Finite Group Databasebeta

← #30647995

Subgroups — #30647995

Order 16256.

784 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861255C211yesmaximal, characteristic, complemented
2C2×#16861236C211yesmaximal, characteristic, complemented
3C2×D8×D127C211yesmaximal, characteristic, complemented
4#22550442C211yesmaximal, characteristic, complemented
5#22550024C211yesmaximal, characteristic, complemented
6#22550021C211yesmaximal, characteristic, complemented
7#22550020C211yesmaximal, characteristic, complemented
8#22550012C211yesmaximal, characteristic, complemented
9#22550011C211yesmaximal, characteristic, complemented
10#22550009C211yesmaximal, characteristic, complemented
11#22550008C211yesmaximal, characteristic, complemented
12#22549879C211yesmaximal, characteristic, complemented
13#22549871C211yesmaximal, characteristic, complemented
14#22549841C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16#16861255—24no—
17#16861255—24no—
18C2×#14523249C2211yescharacteristic
19C2×#14523249C2211yescharacteristic, complemented
20C2×#14523248—21no—
21C2×#14523248C2211yescharacteristic
22C22×D508—21no—
23C22×D508C2211yescharacteristic, complemented
24C2×D4×D127—21no—
25C2×D4×D127—21no—
26C2×D4×D127C2211yescharacteristic
27C2×D4×D127C2211yescharacteristic, complemented
28C2×D4×D127C2211yescharacteristic, complemented
29#16861236—24no—
30C2×#14523235C2211yescharacteristic, complemented
31C2×D1016C2211yescharacteristic, complemented
32C2×D1016C2211yescharacteristic, complemented
33C2×#14523233C2211yescharacteristic, complemented
34C2×#14523233C2211yescharacteristic, complemented
35C2×#14523233C2211yescharacteristic, complemented
36D8×D127—24no—
37C2×#14523231C2211yescharacteristic, complemented
38C2×C8×D127C2211yescharacteristic, complemented
39#16861205—21no—
40#16861195—21no—
41#16861184C2211yescharacteristic, complemented
42#16861183C2211yescharacteristic, complemented
43#16861180C2211yescharacteristic, complemented
44#16861179—21no—
45#16861177—21no—
46#16861167C2211yescharacteristic, complemented
47#16861133C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic, complemented
49#16861131C2211yescharacteristic, complemented
50#16861130C2211yescharacteristic, complemented
51#16861129C2211yescharacteristic, complemented
52#16861129C2211yescharacteristic, complemented
53C22⋊(C127⋊C8)C2211yescharacteristic, complemented
54(C2×D127)⋊C8C2211yescharacteristic, complemented
55(C2×D127)⋊C8C2211yescharacteristic, complemented
56C2×#41×C127C2211yescharacteristic, complemented
57C2×SD16×C127C2211yescharacteristic, complemented
58C2×D8×C127C2211yescharacteristic, complemented
59#121×C127C2211yescharacteristic, complemented
60#103×C127C2211yescharacteristic, complemented
61#102×C127C2211yescharacteristic, complemented
62C22⋊C8×C127C2211yescharacteristic, complemented
63C23×D127—21no—
64C23×D127—21no—
65C23×D127—21no—
66C23×D127—21no—
67#14523249—22no—
68#14523249—22no—
69#14523249—22no—
70#14523249D414yescomplemented
71#14523248—28no—
72#14523248—22no—
73#14523248—22no—
74C2×Dic254D411yescharacteristic
75C2×D508—28no—
76C2×D508—21no—
77C2×D508—21no—
78C2×D508—22no—
79C2×D508—22no—
80C2×D508—21no—
81C2×D508—21no—
82C2×D508D411yescharacteristic
83C2×D508C2311yescharacteristic
84C2×D508C2311yescharacteristic
85C2×D508C2311yescharacteristic
