Finite Group Databasebeta

← #30653515

Subgroups — #30653515

Order 16256.

1412 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
1#22550713C211yesmaximal, characteristic, complemented
22−1+4×D127C211yesmaximal, characteristic, complemented
3#22550692C213yesmaximal, complemented
4#22550691C211yesmaximal, characteristic, complemented
5#22550689C213yesmaximal, complemented
6#22550688C211yesmaximal, characteristic, complemented
7#22550685C213yesmaximal, complemented
8#22550684C211yesmaximal, characteristic, complemented
9#22550683C213yesmaximal, complemented
10#22550675C213yesmaximal, complemented
11#22550673C213yesmaximal, complemented
12C2×#16861241C213yesmaximal, complemented
13#137×D127C213yesmaximal, complemented
14#22550641C211yesmaximal, characteristic, complemented
15#578×C127C211yesmaximal, characteristic, complemented
16#16861257—21no—
17#16861257—21no—
18#16861257C2213yescomplemented
19#16861257C2213yescomplemented
20#16861256—23no—
21#16861256C2211yescharacteristic, complemented
22#16861256C2213yescomplemented
23#16861255—23no—
24#16861255C2211yescharacteristic, complemented
25#16861255C2213yescomplemented
26C2×#14523249—21no—
27C2×#14523249C2213yes—
28#41×D127—23no—
29#41×D127—23no—
30#41×D127C2213yes—
31#41×D127C2211yescharacteristic
32#41×D127C2213yescomplemented
33C2×Q8×D127—21no—
34C2×Q8×D127C2213yescomplemented
35#16861242C2213yescomplemented
36#16861241C2216yescomplemented
37#16861241C2213yescomplemented
38#16861241C2216yescomplemented
39#16861241C2216yescomplemented
40#16861241C2213yescomplemented
41#16861240C2213yescomplemented
42#16861240C2213yescomplemented
43#16861239C2213yescomplemented
44#16861238C2213yescomplemented
45#16861238C2213yescomplemented
46#16861237C2213yescomplemented
47#16861236C2213yescomplemented
48#16861236C2213yescomplemented
49#16861234C2213yescomplemented
50#16861233C2211yescharacteristic, complemented
51#16861233C2213yescomplemented
52#16861232C2216yescomplemented
53#16861231C2213yescomplemented
54#16861230C2216yescomplemented
55#16861229C2213yescomplemented
56#16861229C2213yescomplemented
57C2×#14523238C2213yescomplemented
58Q16×D127C2216yescomplemented
59C2×#14523237C2213yescomplemented
60C2×#14523235C2213yescomplemented
61SD16×D127C2216yescomplemented
62#16861217C2213yescomplemented
63#16861216C2211yescharacteristic, complemented
64#16861215C2213yescomplemented
65C2×#14523231C2213yescomplemented
66M16×D127C2213yescomplemented
672−1+4×C127C2211yescharacteristic, complemented
682−1+4×C127C2211yescharacteristic, complemented
69#137×C127C2213yescomplemented
70#137×C127C2213yescomplemented
71#135×C127C2213yescomplemented
72C2×Q16×C127C2213yescomplemented
73#131×C127C2211yescharacteristic, complemented
74#14523249—23no—
75#14523249—23no—
76#14523249—23no—
77#14523249—23no—
78#14523249—21no—
79#14523249D411yescharacteristic
80#14523249C2316yes—
81#14523249C2313yes—
82#14523249C2313yes—
83#14523248—23no—
84#14523248—23no—
85#14523248—26no—
86#14523248—26no—
87#14523248—23no—
88#14523248—23no—
89#14523248C2313yes—
90#14523248C2313yes—
91#14523248C2313yes—
92#14523247—26no—
93#14523247—23no—
94#14523247—23no—
95#14523247—23no—
96#14523247D413yes—
97#14523247C2313yes—
98#14523247C2311yescharacteristic
99C2×Dic254—23no—
100C2×Dic254—23no—
101C2×Dic254C2313yes—
102Q8×D127—23no—
103Q8×D127—23no—
104Q8×D127—23no—
