Finite Group Databasebeta

← #30653502

Subgroups — #30653502

Order 16256.

1424 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550713C211yesmaximal, characteristic, complemented
2#22550711C211yesmaximal, characteristic, complemented
3#22550692C213yesmaximal, complemented
4#22550691C211yesmaximal, characteristic, complemented
5#22550688C211yesmaximal, characteristic, complemented
6#22550683C213yesmaximal, complemented
7#22550682C213yesmaximal, complemented
8#22550680C213yesmaximal, complemented
9#22550679C211yesmaximal, characteristic, complemented
10#22550675C213yesmaximal, complemented
11#22550672C213yesmaximal, complemented
12C2×#16861230C213yesmaximal, complemented
13#135×D127C213yesmaximal, complemented
14#131×D127C211yesmaximal, characteristic, complemented
15#577×C127C211yesmaximal, characteristic, complemented
16#16861257—21no—
17#16861257C2213yescomplemented
18#16861256—21no—
19#16861256C2211yescharacteristic, complemented
20#16861256C2213yescomplemented
21#16861255—23no—
22#16861255C2211yescharacteristic, complemented
23#16861255C2213yescomplemented
24#16861254—23no—
25#16861254C2213yescomplemented
26C2×#14523249—21no—
27C2×#14523249C2213yescomplemented
28C2×#14523247—23no—
29C2×#14523247C2213yescomplemented
30#41×D127—21no—
31#41×D127—23no—
32#41×D127C2211yescharacteristic
33#41×D127C2213yescomplemented
34#41×D127C2213yescomplemented
35#16861242C2213yescomplemented
36#16861241C2216yescomplemented
37#16861240C2213yescomplemented
38#16861239C2213yescomplemented
39#16861238C2216yescomplemented
40#16861237C2213yescomplemented
41#16861236C2216yescomplemented
42#16861235C2213yescomplemented
43#16861234C2213yescomplemented
44#16861233C2216yescomplemented
45#16861232C2213yescomplemented
46#16861231C2213yescomplemented
47#16861230C2211yescharacteristic, complemented
48#16861230C2213yescomplemented
49#16861230C2216yescomplemented
50#16861230C2216yescomplemented
51#16861230C2216yescomplemented
52#16861229C2213yescomplemented
53#16861229C2213yescomplemented
54#16861228C2213yescomplemented
55C2×#14523238C2213yescomplemented
56Q16×D127C2213yescomplemented
57C2×#14523237C2213yescomplemented
58SD16×D127C2213yescomplemented
59SD16×D127C2213yescomplemented
60C2×#14523233C2213yescomplemented
61D8×D127C2213yescomplemented
62#16861217C2213yescomplemented
63#16861216C2211yescharacteristic, complemented
64#16861215C2213yescomplemented
65M16×D127C2213yescomplemented
66C2×C8×D127C2213yescomplemented
672−1+4×C127C2211yescharacteristic, complemented
68H(5,𝔽2)×C127C2211yescharacteristic, complemented
69#137×C127C2213yescomplemented
70AGL(1,ℤ/8)×C127C2213yescomplemented
71#135×C127C2213yescomplemented
72C2×SD16×C127C2213yescomplemented
73#131×C127C2211yescharacteristic, complemented
74#14523249—23no—
75#14523249—23no—
76#14523249—23no—
77#14523249—21no—
78#14523249D411yescharacteristic
79#14523249C2313yes—
80#14523249C2316yes—
81#14523249C2311yescharacteristic
82#14523248—23no—
83#14523248—23no—
84#14523248—26no—
85#14523248—23no—
86#14523248—23no—
87#14523248C2313yes—
88#14523248C2313yes—
89#14523248C2313yes—
90#14523247—23no—
91#14523247—26no—
92#14523247—23no—
93#14523247—23no—
94#14523247—23no—
95#14523247—23no—
96#14523247—23no—
97#14523247D413yes—
98#14523247C2311yescharacteristic
99#14523247C2313yes—
100#14523247C2316yes—
101C2×Dic254—23no—
102C2×Dic254C2313yes—
103Q8×D127—23no—
104Q8×D127—21no—
