Finite Group Databasebeta

← #30650492

Subgroups — #30650492

Order 16256.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550684C211yesmaximal, characteristic, complemented
2#22550679C211yesmaximal, characteristic, complemented
3#22550678C211yesmaximal, characteristic, complemented
4#22550262C211yesmaximal, characteristic, complemented
5#22550146C211yesmaximal, characteristic, complemented
6#22550144C211yesmaximal, characteristic, complemented
7#22550141C211yesmaximal, characteristic, complemented
8#22550136C211yesmaximal, characteristic, complemented
9#22550052C211yesmaximal, characteristic, complemented
10#22550049C211yesmaximal, characteristic, complemented
11#22550044C211yesmaximal, characteristic, complemented
12#22550042C211yesmaximal, characteristic, complemented
13#22549998C211yesmaximal, characteristic, complemented
14#22549996C211yesmaximal, characteristic, complemented
15#577×C127C211yesmaximal, characteristic, complemented
16#16861255C2211yescharacteristic
17#16861255C2211yescharacteristic, complemented
18#16861240—21no—
19#16861240C2211yescharacteristic, complemented
20#16861236—21no—
21#16861236—21no—
22#16861235—21no—
23#16861235C2211yescharacteristic
24#16861234—21no—
25#16861234—21no—
26#16861234C2211yescharacteristic
27#16861234C2211yescharacteristic, complemented
28#16861234C2211yescharacteristic, complemented
29#16861234C2211yescharacteristic, complemented
30#16861232—21no—
31#16861229—21no—
32#16861229—21no—
33#16861229C2211yescharacteristic
34C2×#14523235—21no—
35C2×#14523235C2211yescharacteristic, complemented
36C2×D1016C2211yescharacteristic, complemented
37C2×#14523233—21no—
38C2×#14523233C2211yescharacteristic, complemented
39D8×D127—21no—
40#16861216C2211yescharacteristic, complemented
41#16861216C2211yescharacteristic, complemented
42#16861215C2211yescharacteristic, complemented
43#16861205C2211yescharacteristic, complemented
44#16861198C2211yescharacteristic, complemented
45#16861145C2211yescharacteristic, complemented
46#16861144C2211yescharacteristic, complemented
47#16861136C2211yescharacteristic, complemented
48#16861136C2211yescharacteristic, complemented
49#16861136C2211yescharacteristic, complemented
50#16861135C2211yescharacteristic, complemented
51#16861128C2211yescharacteristic, complemented
52#16861126C2211yescharacteristic, complemented
53#16861125C2211yescharacteristic, complemented
54#16861125C2211yescharacteristic, complemented
55#16861119C2211yescharacteristic, complemented
562−1+4×C127C2211yescharacteristic, complemented
57H(5,𝔽2)×C127C2211yescharacteristic, complemented
58#137×C127—21no—
59#137×C127C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C127—21no—
61AGL(1,ℤ/8)×C127C2211yescharacteristic, complemented
62#135×C127—21no—
63#135×C127C2211yescharacteristic
64C2×SD16×C127—21no—
65C2×SD16×C127C2211yescharacteristic, complemented
66#131×C127C2211yescharacteristic, complemented
67#14523249—21no—
68#14523249—21no—
69#14523248—21no—
70#14523248—21no—
71#14523248C2311yescharacteristic
72C2×D508—21no—
73C2×D508—21no—
74C2×D508C2311yescharacteristic
75C2×D508C2311yescharacteristic
76C2×C22⋊D127—41no—
77D4×D127—21no—
78D4×D127—21no—
79D4×D127—41no—
80D4×D127—41no—
81#14523238—21no—
82#14523238D411yescharacteristic, complemented
