Finite Group Databasebeta

← #30650491

Subgroups — #30650491

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#22550687C211yesmaximal, characteristic, complemented
2#22550686C211yesmaximal, characteristic, complemented
3#22550680C211yesmaximal, characteristic, complemented
4#22550679C211yesmaximal, characteristic, complemented
5#22550263C211yesmaximal, characteristic, complemented
6#22550145C211yesmaximal, characteristic, complemented
7#22550144C211yesmaximal, characteristic, complemented
8#22550138C211yesmaximal, characteristic, complemented
9#22550137C211yesmaximal, characteristic, complemented
10#22550055C211yesmaximal, characteristic, complemented
11#22550050C211yesmaximal, characteristic, complemented
12#22550043C211yesmaximal, characteristic, complemented
13#22549995C211yesmaximal, characteristic, complemented
14#22549995C211yesmaximal, characteristic, complemented
15#453×C127C211yesmaximal, characteristic, complemented
16#16861256C2211yescharacteristic, complemented
17#16861255C2211yescharacteristic
18#16861254C2211yescharacteristic, complemented
19#16861242C2211yescharacteristic, complemented
20#16861239—21no—
21#16861239C2211yescharacteristic, complemented
22#16861238—21no—
23#16861237C2211yescharacteristic, complemented
24#16861236—21no—
25#16861235—21no—
26#16861235—21no—
27#16861235C2211yescharacteristic
28#16861235C2211yescharacteristic, complemented
29#16861234—21no—
30#16861234C2211yescharacteristic, complemented
31#16861233—21no—
32#16861233—21no—
33#16861230—21no—
34#16861229—21no—
35#16861229—21no—
36#16861229C2211yescharacteristic
37#16861229C2211yescharacteristic
38#16861228—21no—
39C2×#14523236—21no—
40C2×#14523236C2211yescharacteristic, complemented
41C2×#14523236C2211yescharacteristic, complemented
42SD16×D127—21no—
43C2×#14523233—21no—
44C2×#14523233C2211yescharacteristic, complemented
45C2×#14523233C2211yescharacteristic, complemented
46D8×D127—21no—
47#16861217C2211yescharacteristic, complemented
48#16861217C2211yescharacteristic, complemented
49#16861216C2211yescharacteristic, complemented
50#16861216C2211yescharacteristic, complemented
51#16861197C2211yescharacteristic, complemented
52#16861144C2211yescharacteristic, complemented
53#16861144C2211yescharacteristic, complemented
54#16861137C2211yescharacteristic, complemented
55#16861137C2211yescharacteristic, complemented
56#16861127C2211yescharacteristic, complemented
57#16861126C2211yescharacteristic, complemented
58#16861125C2211yescharacteristic, complemented
59#16861120C2211yescharacteristic, complemented
60H(5,𝔽2)×C127C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C127C2211yescharacteristic
62AGL(1,ℤ/8)×C127C2211yescharacteristic, complemented
63#127×C127C2211yescharacteristic, complemented
64#104×C127C2211yescharacteristic, complemented
65#104×C127C2211yescharacteristic, complemented
66#100×C127C2211yescharacteristic, complemented
67#14523249—21no—
68#14523248—21no—
69#14523248—21no—
70#14523248C2311yescharacteristic
71#14523248C2311yescharacteristic
72#14523247—21no—
73#14523247—21no—
74#14523247—21no—
75#14523247—41no—
76C2×Dic254—21no—
77C2×Dic254C2311yescharacteristic
78Q8×D127—21no—
79C2×D508—21no—
80C2×D508C2311yescharacteristic