86C2×C22⋊D127—21no—
87C2×C22⋊D127—21no—
88C2×C22⋊D127—21no—
89C2×C22⋊D127—21no—
90C2×C22⋊D127—21no—
91C2×C22⋊D127D411yescharacteristic
92C2×C22⋊D127D411yescharacteristic
93D4×D127—28no—
94D4×D127—28no—
95D4×D127—22no—
96D4×D127—22no—
97D4×D127—22no—
98D4×D127—22no—
99D4×D127—22no—
100D4×D127D414yescomplemented
101C2×C4×D127—21no—
102C2×C4×D127—21no—
103C2×C4×D127C2311yescharacteristic
104C2×C4×D127C2311yescharacteristic
105#14523235—22no—
106D1016—22no—
107D1016—22no—
108#14523233—22no—
109#14523233—22no—
110#14523233—22no—
111#14523231—22no—
112C2×C127⋊C8C2311yescharacteristic
113C2×C127⋊C8C2311yescharacteristic
114C8×D127—22no—
115C4⋊Dic127C2311yescharacteristic
116C22⋊Dic127—21no—
117(C2×D127)⋊C4—21no—
118(C2×D127)⋊C4—21no—
119(C2×D127)⋊C4—21no—
120C4×Dic127—21no—
121#41×C127—22no—
122#41×C127—22no—
123C2×Q8×C127C2311yescharacteristic
124C2×D4×C127—21no—
125C2×D4×C127—21no—
126C2×D4×C127C2311yescharacteristic
127C2×D4×C127C2311yescharacteristic
128C22×C4×C127—21no—
129C22×C4×C127C2311yescharacteristic
130SD16×C127—22no—
131D8×C127—22no—
132C2×C8×C127C2311yescharacteristic
133C2×C8×C127C2311yescharacteristic
134C4⋊C4×C127C2311yescharacteristic
135C22⋊C4×C127—21no—
136C22×D127—21no—
137C22×D127—21no—
138C22×D127—24no—
139C22×D127—24no—
140C22×D127—24no—
141C22×D127—21no—
142C22×D127—42no—
143C22×D127—81no—
144C22×D127—81no—
145C22×D127—81no—
146C22×D127—81no—
147C22×D127—81no—
148C22×D127—81no—
149C22×D127—81no—
150C22×D127—81no—
151C22×D127C2×D411yescharacteristic
152C22×D127C2×D411yescharacteristic
153Dic254—22no—
154D508—24no—
155D508—24no—
156D508—24no—
157D508—22no—
158D508—21no—
159D508—21no—
160D508—21no—
161D508—21no—
162D508—81no—
163D508C2×D412yes—
164D508C2×D412yes—
165C22⋊D127—24no—
166C22⋊D127—22no—
167C22⋊D127—22no—
168C22⋊D127—24no—
169C22⋊D127—42no—
170C22⋊D127—81no—
171C22⋊D127—81no—
172C2×Dic127—21no—
173C2×Dic127C2×D411yescharacteristic
174C2×Dic127C2×D411yescharacteristic
175C4×D127—24no—
176C4×D127—21no—
177C4×D127—21no—
178C4×D127—24no—
179C4×D127C2×D412yes—
180C4×D127C2×D412yes—
181C127⋊C8—21no—
182C127⋊C8—21no—
183C23×C127—21no—
184C23×C127—21no—
185C23×C127C2×D411yescharacteristic
186Q8×C127—21no—
187Q8×C127C2×D412yes—
188D4×C127—24no—
189D4×C127—21no—
190D4×C127—21no—
191D4×C127—21no—
192D4×C127—81no—
193D4×C127C2×D412yes—
194C2×C4×C127—24no—
195C2×C4×C127—21no—
196C2×C4×C127—21no—
197C2×C4×C127—21no—
198C2×C4×C127—21no—
199C2×C4×C127C2×D411yescharacteristic
200C2×C4×C127C2411yescharacteristic
201C8×C127—21no—
202C8×C127—21no—
203C22×C127—21no—
204C22×C127—22no—
205C22×C127—21no—
206C22×C127—21no—
207C22×C127—41no—
208C22×C127—41no—
209C22×C127—41no—
210C22×C127—41no—
211C22×C127—41no—
212C22×C127C22×D411yescharacteristic
213C2×D127—21no—
214C2×D127—21no—
215C2×D127—21no—
216C2×D127—22no—
217C2×D127—22no—
218C2×D127—21no—
219C2×D127—21no—
220C2×D127—41no—
221C2×D127—41no—
222C2×D127—41no—
223C2×D127—41no—
224C2×D127—41no—
225C2×D127—41no—
226C2×D127—41no—
227C2×D127—84no—
228C2×D127—84no—
229C2×D127—84no—
230C2×D127—84no—
231C2×D127#12012yes—
232C4×C127—21no—
233C4×C127—22no—
234C4×C127—41no—
235C4×C127C22×D411yescharacteristic
236C4×C127C22×D411yescharacteristic
237Dic127—21no—
238Dic127—41no—
239Dic127#12012yes—
240C2×C127—41no—
241C2×C127—42no—
242C2×C127—81no—
243C2×C127C2×#12011yescharacteristic
