Finite Group Databasebeta

← #30650209

Subgroups — #30650209

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
2C2×D8×D127C211yesmaximal, characteristic, complemented
3#22550519C211yesmaximal, characteristic, complemented
4#22550511C211yesmaximal, characteristic, complemented
5#22550126C211yesmaximal, characteristic, complemented
6#22550102C211yesmaximal, characteristic, complemented
7#22550088C211yesmaximal, characteristic, complemented
8#22550068C211yesmaximal, characteristic, complemented
9#22550025C211yesmaximal, characteristic, complemented
10#22550020C211yesmaximal, characteristic, complemented
11#22550008C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549989C211yesmaximal, characteristic, complemented
14#22549949C211yesmaximal, characteristic, complemented
15#496×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic, complemented
17C2×D4×D127—21no—
18C2×D4×D127C2211yescharacteristic
19C2×D4×D127C2211yescharacteristic, complemented
20C2×D4×D127C2211yescharacteristic, complemented
21#16861236—24no—
22C2×#14523235C2211yescharacteristic, complemented
23C2×D1016C2211yescharacteristic
24C2×#14523233C2211yescharacteristic
25C2×#14523233C2211yescharacteristic, complemented
26C2×#14523233C2211yescharacteristic, complemented
27D8×D127—24no—
28C2×#14523231C2211yescharacteristic, complemented
29C2×C8×D127C2211yescharacteristic, complemented
30#16861191—21no—
31#16861185C2211yescharacteristic, complemented
32#16861182—21no—
33#16861180—21no—
34#16861180C2211yescharacteristic
35#16861180C2211yescharacteristic, complemented
36#16861180C2211yescharacteristic, complemented
37#16861179—21no—
38#16861177—21no—
39#16861177—21no—
40C4×D508C2211yescharacteristic
41C4×D508C2211yescharacteristic, complemented
42C4×D508C2211yescharacteristic, complemented
43C22⋊C4×D127—21no—
44#16861141C2211yescharacteristic, complemented
45C4⋊(C127⋊C8)C2211yescharacteristic, complemented
46#16861133C2211yescharacteristic, complemented
47#16861131C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic, complemented
49#16861130C2211yescharacteristic, complemented
50#16861130C2211yescharacteristic, complemented
51#16861130C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53(C2×D127)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C2×SD16×C127C2211yescharacteristic, complemented
56C2×D8×C127C2211yescharacteristic, complemented
57C2×D8×C127C2211yescharacteristic, complemented
58#127×C127C2211yescharacteristic, complemented
59#124×C127C2211yescharacteristic, complemented
60#97×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63C23×D127—21no—
64#14523249—22no—
65#14523249—41no—
66C2×D508—21no—
67C2×D508D411yescharacteristic
68C2×D508C2311yescharacteristic
69C2×D508C2311yescharacteristic
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—21no—
73C2×C22⋊D127—21no—
74C2×C22⋊D127—21no—
75C2×C22⋊D127—21no—
76D4×D127—22no—
77D4×D127—22no—
78D4×D127—22no—
79D4×D127—41no—
80D4×D127—41no—
81D4×D127—41no—
82D4×D127—44no—
83C2×C4×D127—21no—
84C2×C4×D127—21no—
85C2×C4×D127—21no—
86C2×C4×D127C2311yescharacteristic
87C2×C4×D127C2311yescharacteristic
88#14523235—22no—
89D1016D414yescomplemented
90#14523233—22no—
91#14523233—22no—
92#14523233—22no—
93#14523231—22no—
94C2×C127⋊C8C2311yescharacteristic
95C2×C127⋊C8C2311yescharacteristic
96C8×D127—22no—
97C4⋊Dic127D411yescharacteristic
98C4⋊Dic127C2311yescharacteristic
99C4⋊Dic127C2311yescharacteristic
100Dic127⋊C4—21no—
101C22⋊Dic127—21no—
102C22⋊Dic127—21no—
103C22⋊Dic127—21no—
104(C2×D127)⋊C4—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4D411yescharacteristic
108(C2×D127)⋊C4D411yescharacteristic
109C2×Q8×C127C2311yescharacteristic
110C2×D4×C127—21no—
111C2×D4×C127—21no—
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115SD16×C127—22no—
116SD16×C127—41no—
117D8×C127—22no—
118D8×C127—41no—
119C2×C8×C127C2311yescharacteristic
120C2×C8×C127C2311yescharacteristic
121C22⋊C4×C127—21no—
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
137D508—22no—
138D508—21no—
139D508—21no—
140D508—42no—
141D508C2×D412yes—
142D508C2×D412yescomplemented
143C22⋊D127—42no—
144C22⋊D127—42no—
145C22⋊D127—42no—
146C22⋊D127—42no—
147C22⋊D127—42no—
148C22⋊D127—81no—
149C2×Dic127—21no—
150C2×Dic127—21no—
151C2×Dic127C2×D411yescharacteristic
152C2×Dic127C2×D411yescharacteristic
153C4×D127—21no—
154C4×D127—21no—
155C4×D127—21no—
156C4×D127—21no—
157C4×D127—42no—
158C4×D127—42no—
159C4×D127—81no—
160C127⋊C8—21no—
161C127⋊C8—21no—
162C23×C127—21no—
163C23×C127—21no—
164C23×C127—21no—
165Q8×C127—21no—
166Q8×C127—21no—
167D4×C127—21no—
168D4×C127—21no—
169D4×C127—21no—
170D4×C127—21no—
171D4×C127—21no—
172D4×C127—21no—
