Finite Group Databasebeta

← #30650204

Subgroups — #30650204

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×D8×D127C211yesmaximal, characteristic, complemented
2#22550514C211yesmaximal, characteristic, complemented
3D4×C22⋊D127C211yesmaximal, characteristic, complemented
4#22550262C211yesmaximal, characteristic, complemented
5#22550129C211yesmaximal, characteristic, complemented
6#22550102C211yesmaximal, characteristic, complemented
7#22550093C211yesmaximal, characteristic, complemented
8#22550069C211yesmaximal, characteristic, complemented
9#22550025C211yesmaximal, characteristic, complemented
10#22550021C211yesmaximal, characteristic, complemented
11#22550010C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549988C211yesmaximal, characteristic, complemented
14#22549945C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic, complemented
16C22×C22⋊D127—21no—
17C2×D4×D127C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic, complemented
19C2×#14523235C2211yescharacteristic, complemented
20C2×D1016C2211yescharacteristic, complemented
21C2×#14523233C2211yescharacteristic
22C2×#14523233C2211yescharacteristic, complemented
23C2×#14523233C2211yescharacteristic, complemented
24D8×D127—24no—
25C2×C8×D127C2211yescharacteristic, complemented
26#16861205C2211yescharacteristic, complemented
27#16861198C2211yescharacteristic, complemented
28#16861192—21no—
29#16861185C2211yescharacteristic, complemented
30#16861183C2211yescharacteristic, complemented
31#16861183C2211yescharacteristic, complemented
32#16861181—21no—
33#16861181—21no—
34#16861180C2211yescharacteristic
35#16861178—21no—
36#16861177—21no—
37#16861177—21no—
38D4×Dic127C2211yescharacteristic, complemented
39D4×Dic127C2211yescharacteristic, complemented
40C4×C22⋊D127C2211yescharacteristic
41C2×(C2×D127)⋊C4—21no—
42#16861141C2211yescharacteristic, complemented
43Dic127⋊C8C2211yescharacteristic, complemented
44#16861133C2211yescharacteristic, complemented
45#16861131C2211yescharacteristic, complemented
46#16861131C2211yescharacteristic, complemented
47#16861130C2211yescharacteristic, complemented
48#16861130C2211yescharacteristic, complemented
49#16861130C2211yescharacteristic, complemented
50C22⋊(C127⋊C8)C2211yescharacteristic, complemented
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52#16861119C2211yescharacteristic, complemented
53C2×#41×C127C2211yescharacteristic, complemented
54C22×D4×C127C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C127—24no—
56C2×SD16×C127C2211yescharacteristic, complemented
57C2×SD16×C127C2211yescharacteristic, complemented
58C2×D8×C127C2211yescharacteristic
59C2×D8×C127C2211yescharacteristic, complemented
60C2×M16×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63C2×D508C2311yescharacteristic
64C2×D508C2311yescharacteristic
65C2×C22⋊D127—21no—
66C2×C22⋊D127—21no—
67C2×C22⋊D127—21no—
68C2×C22⋊D127—21no—
69C2×C22⋊D127—21no—
70C2×C22⋊D127—21no—
71C2×C22⋊D127—21no—
72C2×C22⋊D127—44no—
73C2×C22⋊D127D411yescharacteristic
74D4×D127—22no—
75D4×D127—22no—
76D4×D127—41no—
77D4×D127—41no—
78C22×Dic127—21no—
79C22×Dic127—21no—
80C2×C4×D127C2311yescharacteristic
81#14523235—41no—
82D1016—22no—
83#14523233—22no—
84#14523233—22no—
85#14523233—41no—
86C2×C127⋊C8C2311yescharacteristic
87C2×C127⋊C8C2311yescharacteristic
88C8×D127—22no—
89C4⋊Dic127C2311yescharacteristic
90C4⋊Dic127C2311yescharacteristic
91Dic127⋊C4—21no—
92Dic127⋊C4D411yescharacteristic
93C22⋊Dic127—21no—
94C22⋊Dic127—21no—
95C22⋊Dic127—21no—
96C22⋊Dic127D411yescharacteristic
97(C2×D127)⋊C4—21no—
98(C2×D127)⋊C4—21no—
99(C2×D127)⋊C4—21no—
100(C2×D127)⋊C4—21no—
101(C2×D127)⋊C4—21no—
102(C2×D127)⋊C4D411yescharacteristic
103C4×Dic127C2311yescharacteristic
104C24×C127—21no—
105#41×C127—22no—
106#41×C127—41no—
107C2×Q8×C127C2311yescharacteristic
108C2×D4×C127—22no—