105Q8×D127D411yescharacteristic
106Q8×D127C2313yes—
107Q8×D127C2313yes—
108C2×D508—23no—
109C2×D508—23no—
110C2×D508C2313yes—
111C2×C22⋊D127—23no—
112C2×C22⋊D127—26no—
113D4×D127—23no—
114D4×D127—23no—
115D4×D127—23no—
116D4×D127—26no—
117D4×D127—23no—
118D4×D127D413yes—
119D4×D127C2313yes—
120D4×D127C2313yes—
121C22×Dic127—23no—
122C2×C4×D127—23no—
123C2×C4×D127—23no—
124C2×C4×D127—23no—
125C2×C4×D127C2313yes—
126Dic508C2313yes—
127#14523238C2311yescharacteristic
128#14523238C2313yes—
129#14523238C2316yes—
130#14523237C2311yescharacteristic
131#14523237C2313yes—
132#14523237C2316yes—
133#14523236C2313yes—
134#14523236C2313yes—
135#14523235C2313yes—
136#14523235C2313yes—
137D1016C2313yes—
138#14523233C2311yescharacteristic
139#14523233C2313yes—
140#14523233C2316yes—
141#14523232C2313yes—
142#14523231C2316yes—
143C2×C127⋊C8C2313yes—
144C8×D127C2311yescharacteristic
145C8×D127C2313yes—
146C8×D127C2313yes—
147C8×D127C2313yes—
148#41×C127—21no—
149#41×C127—23no—
150#41×C127C2313yes—
151#41×C127C2313yes—
152#41×C127C2311yescharacteristic
153C2×Q8×C127—21no—
154C2×Q8×C127C2313yes—
155C2×D4×C127—23no—
156C2×D4×C127C2313yes—
157Q16×C127C2313yes—
158SD16×C127C2311yescharacteristic
159SD16×C127C2313yes—
160SD16×C127C2316yes—
161D8×C127C2313yes—
162M16×C127C2313yes—
163C2×C8×C127C2313yes—
164C22×D127—23no—
165C22×D127—23no—
166C22×D127—23no—
167C22×D127C2×D413yes—
168Dic254—23no—
169Dic254—23no—
170Dic254—23no—
171Dic254C2×D413yes—
172Dic254C2411yescharacteristic
173Dic254C2413yes—
174D508—21no—
175D508—26no—
176D508—23no—
177D508—23no—
178D508C2×D413yes—
179D508C2413yes—
180D508C2411yescharacteristic
181C22⋊D127—23no—
182C22⋊D127—23no—
183C22⋊D127—21no—
184C22⋊D127—26no—
185C22⋊D127—23no—
186C22⋊D127—23no—
187C22⋊D127—23no—
188C22⋊D127C2×D416yes—
189C2×Dic127—23no—
190C2×Dic127—21no—
191C2×Dic127—23no—
192C2×Dic127—26no—
193C2×Dic127C2×D413yes—
194C4×D127—23no—
195C4×D127—23no—
196C4×D127—23no—
197C4×D127—23no—
198C4×D127—21no—
199C4×D127—23no—
200C4×D127—26no—
201C4×D127C2×D413yes—
202C4×D127C2×D413yes—
203C4×D127C2411yescharacteristic
204C4×D127C2413yes—
205C127⋊C8C2411yescharacteristic
206C127⋊C8C2413yes—
207C23×C127—23no—
208Q8×C127—23no—
209Q8×C127C2×D411yescharacteristic
210Q8×C127C2413yes—
211Q8×C127C2411yescharacteristic
212D4×C127—23no—
213D4×C127—23no—
214D4×C127—23no—
215D4×C127C2×D413yes—
216D4×C127C2411yescharacteristic
217D4×C127C2413yes—
218C2×C4×C127—23no—
219C2×C4×C127—23no—
220C2×C4×C127—23no—
221C2×C4×C127C2413yes—
222C8×C127C2411yescharacteristic
223C8×C127C2413yes—
224C22×C127—21no—
225C22×C127—23no—
226C22×C127—83no—
227C22×C127C22×D413yes—
228C2×D127—21no—
229C2×D127—23no—
230C2×D127—43no—
231C2×D127—83no—
232C2×D127—83no—
233C2×D127—83no—
234C2×D127C22×D413yes—
235C2×D127C22×D411yescharacteristic
236C4×C127—21no—
237C4×C127—23no—
238C4×C127C22×D413yes—
239C4×C127C2511yescharacteristic
240Dic127—21no—
241Dic127—23no—
242Dic127C22×D411yescharacteristic
243Dic127C22×D413yes—
244C2×C127—23no—
245C2×C127—41no—
246C2×C127—43no—
247C2×C127C23×D411yescharacteristic
248D127—21no—
249D127—23no—
250D127—41no—
251D127—43no—
252#3584—1271nomaximal, Sylow
253C127#358411yescharacteristic, complemented, Sylow
254#585—1271no—
255#585—1271no—
256#578—1271no—
257#578—1273no—
258#577—1273no—
259#577—1273no—
260#577—1271no—
261#577D12711yescharacteristic, complemented
262#576—1271no—
263#576—1273no—