173D4×C127—42no—
174D4×C127—81no—
175C2×C4×C127—21no—
176C2×C4×C127C2×D411yescharacteristic
177C2×C4×C127C2×D411yescharacteristic
178C2×C4×C127C2411yescharacteristic
179C8×C127—21no—
180C8×C127C2×D412yescomplemented
181C22×C127—41no—
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—22no—
196C2×D127—21no—
197C2×D127—21no—
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—81no—
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—42no—
225#3264—1271nomaximal, Sylow
226C127#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1271no—
228C22×D8—1271no—
229#546—1271no—
230D42—1271no—
231#496D12711yescharacteristic, complemented
232#469—1271no—
233#465—1271no—
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×SD16C2×D12711yescharacteristic, complemented
250C2×SD16C2×D12711yescharacteristic, complemented
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#124C2×D12711yescharacteristic, complemented
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×D4—1271no—
275C2×C22⋊C4—2541no—
276#106—1271no—
277C4⋊C8—1271no—
278#103—1271no—
279#102—1271no—
280#102—1271no—
281#102—1271no—
282#102—1271no—
283#102—1271no—
284C22⋊C8—1271no—
285C22⋊C8—1271no—
286#97C2×D12711yescharacteristic, complemented
287C24—2541no—
288C24—2541no—
289C24—2541no—
290#41—5081no—
291#41—2542no—
292C2×Q8C22×D12711yescharacteristic
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—2542no—
309C2×D4—2541no—
310C2×D4—2541no—
311C2×D4—2541no—
312C2×D4C22×D12711yescharacteristic
313C2×D4C22×D12711yescharacteristic
314C2×D4C22×D12711yescharacteristic
315C22×C4—1271no—
316C22×C4—1271no—
317C22×C4—2541no—
318C22×C4—2541no—
319C22×C4—2541no—
320SD16—22no—
321SD16—41no—
322SD16—2542no—
323D8—22no—
324D8—41no—
325D8—1274no—
326D8—2542no—
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—1271no—
338C4⋊C4—2541no—
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—
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—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—1272no—
377D4—1272no—
378D4—10161no—
379D4—5082no—
380D4—5082no—
381D4—5082no—
382D4—5082no—
383D4—5082no—
384D4—5082no—
385D4—2542no—
386D4—2541no—
387D4—2541no—
388C2×C4—21no—
389C2×C4—1271no—
390C2×C4—1271no—
391C2×C4—10161no—
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—2541no—
405C8—2541no—
406C8D4×D12712yescomplemented
407C22—41no—
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—2541no—
425C22—2541no—
426C22—2542no—
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#1686125411yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—5082no—
445C2—5081no—
446C2—5082no—
447C2—2542no—
448C2#2255051111yescharacteristic
449C2AGL(1,ℤ/8)×D12711yescharacteristic
450C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#496×C127C2
socleC22×C127C22×D4
hypercenter#496D127
hypocenterC127#3264
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×D8×D127C2semidirect
3. #22550519C2semidirect
4. #22550511C2semidirect
5. #22550126C2semidirect
6. #22550102C2semidirect
7. #22550088C2semidirect
8. #22550068C2semidirect
9. #22550025C2semidirect
10. #22550020C2semidirect
11. #22550008C2semidirect
12. #22550008C2semidirect
13. #22549989C2semidirect
14. #22549949C2semidirect
15. #496×C127C2semidirect
16. C2×#14523249C22semidirect
19. C2×D4×D127C22semidirect
20. C2×D4×D127C22semidirect
22. C2×#14523235C22semidirect
25. C2×#14523233C22semidirect
26. C2×#14523233C22semidirect
28. C2×#14523231C22semidirect
29. C2×C8×D127C22semidirect
31. #16861185C22semidirect
35. #16861180C22semidirect
36. #16861180C22semidirect
41. C4×D508C22semidirect
42. C4×D508C22semidirect
44. #16861141C22semidirect
45. C4⋊(C127⋊C8)C22semidirect
46. #16861133C22semidirect
47. #16861131C22semidirect
48. #16861131C22semidirect
49. #16861130C22semidirect
50. #16861130C22semidirect
51. #16861130C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. (C2×D127)⋊C8C22semidirect
54. C2×SD16×C127C22semidirect
55. C2×SD16×C127C22semidirect
56. C2×D8×C127C22semidirect
57. C2×D8×C127C22semidirect
58. #127×C127C22semidirect
59. #124×C127C22semidirect
60. #97×C127C22semidirect
89. D1016D4semidirect
142. D508C2×D4semidirect
180. C8×C127C2×D4semidirect
205. C2×D127AGL(1,ℤ/8)semidirect
226. C127#3264semidirect
231. #496D127semidirect
249. C2×SD16C2×D127semidirect
250. C2×SD16C2×D127semidirect
256. C2×D8C2×D127semidirect
257. C2×D8C2×D127semidirect
260. #127C2×D127semidirect
261. #124C2×D127semidirect
286. #97C2×D127semidirect
406. C8D4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C2×D8×D1272
#225505192
#225505112
#225501262
#225501022
#225500882
#225500682
#225500252
#225500202
#225500082
#225499892
#225499492
#496×C1272
#3264127