83#14523236—21no—
84#14523236D411yescharacteristic, complemented
85#14523235—21no—
86#14523235—21no—
87#14523235—21no—
88#14523235—21no—
89#14523235—21no—
90#14523235—21no—
91#14523235—21no—
92#14523235D411yescharacteristic, complemented
93D1016—21no—
94D1016—21no—
95#14523233—21no—
96#14523233—21no—
97#14523233—21no—
98#14523233—21no—
99#14523233—21no—
100#14523233—21no—
101#14523233—21no—
102#14523233D411yescharacteristic, complemented
103#14523232—21no—
104#14523232—21no—
105#14523232C2311yescharacteristic
106#14523232C2311yescharacteristic
107#14523232C2311yescharacteristic
108#14523231—21no—
109C2×C127⋊C8—21no—
110C2×C127⋊C8—21no—
111C2×C127⋊C8C2311yescharacteristic
112C8×D127—21no—
113(C2×D127)⋊C4—41no—
114C4×Dic127C2311yescharacteristic
115#41×C127—21no—
116#41×C127—21no—
117#41×C127—21no—
118#41×C127—21no—
119#41×C127—41no—
120#41×C127C2311yescharacteristic
121#41×C127C2311yescharacteristic
122#41×C127C2311yescharacteristic
123C2×Q8×C127—21no—
124C2×Q8×C127—21no—
125C2×Q8×C127C2311yescharacteristic
126C2×D4×C127—21no—
127C2×D4×C127—21no—
128C2×D4×C127—41no—
129C2×D4×C127C2311yescharacteristic
130Q16×C127—21no—
131Q16×C127D411yescharacteristic, complemented
132SD16×C127—21no—
133SD16×C127—21no—
134SD16×C127—21no—
135SD16×C127—21no—
136SD16×C127D411yescharacteristic, complemented
137SD16×C127D411yescharacteristic, complemented
138D8×C127—21no—
139D8×C127D411yescharacteristic, complemented
140M16×C127—21no—
141M16×C127C2311yescharacteristic
142C2×C8×C127—21no—
143C2×C8×C127C2311yescharacteristic
144C22×D127—21no—
145C22×D127—21no—
146C22×D127—41no—
147C22×D127—41no—
148Dic254C2×D411yescharacteristic
149D508—21no—
150D508—21no—
151D508—21no—
152D508—21no—
153D508—41no—
154D508—41no—
155D508C2×D411yescharacteristic
156C22⋊D127—21no—
157C22⋊D127—41no—
158C22⋊D127—41no—
159C22⋊D127—82no—
160C2×Dic127—21no—
161C4×D127—21no—
162C4×D127—41no—
163C4×D127—41no—
164C127⋊C8—21no—
165C127⋊C8—21no—
166C127⋊C8—21no—
167C127⋊C8C2×D411yescharacteristic
168C127⋊C8C2×D411yescharacteristic
169C23×C127—21no—
170C23×C127—41no—
171Q8×C127—21no—
172Q8×C127—21no—
173Q8×C127—41no—
174Q8×C127C2×D411yescharacteristic
175Q8×C127C2×D411yescharacteristic
176Q8×C127C2×D411yescharacteristic
177D4×C127—21no—
178D4×C127—21no—
179D4×C127—21no—
180D4×C127—21no—
181D4×C127—21no—
182D4×C127—41no—
183D4×C127—41no—
184D4×C127—41no—
185D4×C127C2×D411yescharacteristic
186D4×C127C2×D411yescharacteristic
187D4×C127C2×D411yescharacteristic
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127—41no—
193C2×C4×C127—41no—
194C2×C4×C127—41no—
195C2×C4×C127C2411yescharacteristic
196C8×C127—21no—
197C8×C127C2×D411yescharacteristic
198C8×C127C2×D411yescharacteristic
199C22×C127—21no—
200C22×C127—21no—
201C22×C127—21no—
202C22×C127—41no—
203C22×C127—81no—
204C22×C127—161no—
205C22×C127H(5,𝔽2)11yescharacteristic
206C2×D127—21no—
207C2×D127—41no—
208C2×D127—41no—
209C2×D127—81no—
210C2×D127—81no—
211C2×D127—161no—
212C2×D127—161no—
213C4×C127—21no—
214C4×C127—21no—
215C4×C127—21no—
216C4×C127—41no—
217C4×C127C22×D411yescharacteristic
218C4×C127C22×D411yescharacteristic
219Dic127—21no—
220Dic127—41no—
221C2×C127—21no—
222C2×C127—41no—
223C2×C127—41no—
224C2×C127—41no—
225C2×C127—81no—
226C2×C127D4211yescharacteristic