264#575—1273no—
265#575—1273no—
266C2×#135—1273no—
267C2×#135—1273no—
268C2×#131—1271no—
2692−1+4—1273no—
2702−1+4—1271no—
2712−1+4—1273no—
2722−1+4—2541no—
2732−1+4—2541no—
2742−1+4C2×D12711yescharacteristic, complemented
275H(5,𝔽2)—1273no—
276H(5,𝔽2)—1271no—
277H(5,𝔽2)—1273no—
278H(5,𝔽2)—2543no—
279H(5,𝔽2)—2543no—
280H(5,𝔽2)C2×D12711yescharacteristic, complemented
281C2×#41—1271no—
282C2×#41—1273no—
283C2×#41—1273no—
284C2×#41—1273no—
285C2×#41—1273no—
286C2×#41—2543no—
287C2×#41—2541no—
288C2×#41—2541no—
289C2×#41—2543no—
290#137—1273no—
291#137—1273no—
292#137—1276no—
293#137—1276no—
294#137—1273no—
295#137C2×D12713yescomplemented
296AGL(1,ℤ/8)—1273no—
297AGL(1,ℤ/8)—1273no—
298AGL(1,ℤ/8)—1276no—
299AGL(1,ℤ/8)—1276no—
300AGL(1,ℤ/8)—1273no—
301AGL(1,ℤ/8)C2×D12713yescomplemented
302#135—1273no—
303#135—1273no—
304#135—1273no—
305#135—1271no—
306#135—1273no—
307#135—1276no—
308#135—1276no—
309#135—1276no—
310#135—1273no—
311#135—1273no—
312#135C2×D12713yescomplemented
313C2×Q16—1273no—
314C2×Q16—1273no—
315C2×SD16—1273no—
316C2×SD16—1273no—
317C2×SD16—1273no—
318C2×SD16C2×D12713yescomplemented
319C2×D8—1273no—
320C2×D8—1273no—
321#131—1271no—
322#131—1273no—
323#131—1273no—
324#131C2×D12711yescharacteristic, complemented
325C2×M16—1273no—
326C22×C8—1273no—
327#41—21no—
328#41—23no—
329#41—1271no—
330#41—1273no—
331#41—1273no—
332#41—1271no—
333#41—1273no—
334#41—1276no—
335#41—1273no—
336#41—1271no—
337#41—1276no—
338#41—1273no—
339#41—1273no—
340#41—2543no—
341#41—2543no—
342#41—2546no—
343#41—2543no—
344#41—2543no—
345#41—2543no—
346#41—2543no—
347#41—2543no—
348#41—2543no—
349#41—2543no—
350#41—2543no—
351#41—2546no—
352#41—2543no—
353#41—2543no—
354#41—2541no—
355#41—2543no—
356#41C22×D12713yes—
357#41C22×D12713yes—
358#41C22×D12711yescharacteristic
359C2×Q8—21no—
360C2×Q8—1271no—
361C2×Q8—1273no—
362C2×Q8—1273no—
363C2×Q8—1273no—
364C2×Q8—2543no—
365C2×Q8—2543no—
366C2×Q8—2541no—
367C2×Q8C22×D12713yes—
368C2×D4—23no—
369C2×D4—1273no—
370C2×D4—1273no—
371C2×D4—1273no—
372C2×D4—1273no—
373C2×D4—2543no—
374C2×D4—2543no—
375C2×D4—2546no—
376C2×D4—2543no—
377C2×D4—2543no—
378C2×D4—2543no—
379C2×D4—2546no—
380C2×D4—2543no—
381C2×D4—2543no—
382C2×D4C22×D12713yes—
383C22×C4—1273no—
384C22×C4—2543no—
385C22×C4—2543no—
386C22×C4—2543no—
387C22×C4—2543no—
388Q16—1271no—
389Q16—1273no—
390Q16—1276no—
391Q16—1273no—
392Q16C22×D12713yes—
393SD16—1273no—
394SD16—1273no—
395SD16—1273no—
396SD16—1273no—
397SD16—1271no—
398SD16—1273no—
399SD16—1276no—
400SD16C22×D12711yescharacteristic
401SD16C22×D12713yes—
402SD16C22×D12716yes—
403D8—1273no—
404D8—1271no—
405D8—1276no—
406D8—1273no—
407D8C22×D12713yes—
408M16—1273no—
409M16—1276no—
410M16C22×D12713yes—
411C2×C8—1273no—
412C2×C8—1271no—
413C2×C8—1273no—
414C2×C8—1273no—
415C2×C8—1273no—
416C2×C8C22×D12713yes—
417C23—23no—
418C23—1273no—
419C23—2543no—
420C23—2543no—
421C23—2543no—
422Q8—23no—
423Q8—1273no—
424Q8—1271no—
425Q8—1273no—
426Q8—2543no—
427Q8—2543no—
428Q8—2543no—
429Q8D4×D12711yescharacteristic
430Q8C23×D12711yescharacteristic
431Q8C23×D12713yes—
432D4—23no—
433D4—23no—
434D4—23no—
435D4—1276no—
436D4—1273no—
437D4—1273no—
438D4—1271no—
439D4—2541no—
440D4—2543no—
441D4—2546no—
442D4—2543no—
443D4—2543no—
444D4—2541no—
445D4—2543no—
446D4—2546no—
447D4—2543no—
448D4—2543no—
449D4—2543no—
450D4D4×D12713yes—
451D4C23×D12713yes—