105Q8×D127—23no—
106Q8×D127—21no—
107Q8×D127D411yescharacteristic
108Q8×D127C2316yes—
109Q8×D127C2313yes—
110Q8×D127C2313yes—
111C2×D508—23no—
112C2×D508—23no—
113C2×D508C2313yes—
114D4×D127—23no—
115D4×D127—23no—
116D4×D127—26no—
117D4×D127—23no—
118D4×D127D413yes—
119D4×D127C2313yes—
120D4×D127C2311yescharacteristic
121C2×C4×D127—23no—
122C2×C4×D127—23no—
123C2×C4×D127—23no—
124C2×C4×D127—23no—
125C2×C4×D127C2313yes—
126Dic508C2313yes—
127#14523238C2313yes—
128#14523238C2316yes—
129#14523238C2313yes—
130#14523237C2316yes—
131#14523237C2313yes—
132#14523237C2311yescharacteristic
133#14523236C2313yes—
134#14523236C2311yescharacteristic
135#14523235C2316yes—
136#14523235C2313yes—
137#14523235C2313yes—
138D1016C2313yes—
139#14523233C2311yescharacteristic
140#14523233C2313yes—
141#14523232C2313yes—
142#14523231C2313yes—
143#14523231C2311yescharacteristic
144#14523231C2313yes—
145#14523231C2313yes—
146C2×C127⋊C8C2313yes—
147C8×D127C2316yes—
148#41×C127—23no—
149#41×C127—23no—
150#41×C127C2313yes—
151#41×C127C2313yes—
152#41×C127C2311yescharacteristic
153C2×Q8×C127—21no—
154C2×Q8×C127—21no—
155C2×Q8×C127C2313yes—
156C2×Q8×C127C2313yes—
157Q16×C127C2316yes—
158Q16×C127C2313yes—
159SD16×C127C2313yes—
160SD16×C127C2313yes—
161D8×C127C2311yescharacteristic
162M16×C127C2313yes—
163C2×C8×C127C2313yes—
164C22×D127—23no—
165C22×D127—21no—
166C22×D127—23no—
167C22×D127C2×D413yes—
168Dic254—23no—
169Dic254—23no—
170Dic254—26no—
171Dic254—23no—
172Dic254C2×D413yes—
173Dic254C2413yes—
174Dic254C2411yescharacteristic
175D508—23no—
176D508—26no—
177D508—23no—
178D508—23no—
179D508C2×D413yes—
180D508C2411yescharacteristic
181D508C2413yes—
182C22⋊D127—23no—
183C22⋊D127—23no—
184C22⋊D127—21no—
185C22⋊D127—23no—
186C22⋊D127—23no—
187C22⋊D127—21no—
188C22⋊D127C2×D416yes—
189C2×Dic127—23no—
190C2×Dic127—21no—
191C2×Dic127—23no—
192C2×Dic127C2×D413yes—
193C4×D127—23no—
194C4×D127—23no—
195C4×D127—26no—
196C4×D127—23no—
197C4×D127—23no—
198C4×D127—23no—
199C4×D127—26no—
200C4×D127—23no—
201C4×D127C2×D413yes—
202C4×D127C2×D413yes—
203C4×D127C2413yes—
204C4×D127C2411yescharacteristic
205C127⋊C8C2413yes—
206C127⋊C8C2411yescharacteristic
207Q8×C127—23no—
208Q8×C127—23no—
209Q8×C127C2×D411yescharacteristic
210Q8×C127C2413yes—
211Q8×C127C2413yes—
212D4×C127—23no—
213D4×C127—23no—
214D4×C127C2×D413yes—
215D4×C127C2411yescharacteristic
216D4×C127C2411yescharacteristic
217C2×C4×C127—23no—
218C2×C4×C127—23no—
219C2×C4×C127—23no—
220C2×C4×C127—23no—
221C2×C4×C127C2413yes—
222C8×C127C2411yescharacteristic
223C8×C127C2413yes—
224C22×C127—21no—
225C22×C127—21no—
226C22×C127C22×D413yes—
227C2×D127—21no—
228C2×D127—23no—
229C2×D127—43no—
230C2×D127—42no—
231C2×D127—83no—
232C2×D127—83no—
233C2×D127C22×D411yescharacteristic
234C2×D127C22×D413yes—
235C4×C127—23no—
236C4×C127—23no—
237C4×C127C22×D413yes—
238C4×C127C2511yescharacteristic
239Dic127—21no—
240Dic127—23no—
241Dic127C22×D411yescharacteristic
242Dic127C22×D413yes—
243C2×C127—23no—
244C2×C127—41no—
245C2×C127—41no—
246C2×C127C23×D411yescharacteristic
247D127—21no—
248D127—23no—
249D127—41no—
250D127—43no—
251#3586—1271nomaximal, Sylow
252C127#358611yescharacteristic, complemented, Sylow
253#585—1271no—
254C2×2−1+4—1271no—
255#578—1271no—
256#578—1273no—
257#578—1273no—
258#578D12711yescharacteristic, complemented
259#577—1271no—
260#577—1273no—