227D127—41no—
228D127—81no—
229D127—81no—
230#3288—1271nomaximal, Sylow
231C127#328811yescharacteristic, complemented, Sylow
232#577—1271no—
233#577D12711yescharacteristic, complemented
234#576—1271no—
235#576—1271no—
236#496—1271no—
237#472—1271no—
238#472—1271no—
239#471—1271no—
240#471—1271no—
241#455—1271no—
242#454—1271no—
243#453—1271no—
244#453—1271no—
245#444—1271no—
246#444—1271no—
2472−1+4C2×D12711yescharacteristic, complemented
248H(5,𝔽2)—1271no—
249H(5,𝔽2)—1271no—
250H(5,𝔽2)C2×D12711yescharacteristic, complemented
251#137—21no—
252#137—1271no—
253#137—2541no—
254#137C2×D12711yescharacteristic, complemented
255AGL(1,ℤ/8)—21no—
256AGL(1,ℤ/8)—1271no—
257AGL(1,ℤ/8)—1271no—
258AGL(1,ℤ/8)—1271no—
259AGL(1,ℤ/8)—1271no—
260AGL(1,ℤ/8)—1271no—
261AGL(1,ℤ/8)—2541no—
262AGL(1,ℤ/8)—2541no—
263AGL(1,ℤ/8)—2541no—
264AGL(1,ℤ/8)—2541no—
265AGL(1,ℤ/8)—2541no—
266AGL(1,ℤ/8)C2×D12711yescharacteristic, complemented
267#135—21no—
268#135—1271no—
269#135—2541no—
270#135—2541no—
271#135—2541no—
272#135C2×D12711yescharacteristic
273C2×SD16—21no—
274C2×SD16—1271no—
275C2×SD16—2541no—
276C2×SD16C2×D12711yescharacteristic, complemented
277C2×D8—1271no—
278C2×D8—1271no—
279C2×D8—2541no—
280C2×D8—2541no—
281#131—1271no—
282#131—1271no—
283#131—1271no—
284#131C2×D12711yescharacteristic, complemented
285#127—1271no—
286#124—1271no—
287#108—1271no—
288#108—1271no—
289#104—1271no—
290#104—1271no—
291#104—1271no—
292#104—1271no—
293#101—1271no—
294#100—1271no—
295#100—1271no—
296#100—1271no—
297#97—1271no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—41no—
303#41—1271no—
304#41—2541no—
305#41—2541no—
306#41—2541no—
307#41—2541no—
308#41C22×D12711yescharacteristic
309#41C22×D12711yescharacteristic
310#41C22×D12711yescharacteristic
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8C22×D12711yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—1271no—
318C2×D4—1271no—
319C2×D4—5081no—
320C2×D4—5081no—
321C2×D4—5081no—
322C2×D4—2541no—
323C2×D4—2541no—
324C2×D4—2541no—
325C2×D4—2541no—
326C2×D4C22×D12711yescharacteristic
327Q16—21no—
328Q16—1271no—
329Q16—2541no—
330Q16C22⋊D12711yescharacteristic, complemented
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—21no—
335SD16—1271no—
336SD16—1271no—
337SD16—2541no—
338SD16—2541no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342SD16—2541no—
343SD16—2541no—
344SD16—2541no—
345SD16C22⋊D12711yescharacteristic, complemented
346SD16C22⋊D12711yescharacteristic, complemented
347D8—21no—
348D8—1271no—
349D8—2541no—
350D8—2541no—
351D8—2541no—
352D8—2541no—
353D8—2541no—
354D8—2541no—
355D8—2541no—
356D8—2541no—
357D8—2541no—
358D8C22⋊D12711yescharacteristic, complemented
359M16—21no—
360M16—1271no—
361M16—1271no—
362M16—1271no—
363M16—2541no—
364M16—2541no—
365M16—2541no—
366M16C22×D12711yescharacteristic
367C2×C8—21no—
368C2×C8—1271no—
369C2×C8—2541no—
370C2×C8—2541no—
371C2×C8—2541no—
372C2×C8C22×D12711yescharacteristic
373C22⋊C4—5081no—
374C42—1271no—
375C23—21no—
376C23—41no—
377C23—5081no—
378C23—5081no—
379C23—2541no—
380C23—2541no—
381Q8—21no—
382Q8—21no—
383Q8—41no—
384Q8—1271no—
385Q8D4×D12711yescharacteristic
386Q8C2×C22⋊D12711yescharacteristic
387Q8C2×C22⋊D12711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—1271no—