81C2×C22⋊D127—41no—
82D4×D127—21no—
83D4×D127—21no—
84D4×D127—21no—
85D4×D127—41no—
86Dic508—21no—
87#14523238—21no—
88#14523238D411yescharacteristic, complemented
89#14523238D411yescharacteristic, complemented
90#14523237—21no—
91#14523237—21no—
92#14523236—21no—
93#14523236—21no—
94#14523236—21no—
95#14523236—21no—
96#14523236—21no—
97#14523236—21no—
98#14523236—21no—
99#14523236D411yescharacteristic, complemented
100#14523236D411yescharacteristic, complemented
101#14523235—21no—
102#14523235D411yescharacteristic, complemented
103#14523235D411yescharacteristic, complemented
104D1016—21no—
105#14523233—21no—
106#14523233—21no—
107#14523233—21no—
108#14523233—21no—
109#14523233—21no—
110#14523233—21no—
111#14523233—21no—
112#14523233D411yescharacteristic, complemented
113#14523233D411yescharacteristic, complemented
114#14523232—21no—
115#14523232—21no—
116#14523232C2311yescharacteristic
117#14523232C2311yescharacteristic
118#14523231—21no—
119#14523231—21no—
120C2×C127⋊C8—21no—
121C2×C127⋊C8—21no—
122C2×C127⋊C8C2311yescharacteristic
123C2×C127⋊C8C2311yescharacteristic
124C8×D127—21no—
125C8×D127—21no—
126(C2×D127)⋊C4—41no—
127#41×C127—21no—
128#41×C127—21no—
129#41×C127C2311yescharacteristic
130#41×C127C2311yescharacteristic
131C2×D4×C127—21no—
132C2×D4×C127—21no—
133C2×D4×C127—41no—
134C2×D4×C127—41no—
135C2×D4×C127C2311yescharacteristic
136C2×D4×C127C2311yescharacteristic
137SD16×C127—21no—
138SD16×C127—21no—
139D8×C127—21no—
140D8×C127—21no—
141M16×C127C2311yescharacteristic
142M16×C127C2311yescharacteristic
143C42×C127C2311yescharacteristic
144C22×D127—21no—
145C22×D127—41no—
146C22×D127—41no—
147Dic254—21no—
148Dic254—21no—
149Dic254—41no—
150Dic254C2×D411yescharacteristic
151Dic254C2×D411yescharacteristic
152D508—21no—
153D508—21no—
154D508—41no—
155D508C2×D411yescharacteristic
156D508C2×D411yescharacteristic
157C22⋊D127—21no—
158C22⋊D127—21no—
159C22⋊D127—41no—
160C22⋊D127—41no—
161C22⋊D127—41no—
162C22⋊D127—41no—
163C2×Dic127—21no—
164C2×Dic127—41no—
165C2×Dic127—41no—
166C4×D127—21no—
167C4×D127—21no—
168C4×D127—41no—
169C4×D127—41no—
170C127⋊C8—21no—
171C127⋊C8—21no—
172C127⋊C8C2×D411yescharacteristic
173C127⋊C8C2×D411yescharacteristic
174C127⋊C8C2×D411yescharacteristic
175C127⋊C8C2×D411yescharacteristic
176C23×C127—21no—
177C23×C127—21no—
178C23×C127—41no—
179Q8×C127C2×D411yescharacteristic
180Q8×C127C2×D411yescharacteristic
181D4×C127—21no—
182D4×C127—21no—
183D4×C127—21no—
184D4×C127—21no—
185D4×C127—21no—
186D4×C127—21no—
187D4×C127—41no—
188D4×C127—41no—
189D4×C127—82no—
190D4×C127C2×D411yescharacteristic
191D4×C127C2×D411yescharacteristic
192C2×C4×C127—21no—
193C2×C4×C127—21no—
194C2×C4×C127—21no—
195C2×C4×C127—41no—
196C2×C4×C127C2411yescharacteristic
197C8×C127—21no—
198C8×C127—21no—
199C22×C127—21no—
200C22×C127—21no—
201C22×C127—41no—
202C22×C127—41no—
203C22×C127—81no—
204C22×C127—81no—
205C22×C127—161no—
206C22×C127H(5,𝔽2)11yescharacteristic
207C2×D127—21no—
208C2×D127—21no—
209C2×D127—41no—
210C2×D127—81no—
211C2×D127—161no—
212C2×D127—161no—
213C4×C127—21no—
214C4×C127—21no—
215C4×C127—41no—
216C4×C127C22×D411yescharacteristic
217C4×C127C22×D411yescharacteristic
218Dic127—21no—
219Dic127—21no—
220Dic127—41no—
