C2semidirect
5. #22550263C2semidirect
6. #22550128C2semidirect
7. #22550104C2semidirect
8. #22550092C2semidirect
9. #22550028C2semidirect
10. #22550018C2semidirect
11. #22550011C2semidirect
12. #22550008C2semidirect
13. #22549986C2semidirect
14. #22549946C2semidirect
15. #459×C127C2semidirect
16. C2×#14523247C22semidirect
19. C2×D4×D127C22semidirect
20. C2×D4×D127C22semidirect
22. C2×#14523237C22semidirect
23. C2×#14523236C22semidirect
24. C2×#14523236C22semidirect
27. C2×#14523233C22semidirect
28. C2×#14523233C22semidirect
30. C2×#14523231C22semidirect
31. C2×C8×D127C22semidirect
32. #16861197C22semidirect
34. #16861189C22semidirect
35. #16861184C22semidirect
39. #16861180C22semidirect
44. C4×D508C22semidirect
46. #16861140C22semidirect
47. #16861134C22semidirect
48. #16861131C22semidirect
49. #16861130C22semidirect
50. #16861129C22semidirect
51. (C2×D127)⋊C8C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861120C22semidirect
54. C2×D8×C127C22semidirect
55. C2×D8×C127C22semidirect
56. #127×C127C22semidirect
57. C4×D4×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #102×C127C22semidirect
60. #102×C127C22semidirect
94. #14523233D4semidirect
162. C127⋊C8C2×D4semidirect
173. D4×C127C2×D4semidirect
205. C2×D127AGL(1,ℤ/8)semidirect
226. C127#3264semidirect
235. #459D127semidirect
256. C2×D8C2×D127semidirect
257. C2×D8C2×D127semidirect
260. #127C2×D127semidirect
274. C4×D4C2×D127semidirect
277. C4⋊C8C2×D127semidirect
282. #102C2×D127semidirect
283. #102C2×D127semidirect
387. D4D4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
C2×D8×D1272
#225505212
#225505112
#225502632
#225501282
#225501042
#225500922
#225500282
#225500182
#225500112
#225500082
#225499862
#225499462
#459×C1272
#3264127