452D4C23×D12711yescharacteristic
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—1273no—
457C2×C4—1273no—
458C2×C4—1273no—
459C2×C4—1271no—
460C2×C4—1273no—
461C2×C4—2541no—
462C2×C4—2543no—
463C2×C4—2546no—
464C2×C4—2543no—
465C2×C4—2543no—
466C2×C4—2541no—
467C2×C4—2543no—
468C2×C4—2546no—
469C2×C4—2543no—
470C2×C4—2543no—
471C2×C4—2543no—
472C2×C4C23×D12713yes—
473C8—1271no—
474C8—1273no—
475C8C23×D12711yescharacteristic
476C8C23×D12713yes—
477C22—21no—
478C22—23no—
479C22—83no—
480C22—1271no—
481C22—1273no—
482C22—10163no—
483C22—10163no—
484C22—10163no—
485C22—5083no—
486C22—2543no—
487C22—2541no—
488C22C2×D4×D12713yes—
489C4—21no—
490C4—23no—
491C4—1271no—
492C4—1273no—
493C4—2541no—
494C4—2543no—
495C4C2×D4×D12713yes—
496C4C24×D12711yescharacteristic
497C2—23no—
498C2—43no—
499C2—41no—
500C2—5083no—
501C2—5081no—
502C2—2543no—
503C2—2541no—
504C2C22×D4×D12711yescharacteristic
subgroupG/H
centerC2C22×D4×D127
commutator, cosocleC4×C127C25
FrattiniC4C24×D127
Fitting#577×C127C2
socleC2×C127C23×D4
hypercenter#577D127
hypocenterC127#3584
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ #41 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #22550713C2semidirect
2. #22550711C2semidirect
3. #22550692C2semidirect
4. #22550691C2semidirect
5. #22550688C2semidirect
6. #22550683C2semidirect
7. #22550682C2semidirect
8. #22550680C2semidirect
9. #22550679C2semidirect
10. #22550675C2semidirect
11. #22550672C2semidirect
12. C2×#16861230C2semidirect
13. #135×D127C2semidirect
14. #131×D127C2semidirect
15. #577×C127C2semidirect
17. #16861257C22semidirect
19. #16861256C22semidirect
20. #16861256C22semidirect
22. #16861255C22semidirect
23. #16861255C22semidirect
25. #16861254C22semidirect
27. C2×#14523249C22semidirect
29. C2×#14523247C22semidirect
33. #41×D127C22semidirect
34. #41×D127C22semidirect
35. #16861242C22semidirect
36. #16861241C22semidirect
37. #16861240C22semidirect
38. #16861239C22semidirect
39. #16861238C22semidirect
40. #16861237C22semidirect
41. #16861236C22semidirect
42. #16861235C22semidirect
43. #16861234C22semidirect
44. #16861233C22semidirect
45. #16861232C22semidirect
46. #16861231C22semidirect
47. #16861230C22semidirect
48. #16861230C22semidirect
49. #16861230C22semidirect
50. #16861230C22semidirect
51. #16861230C22semidirect
52. #16861229C22semidirect
53. #16861229C22semidirect
54. #16861228C22semidirect
55. C2×#14523238C22semidirect
56. Q16×D127C22semidirect
57. C2×#14523237C22semidirect
58. SD16×D127C22semidirect
59. SD16×D127C22semidirect
60. C2×#14523233C22semidirect
61. D8×D127C22semidirect
62. #16861217C22semidirect
63. #16861216C22semidirect
64. #16861215C22semidirect
65. M16×D127C22semidirect
66. C2×C8×D127C22semidirect
67. 2−1+4×C127C22semidirect
68. H(5,𝔽2)×C127C22semidirect
69. #137×C127C22semidirect
70. AGL(1,ℤ/8)×C127C22semidirect
71. #135×C127C22semidirect
72. C2×SD16×C127C22semidirect
73. #131×C127C22semidirect
253. C127#3584semidirect
261. #577D127semidirect
274. 2−1+4C2×D127semidirect
280. H(5,𝔽2)C2×D127semidirect
295. #137C2×D127semidirect
301. AGL(1,ℤ/8)C2×D127semidirect
312. #135C2×D127semidirect
318. C2×SD16C2×D127semidirect
324. #131C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507132
#225507112
#225506922
#225506912
#225506882
#225506832
#225506822
#225506802
#225506792
#225506752
#225506722
C2×#168612302
#135×D1272
#131×D1272
#577×C1272
#3584127