397D4—10162no—
398D4—5081no—
399D4—5081no—
400D4—5081no—
401D4—5081no—
402D4—2541no—
403D4—2541no—
404D4—2541no—
405D4—2541no—
406D4—2541no—
407D4D4×D12711yescharacteristic
408D4C2×C22⋊D12711yescharacteristic
409D4C2×C22⋊D12711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—5081no—
418C2×C4—5081no—
419C2×C4—2541no—
420C2×C4—2541no—
421C2×C4C23×D12711yescharacteristic
422C8—21no—
423C8—1271no—
424C8—1271no—
425C8—2541no—
426C8—2541no—
427C8—2541no—
428C8C2×C22⋊D12711yescharacteristic
429C8C2×C22⋊D12711yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—41no—
434C22—81no—
435C22—161no—
436C22—20321no—
437C22—20321no—
438C22—10161no—
439C22—10161no—
440C22—5081no—
441C22—5081no—
442C22—2541no—
443C22#1686125411yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—41no—
448C4—5081no—
449C4—2541no—
450C4C2×D4×D12711yescharacteristic
451C4C22×C22⋊D12711yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2—81no—
457C2—10161no—
458C2—10161no—
459C2—5081no—
460C2D4×C22⋊D12711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#577×C127C2
socleC2×C127D42
hypercenter#577D127
hypocenterC127#3288
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #577

Complements

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

NH ≅ G/Ntype
1. #22550684C2semidirect
2. #22550679C2semidirect
3. #22550678C2semidirect
4. #22550262C2semidirect
5. #22550146C2semidirect
6. #22550144C2semidirect
7. #22550141C2semidirect
8. #22550136C2semidirect
9. #22550052C2semidirect
10. #22550049C2semidirect
11. #22550044C2semidirect
12. #22550042C2semidirect
13. #22549998C2semidirect
14. #22549996C2semidirect
15. #577×C127C2semidirect
17. #16861255C22semidirect
19. #16861240C22semidirect
27. #16861234C22semidirect
28. #16861234C22semidirect
29. #16861234C22semidirect
35. C2×#14523235C22semidirect
36. C2×D1016C22semidirect
38. C2×#14523233C22semidirect
40. #16861216C22semidirect
41. #16861216C22semidirect
42. #16861215C22semidirect
43. #16861205C22semidirect
44. #16861198C22semidirect
45. #16861145C22semidirect
46. #16861144C22semidirect
47. #16861136C22semidirect
48. #16861136C22semidirect
49. #16861136C22semidirect
50. #16861135C22semidirect
51. #16861128C22semidirect
52. #16861126C22semidirect
53. #16861125C22semidirect
54. #16861125C22semidirect
55. #16861119C22semidirect
56. 2−1+4×C127C22semidirect
57. H(5,𝔽2)×C127C22semidirect
59. #137×C127C22semidirect
61. AGL(1,ℤ/8)×C127C22semidirect
65. C2×SD16×C127C22semidirect
66. #131×C127C22semidirect
82. #14523238D4semidirect
84. #14523236D4semidirect
92. #14523235D4semidirect
102. #14523233D4semidirect
131. Q16×C127D4semidirect
136. SD16×C127D4semidirect
137. SD16×C127D4semidirect
139. D8×C127D4semidirect
231. C127#3288semidirect
233. #577D127semidirect
247. 2−1+4C2×D127semidirect
250. H(5,𝔽2)C2×D127semidirect
254. #137C2×D127semidirect
266. AGL(1,ℤ/8)C2×D127semidirect
276. C2×SD16C2×D127semidirect
284. #131C2×D127semidirect
330. Q16C22⋊D127semidirect
345. SD16C22⋊D127semidirect
346. SD16C22⋊D127semidirect
358. D8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506842
#225506792
#225506782
#225502622
#225501462
#225501442
#225501412
#225501362
#225500522
#225500492
#225500442
#225500422
#225499982
#225499962
#577×C1272
#3288127