109C2×D4×C127—21no—
110C2×D4×C127—21no—
111C2×D4×C127—41no—
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C2×D4×C127C2311yescharacteristic
115C22×C4×C127—21no—
116C22×C4×C127C2311yescharacteristic
117SD16×C127—22no—
118SD16×C127—22no—
119D8×C127—22no—
120D8×C127D414yescomplemented
121M16×C127—22no—
122C2×C8×C127C2311yescharacteristic
123C2×C8×C127C2311yescharacteristic
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—42no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127—42no—
130C22×D127—81no—
131C22×D127—81no—
132C22×D127C2×D411yescharacteristic
133D508—21no—
134D508—21no—
135D508—21no—
136D508—21no—
137C22⋊D127—22no—
138C22⋊D127—42no—
139C22⋊D127—42no—
140C22⋊D127—42no—
141C22⋊D127—42no—
142C22⋊D127—42no—
143C22⋊D127—81no—
144C22⋊D127—81no—
145C2×Dic127—21no—
146C2×Dic127—21no—
147C2×Dic127—42no—
148C2×Dic127—81no—
149C2×Dic127C2×D411yescharacteristic
150C2×Dic127C2×D411yescharacteristic
151C2×Dic127C2×D411yescharacteristic
152C4×D127—21no—
153C4×D127—21no—
154C127⋊C8—21no—
155C127⋊C8—21no—
156C23×C127—21no—
157C23×C127—21no—
158C23×C127—21no—
159C23×C127—42no—
160C23×C127—42no—
161C23×C127—81no—
162C23×C127C2×D411yescharacteristic
163Q8×C127—21no—
164Q8×C127—21no—
165D4×C127—21no—
166D4×C127—21no—
167D4×C127—21no—
168D4×C127—21no—
169D4×C127—42no—
170D4×C127—42no—
171D4×C127C2×D412yes—
172D4×C127C2×D412yescomplemented
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—42no—
177C2×C4×C127C2×D411yescharacteristic
178C2×C4×C127C2411yescharacteristic
179C8×C127—21no—
180C8×C127C2×D412yescomplemented
181C22×C127—21no—
182C22×C127—22no—
183C22×C127—22no—
184C22×C127—21no—
185C22×C127—21no—
186C22×C127—41no—
187C22×C127—41no—
188C22×C127—41no—
189C22×C127—41no—
190C22×C127—41no—
191C22×C127—41no—
192C22×C127—41no—
193C22×C127—84no—
194C22×C127C22×D411yescharacteristic
195C2×D127—21no—
196C2×D127—21no—
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×D127AGL(1,ℤ/8)12yescomplemented
206C4×C127—21no—
207C4×C127—41no—
208C4×C127C22×D411yescharacteristic
209C4×C127H(5,𝔽2)11yescharacteristic
210Dic127—21no—
211Dic127—21no—
212Dic127—21no—
213Dic127—41no—
214Dic127—41no—
215C2×C127—42no—
216C2×C127—42no—
217C2×C127—41no—
218C2×C127—81no—
219C2×C127D4211yescharacteristic
220C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C127#57611yescharacteristic
222D127—22no—
223D127—81no—
224D127—81no—
225#3264—1271nomaximal, Sylow
226C127#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)D12711yescharacteristic, complemented
228C22×D8—1271no—
229#546—1271no—
230D42—1271no—
231#496—1271no—
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×#41C2×D12711yescharacteristic, complemented
243C22×D4—1271no—
244C22×D4—1271no—
245C22×D4—2541no—
246C22×D4C2×D12711yescharacteristic, complemented
247AGL(1,ℤ/8)—24no—
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
257C2×D8C2×D12711yescharacteristic, complemented
258C2×M16C2×D12711yescharacteristic, complemented
259C22×C8—1271no—
260#127—1271no—
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×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#97—1271no—
287C24—21no—
288C24—2541no—
289C24—2541no—
290#41—22no—
291#41—41no—
292C2×Q8C22×D12711yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1271no—
298C2×D4—1271no—
299C2×D4—1271no—
300C2×D4—5084no—
301C2×D4—5081no—
302C2×D4—5081no—
303C2×D4—2541no—
304C2×D4—2541no—
305C2×D4—2541no—
306C2×D4—2541no—
307C2×D4—2541no—
308C2×D4—2541no—
309C2×D4—2541no—
310C2×D4—2542no—
311C2×D4—2542no—
312C2×D4C22×D12711yescharacteristic
313C2×D4C22×D12711yescharacteristic
314C2×D4C22×D12711yescharacteristic
315C22×C4—21no—
316C22×C4—1271no—
317C22×C4—2541no—
318C22×C4—2541no—
319C22×C4C22×D12711yescharacteristic
320SD16—22no—
321SD16—22no—
322SD16—5081no—
323D8—22no—
324D8—5081no—
325D8—2542no—
326D8—2542no—
327D8—2542no—