261#577—1273no—
262#577—1271no—
263#575—1273no—
264#575—1273no—
265C2×#137—1273no—
266C2×#137—1273no—
267#568—1271no—
2682−1+4—1273no—
2692−1+4—1271no—
2702−1+4—1273no—
2712−1+4—1273no—
2722−1+4—2541no—
2732−1+4—2541no—
2742−1+4—2543no—
2752−1+4C2×D12711yescharacteristic, complemented
2762−1+4C2×D12711yescharacteristic, complemented
277H(5,𝔽2)—1273no—
278H(5,𝔽2)—1271no—
279H(5,𝔽2)—2543no—
280C2×#41—1271no—
281C2×#41—1273no—
282C2×#41—1273no—
283C2×#41—1273no—
284C2×#41—2543no—
285C2×#41—2541no—
286C2×#41—2543no—
287C22×Q8—1273no—
288C22×Q8—2541no—
289#137—1273no—
290#137—1273no—
291#137—1273no—
292#137—1276no—
293#137—1273no—
294#137—1276no—
295#137—1276no—
296#137—1273no—
297#137—1273no—
298#137—1273no—
299#137—1273no—
300#137C2×D12713yescomplemented
301#137C2×D12713yescomplemented
302AGL(1,ℤ/8)—1273no—
303AGL(1,ℤ/8)—1273no—
304AGL(1,ℤ/8)—1273no—
305AGL(1,ℤ/8)—1271no—
306AGL(1,ℤ/8)—1273no—
307AGL(1,ℤ/8)—1273no—
308#135—1273no—
309#135—1273no—
310#135—1276no—
311#135—1276no—
312#135—1273no—
313#135C2×D12713yescomplemented
314C2×Q16—1273no—
315C2×Q16—1276no—
316C2×Q16C2×D12713yescomplemented
317C2×SD16—1273no—
318C2×SD16—1276no—
319C2×SD16—1273no—
320#131—1271no—
321#131—1273no—
322#131—1273no—
323#131C2×D12711yescharacteristic, complemented
324C2×M16—1273no—
325C2×M16—1273no—
326#41—23no—
327#41—23no—
328#41—1271no—
329#41—1273no—
330#41—1273no—
331#41—1273no—
332#41—1276no—
333#41—1273no—
334#41—1273no—
335#41—1271no—
336#41—1273no—
337#41—1273no—
338#41—2543no—
339#41—2543no—
340#41—2546no—
341#41—2546no—
342#41—2543no—
343#41—2543no—
344#41—2543no—
345#41—2543no—
346#41—2543no—
347#41—2543no—
348#41—2546no—
349#41—2543no—
350#41—2543no—
351#41—2541no—
352#41—2543no—
353#41C22×D12713yes—
354#41C22×D12713yes—
355#41C22×D12711yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—1271no—
359C2×Q8—1276no—
360C2×Q8—1273no—
361C2×Q8—1273no—
362C2×Q8—1273no—
363C2×Q8—2543no—
364C2×Q8—2543no—
365C2×Q8—2543no—
366C2×Q8—2543no—
367C2×Q8—2543no—
368C2×Q8—2543no—
369C2×Q8—2541no—
370C2×Q8C22×D12713yes—
371C2×Q8C22×D12713yes—
372C2×D4—1273no—
373C2×D4—1273no—
374C2×D4—1271no—
375C2×D4—1273no—
376C2×D4—2543no—
377C2×D4—2543no—
378C2×D4—2546no—
379C2×D4—2543no—
380C2×D4—2543no—
381C2×D4—2543no—
382C22×C4—1273no—
383C22×C4—2543no—
384C22×C4—2543no—
385C22×C4—2543no—
386C22×C4—2543no—
387Q16—1276no—
388Q16—1273no—
389Q16—1273no—
390Q16—1273no—
391Q16C22×D12716yes—
392Q16C22×D12713yes—
393SD16—1276no—
394SD16—1273no—
395SD16—1273no—
396SD16—1273no—
397SD16—1271no—
398SD16—1276no—
399SD16—1273no—
400SD16—1271no—
401SD16C22×D12713yes—
402SD16C22×D12713yes—
403D8—1271no—
404D8—1273no—
405D8—1273no—
406D8C22×D12711yescharacteristic
407M16—1273no—
408M16—1273no—
409M16—1271no—
410M16—1273no—
411M16—1273no—
412M16C22×D12713yes—
413C2×C8—1273no—
414C2×C8—1276no—
415C2×C8C22×D12713yes—
416C23—1273no—
417C23—2543no—
418C23—2541no—
419C23—2543no—
420Q8—23no—
421Q8—23no—
422Q8—1273no—
423Q8—1273no—
424Q8—1271no—
425Q8—2543no—
426Q8—2546no—
427Q8—2543no—
428Q8—2543no—
429Q8D4×D12711yescharacteristic
430Q8C23×D12713yes—
431Q8C23×D12713yes—
432D4—23no—
433D4—23no—
434D4—1276no—
435D4—1273no—
436D4—1271no—
437D4—1273no—
438D4—2543no—
439D4—2546no—
440D4—2543no—
441D4—2543no—
442D4—2541no—
443D4—2541no—
444D4—2543no—
445D4—2543no—
446D4—2543no—
447D4—2543no—
448D4D4×D12713yes—
449D4C23×D12711yescharacteristic