244C2×C127#57611yescharacteristic
245C2×C127#57611yescharacteristic
246D127—22no—
247D127—41no—
248D127—42no—
249D127—42no—
250D127—81no—
251#3008—1271nomaximal, Sylow
252C127#300811yescharacteristic, complemented, Sylow
253C2×H(5,𝔽2)—1271no—
254C2×AGL(1,ℤ/8)—1271no—
255C22×D8—1271no—
256#535—1271no—
257#449—1271no—
258#449—1271no—
259#449—1271no—
260#449D12711yescharacteristic, complemented
261#447—1271no—
262#447—1271no—
263#447—1271no—
264#447—1271no—
265#418—1271no—
266#416—1271no—
267#408—1271no—
268H(5,𝔽2)—2544no—
269H(5,𝔽2)—2544no—
270C2×#41—1271no—
271C2×#41—1271no—
272C2×#41—1271no—
273C2×#41—2541no—
274C2×#41C2×D12711yescharacteristic, complemented
275C22×D4—1271no—
276C22×D4—1271no—
277C22×D4—1271no—
278C22×D4—1271no—
279C22×D4—2541no—
280C22×D4—2541no—
281C22×D4—2541no—
282AGL(1,ℤ/8)—2544no—
283C2×SD16—1271no—
284C2×SD16C2×D12711yescharacteristic, complemented
285C2×D8—1271no—
286C2×D8—1271no—
287C2×D8—1271no—
288C2×D8—1271no—
289C2×D8—1271no—
290C2×D8—2544no—
291C2×D8C2×D12711yescharacteristic, complemented
292C2×M16—1271no—
293C22×C8—1271no—
294#127—2541no—
295#124—2541no—
296#121—1271no—
297#121—1271no—
298#121—1271no—
299#121C2×D12711yescharacteristic, complemented
300#120—2541no—
301#120—2541no—
302#117—1271no—
303#103—1271no—
304#103C2×D12711yescharacteristic, complemented
305#102—1271no—
306#102—1271no—
307#102—1271no—
308#102—1271no—
309#102—1271no—
310#102C2×D12711yescharacteristic, complemented
311C22⋊C8—1271no—
312C22⋊C8—1271no—
313C22⋊C8—1271no—
314C22⋊C8C2×D12711yescharacteristic, complemented
315C24—2541no—
316C24—2541no—
317C24—2541no—
318C24—2541no—
319#41—22no—
320#41—22no—
321#41—1274no—
322#41—2542no—
323#41—2542no—
324#41—2548no—
325#41—2542no—
326#41—2542no—
327#41—2542no—
328C2×Q8—1271no—
329C2×Q8C22×D12711yescharacteristic
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—1274no—
333C2×D4—1271no—
334C2×D4—1271no—
335C2×D4—1271no—
336C2×D4—1271no—
337C2×D4—1271no—
338C2×D4—1271no—
339C2×D4—2542no—
340C2×D4—2542no—
341C2×D4—2541no—
342C2×D4—2541no—
343C2×D4—2541no—
344C2×D4—2548no—
345C2×D4—2542no—
346C2×D4—2542no—
347C2×D4—2542no—
348C2×D4—2541no—
349C2×D4—2541no—
350C2×D4—2541no—
351C2×D4—2542no—
352C2×D4—2548no—
353C2×D4—2548no—
354C2×D4—2541no—
355C2×D4—2541no—
356C2×D4—2542no—
357C2×D4—2541no—
358C2×D4C22×D12711yescharacteristic
359C2×D4C22×D12711yescharacteristic
360C22×C4—21no—
361C22×C4—1271no—
362C22×C4—1271no—
363C22×C4—2541no—
364C22×C4—2541no—
365C22×C4C22×D12711yescharacteristic
366SD16—22no—
367SD16—2542no—
368D8—22no—
369D8—2542no—
370D8—2542no—
371D8—2542no—
372D8—2542no—
373D8—2542no—
374M16—2542no—
375C2×C8—1271no—
376C2×C8—1271no—
377C2×C8—2542no—
378C2×C8C22×D12711yescharacteristic
379C2×C8C22×D12711yescharacteristic
380C4⋊C4—1271no—
381C4⋊C4C22×D12711yescharacteristic
382C22⋊C4—21no—
383C22⋊C4—2541no—
384C22⋊C4—2541no—
385C22⋊C4—2541no—
386C22⋊C4—2541no—
387C42—2541no—
388C23—21no—
389C23—21no—
390C23—1271no—
391C23—1271no—
392C23—10161no—
393C23—10161no—
394C23—10161no—
395C23—10161no—
396C23—10161no—
397C23—10161no—
398C23—10161no—
399C23—10161no—
400C23—5082no—
401C23—2544no—
402C23—2544no—
403C23—2544no—
404C23—2541no—
405C23—2541no—
406C23—2541no—
407C23D4×D12711yescharacteristic
408Q8—21no—
409Q8—2542no—
410Q8D4×D12712yes—
411D4—24no—
412D4—21no—