328D8C22⋊D12714yescomplemented
329M16—22no—
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—1271no—
340C22⋊C4—1271no—
341C22⋊C4—2541no—
342C22⋊C4—2541no—
343C22⋊C4—2541no—
344C22⋊C4—2541no—
345C22⋊C4—2541no—
346C22⋊C4—2541no—
347C22⋊C4—2541no—
348C22⋊C4—2541no—
349C42—1271no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—81no—
356C23—1271no—
357C23—10161no—
358C23—10161no—
359C23—5082no—
360C23—5082no—
361C23—5082no—
362C23—5082no—
363C23—2541no—
364C23—2541no—
365C23D4×D12711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—42no—
374D4—10161no—
375D4—10161no—
376D4—5082no—
377D4—5082no—
378D4—5082no—
379D4—5082no—
380D4—5082no—
381D4—2542no—
382D4—2541no—
383D4—2541no—
384D4—2541no—
385D4—2541no—
386D4C2×C22⋊D12712yes—
387D4C2×C22⋊D12712yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—1271no—
393C2×C4—1271no—
394C2×C4—1271no—
395C2×C4—10161no—
396C2×C4—5082no—
397C2×C4—2541no—
398C2×C4—2541no—
399C2×C4—2541no—
400C2×C4—2541no—
401C2×C4D4×D12711yescharacteristic
402C2×C4C23×D12711yescharacteristic
403C8—21no—
404C8—2541no—
405C8—2541no—
406C8C2×C22⋊D12712yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—84no—
420C22—1272no—
421C22—10164no—
422C22—10164no—
423C22—5081no—
424C22—5081no—
425C22—5081no—
426C22—5081no—
427C22—5081no—
428C22—5081no—
429C22—2541no—
430C22—2541no—
431C22C2×D4×D12711yescharacteristic
432C4—21no—
433C4—41no—
434C4—5081no—
435C4—5081no—
436C4—2541no—
437C4—2541no—
438C4—2541no—
439C4C22×C22⋊D12711yescharacteristic
440C4#1686125411yescharacteristic
441C2—41no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—10161no—
446C2—10161no—
447C2—2542no—
448C2D4×C22⋊D12711yescharacteristic
449C2AGL(1,ℤ/8)×D12711yescharacteristic
450C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×AGL(1,ℤ/8)×C127C2
socleC22×C127C22×D4
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#3264
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×D8×D127C2semidirect
2. #22550514C2semidirect
3. D4×C22⋊D127C2semidirect
4. #22550262C2semidirect
5. #22550129C2semidirect
6. #22550102C2semidirect
7. #22550093C2semidirect
8. #22550069C2semidirect
9. #22550025C2semidirect
10. #22550021C2semidirect
11. #22550010C2semidirect
12. #22550008C2semidirect
13. #22549988C2semidirect
14. #22549945C2semidirect
15. C2×AGL(1,ℤ/8)×C127C2semidirect
18. C2×D4×D127C22semidirect
19. C2×#14523235C22semidirect
20. C2×D1016C22semidirect
22. C2×#14523233C22semidirect
23. C2×#14523233C22semidirect
25. C2×C8×D127C22semidirect
26. #16861205C22semidirect
27. #16861198C22semidirect
29. #16861185C22semidirect
30. #16861183C22semidirect
31. #16861183C22semidirect
38. D4×Dic127C22semidirect
39. D4×Dic127C22semidirect
42. #16861141C22semidirect
43. Dic127⋊C8C22semidirect
44. #16861133C22semidirect
45. #16861131C22semidirect
46. #16861131C22semidirect
47. #16861130C22semidirect
48. #16861130C22semidirect
49. #16861130C22semidirect
50. C22⋊(C127⋊C8)C22semidirect
51. (C2×D127)⋊C8C22semidirect
52. #16861119C22semidirect
53. C2×#41×C127C22semidirect
54. C22×D4×C127C22semidirect
56. C2×SD16×C127C22semidirect
57. C2×SD16×C127C22semidirect
59. C2×D8×C127C22semidirect
60. C2×M16×C127C22semidirect
120. D8×C127D4semidirect
172. D4×C127C2×D4semidirect
180. C8×C127C2×D4semidirect
205. C2×D127AGL(1,ℤ/8)semidirect
226. C127#3264semidirect
227. C2×AGL(1,ℤ/8)D127semidirect
242. C2×#41C2×D127semidirect
246. C22×D4C2×D127semidirect
249. C2×SD16C2×D127semidirect
250. C2×SD16C2×D127semidirect
257. C2×D8C2×D127semidirect
258. C2×M16C2×D127semidirect
328. D8C22⋊D127semidirect
387. D4C2×C22⋊D127semidirect
406. C8C2×C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1272
#225505142
D4×C22⋊D1272
#225502622
#225501292
#225501022
#225500932
#225500692
#225500252
#225500212
#225500102
#225500082
#225499882
#225499452
C2×AGL(1,ℤ/8)×C1272
#3264127