450D4C23×D12711yescharacteristic
451C2×C4—23no—
452C2×C4—23no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—1273no—
456C2×C4—1273no—
457C2×C4—1273no—
458C2×C4—1273no—
459C2×C4—1271no—
460C2×C4—2543no—
461C2×C4—2546no—
462C2×C4—2543no—
463C2×C4—2543no—
464C2×C4—2543no—
465C2×C4—2546no—
466C2×C4—2541no—
467C2×C4—2543no—
468C2×C4—2543no—
469C2×C4—2543no—
470C2×C4—2543no—
471C2×C4C23×D12713yes—
472C8—1271no—
473C8—1273no—
474C8C23×D12713yes—
475C8C23×D12711yescharacteristic
476C22—21no—
477C22—21no—
478C22—1271no—
479C22—1273no—
480C22—10163no—
481C22—10163no—
482C22—5083no—
483C22—5082no—
484C22—2541no—
485C22—2543no—
486C22C2×D4×D12713yes—
487C4—23no—
488C4—23no—
489C4—1271no—
490C4—1273no—
491C4—2541no—
492C4—2543no—
493C4C2×D4×D12713yes—
494C4C24×D12711yescharacteristic
495C2—23no—
496C2—41no—
497C2—41no—
498C2—5083no—
499C2—5081no—
500C2—2543no—
501C2—2541no—
502C2C22×D4×D12711yescharacteristic
subgroupG/H
centerC2C22×D4×D127
commutator, cosocleC4×C127C25
FrattiniC4C24×D127
Fitting#578×C127C2
socleC2×C127C23×D4
hypercenter#578D127
hypocenterC127#3586
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ #41 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #22550713C2semidirect
2. 2−1+4×D127C2semidirect
3. #22550692C2semidirect
4. #22550691C2semidirect
5. #22550689C2semidirect
6. #22550688C2semidirect
7. #22550685C2semidirect
8. #22550684C2semidirect
9. #22550683C2semidirect
10. #22550675C2semidirect
11. #22550673C2semidirect
12. C2×#16861241C2semidirect
13. #137×D127C2semidirect
14. #22550641C2semidirect
15. #578×C127C2semidirect
18. #16861257C22semidirect
19. #16861257C22semidirect
21. #16861256C22semidirect
22. #16861256C22semidirect
24. #16861255C22semidirect
25. #16861255C22semidirect
32. #41×D127C22semidirect
34. C2×Q8×D127C22semidirect
35. #16861242C22semidirect
36. #16861241C22semidirect
37. #16861241C22semidirect
38. #16861241C22semidirect
39. #16861241C22semidirect
40. #16861241C22semidirect
41. #16861240C22semidirect
42. #16861240C22semidirect
43. #16861239C22semidirect
44. #16861238C22semidirect
45. #16861238C22semidirect
46. #16861237C22semidirect
47. #16861236C22semidirect
48. #16861236C22semidirect
49. #16861234C22semidirect
50. #16861233C22semidirect
51. #16861233C22semidirect
52. #16861232C22semidirect
53. #16861231C22semidirect
54. #16861230C22semidirect
55. #16861229C22semidirect
56. #16861229C22semidirect
57. C2×#14523238C22semidirect
58. Q16×D127C22semidirect
59. C2×#14523237C22semidirect
60. C2×#14523235C22semidirect
61. SD16×D127C22semidirect
62. #16861217C22semidirect
63. #16861216C22semidirect
64. #16861215C22semidirect
65. C2×#14523231C22semidirect
66. M16×D127C22semidirect
67. 2−1+4×C127C22semidirect
68. 2−1+4×C127C22semidirect
69. #137×C127C22semidirect
70. #137×C127C22semidirect
71. #135×C127C22semidirect
72. C2×Q16×C127C22semidirect
73. #131×C127C22semidirect
252. C127#3586semidirect
258. #578D127semidirect
275. 2−1+4C2×D127semidirect
276. 2−1+4C2×D127semidirect
300. #137C2×D127semidirect
301. #137C2×D127semidirect
313. #135C2×D127semidirect
316. C2×Q16C2×D127semidirect
323. #131C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507132
2−1+4×D1272
#225506922
#225506912
#225506892
#225506882
#225506852
#225506842
#225506832
#225506752
#225506732
C2×#168612412
#137×D1272
#225506412
#578×C1272
#3586127