413D4—21no—
414D4—21no—
415D4—81no—
416D4—1272no—
417D4—1272no—
418D4—10161no—
419D4—10161no—
420D4—10161no—
421D4—5082no—
422D4—2542no—
423D4—2544no—
424D4—2544no—
425D4—2544no—
426D4—2542no—
427D4—2544no—
428D4—2542no—
429D4—2544no—
430D4—2541no—
431D4—2541no—
432D4—2541no—
433D4—2541no—
434D4D4×D12712yes—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—1271no—
441C2×C4—1272no—
442C2×C4—1272no—
443C2×C4—1271no—
444C2×C4—2544no—
445C2×C4—2544no—
446C2×C4—2541no—
447C2×C4—2541no—
448C2×C4—2541no—
449C2×C4D4×D12711yescharacteristic
450C2×C4C23×D12711yescharacteristic
451C8—21no—
452C8—21no—
453C8—2541no—
454C8—2541no—
455C22—21no—
456C22—22no—
457C22—21no—
458C22—21no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—1272no—
465C22—10164no—
466C22—10164no—
467C22—10164no—
468C22—10164no—
469C22—5081no—
470C22—5081no—
471C22—5081no—
472C22—5081no—
473C22—5081no—
474C22—5081no—
475C22—5081no—
476C22—2542no—
477C22—2542no—
478C22—2541no—
479C22—2541no—
480C22—2541no—
481C22—2541no—
482C22—2541no—
483C22C2×D4×D12711yescharacteristic
484C4—21no—
485C4—22no—
486C4—41no—
487C4—1272no—
488C4—5081no—
489C4—2541no—
490C4C2×D4×D12711yescharacteristic
491C4C2×D4×D12711yescharacteristic
492C2—41no—
493C2—42no—
494C2—81no—
495C2—10161no—
496C2—5082no—
497C2—5082no—
498C2—5081no—
499C2—2542no—
500C2#120×D12711yescharacteristic
501C2#2255067811yescharacteristic
502C2#2255068011yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3008
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#16861255C2semidirect
2. C2×#16861236C2semidirect
3. C2×D8×D127C2semidirect
4. #22550442C2semidirect
5. #22550024C2semidirect
6. #22550021C2semidirect
7. #22550020C2semidirect
8. #22550012C2semidirect
9. #22550011C2semidirect
10. #22550009C2semidirect
11. #22550008C2semidirect
12. #22549879C2semidirect
13. #22549871C2semidirect
14. #22549841C2semidirect
15. #449×C127C2semidirect
19. C2×#14523249C22semidirect
23. C22×D508C22semidirect
27. C2×D4×D127C22semidirect
28. C2×D4×D127C22semidirect
30. C2×#14523235C22semidirect
31. C2×D1016C22semidirect
32. C2×D1016C22semidirect
33. C2×#14523233C22semidirect
34. C2×#14523233C22semidirect
35. C2×#14523233C22semidirect
37. C2×#14523231C22semidirect
38. C2×C8×D127C22semidirect
41. #16861184C22semidirect
42. #16861183C22semidirect
43. #16861180C22semidirect
46. #16861167C22semidirect
47. #16861133C22semidirect
48. #16861131C22semidirect
49. #16861131C22semidirect
50. #16861130C22semidirect
51. #16861129C22semidirect
52. #16861129C22semidirect
53. C22⋊(C127⋊C8)C22semidirect
54. (C2×D127)⋊C8C22semidirect
55. (C2×D127)⋊C8C22semidirect
56. C2×#41×C127C22semidirect
57. C2×SD16×C127C22semidirect
58. C2×D8×C127C22semidirect
59. #121×C127C22semidirect
60. #103×C127C22semidirect
61. #102×C127C22semidirect
62. C22⋊C8×C127C22semidirect
70. #14523249D4semidirect
100. D4×D127D4semidirect
252. C127#3008semidirect
260. #449D127semidirect
274. C2×#41C2×D127semidirect
284. C2×SD16C2×D127semidirect
291. C2×D8C2×D127semidirect
299. #121C2×D127semidirect
304. #103C2×D127semidirect
310. #102C2×D127semidirect
314. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
C2×#168612362
C2×D8×D1272
#225504422
#225500242
#225500212
#225500202
#225500122
#225500112
#225500092
#225500082
#225498792
#225498712
#225498412
#449×C1272
#3008127