221C2×C127—21no—
222C2×C127—41no—
223C2×C127—41no—
224C2×C127—81no—
225C2×C127—81no—
226C2×C127D4211yescharacteristic
227D127—41no—
228D127—41no—
229D127—81no—
230#3288—1271nomaximal, Sylow
231C127#328811yescharacteristic, complemented, Sylow
232#577—1271no—
233#577—1271no—
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#453D12711yescharacteristic, complemented
245#444—1271no—
246#444—1271no—
2472−1+4—1271no—
248H(5,𝔽2)—1271no—
249H(5,𝔽2)—1271no—
250H(5,𝔽2)C2×D12711yescharacteristic, complemented
251#137—1271no—
252#137—1271no—
253#137—2541no—
254#137—2541no—
255AGL(1,ℤ/8)—1271no—
256AGL(1,ℤ/8)—1271no—
257AGL(1,ℤ/8)—1271no—
258AGL(1,ℤ/8)—1271no—
259AGL(1,ℤ/8)—2541no—
260AGL(1,ℤ/8)—2541no—
261AGL(1,ℤ/8)—2541no—
262AGL(1,ℤ/8)—2541no—
263AGL(1,ℤ/8)—2541no—
264AGL(1,ℤ/8)—2541no—
265AGL(1,ℤ/8)C2×D12711yescharacteristic
266AGL(1,ℤ/8)C2×D12711yescharacteristic, complemented
267#135—1271no—
268#135—1271no—
269#135—2541no—
270#135—2541no—
271#135—2541no—
272#135—2541no—
273C2×SD16—1271no—
274C2×SD16—1271no—
275C2×SD16—2541no—
276C2×SD16—2541no—
277C2×D8—1271no—
278C2×D8—1271no—
279C2×D8—2541no—
280C2×D8—2541no—
281#131—1271no—
282#131—1271no—
283#131—1271no—
284#131—1271no—
285#127C2×D12711yescharacteristic, complemented
286#124—1271no—
287#108—1271no—
288#108—1271no—
289#104—1271no—
290#104—1271no—
291#104C2×D12711yescharacteristic, complemented
292#104C2×D12711yescharacteristic, complemented
293#101—1271no—
294#100—1271no—
295#100—1271no—
296#100C2×D12711yescharacteristic, complemented
297#97—1271no—
298#41—21no—
299#41—21no—
300#41—1271no—
301#41—1271no—
302#41—5081no—
303#41—2541no—
304#41—2541no—
305#41—2541no—
306#41—2541no—
307#41—2541no—
308#41—2541no—
309#41C22×D12711yescharacteristic
310#41C22×D12711yescharacteristic
311C2×Q8—1271no—
312C2×Q8—2541no—
313C2×Q8—2541no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—1271no—
319C2×D4—5081no—
320C2×D4—5081no—
321C2×D4—2541no—
322C2×D4—2541no—
323C2×D4—2541no—
324C2×D4—2541no—
325C2×D4C22×D12711yescharacteristic
326C2×D4C22×D12711yescharacteristic
327Q16—1271no—
328Q16—1271no—
329Q16—2541no—
330Q16—2541no—
331SD16—21no—
332SD16—21no—
333SD16—1271no—
334SD16—1271no—
335SD16—1271no—
336SD16—1271no—
337SD16—2541no—
338SD16—2541no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342SD16—2541no—
343SD16—2541no—
344SD16—2541no—
345SD16—2541no—
346SD16—2541no—
347D8—21no—
348D8—21no—
349D8—1271no—
350D8—1271no—
351D8—2541no—
352D8—2541no—
353D8—2541no—
354D8—2541no—
355D8—2541no—
356D8—2541no—
357D8—2541no—
358D8—2541no—
359M16—1271no—
360M16—1271no—
361M16—2541no—
362M16—2541no—
363M16—2541no—
364M16—2541no—
365M16C22×D12711yescharacteristic
366M16C22×D12711yescharacteristic
367C2×C8—1271no—
368C2×C8—1271no—
369C2×C8—2541no—
370C2×C8—2541no—
371C2×C8—2541no—
372C2×C8—2541no—
373C22⋊C4—5081no—
374C42C22×D12711yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—5081no—
379C23—5081no—
380C23—2541no—
381Q8—1271no—
382Q8—1271no—
383Q8—5081no—
384Q8—2541no—
385Q8—2541no—
386Q8D4×D12711yescharacteristic
387Q8D4×D12711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—1271no—
398D4—1271no—
399D4—5081no—
400D4—5081no—
401D4—5081no—
402D4—5081no—
403D4—5081no—
404D4—2541no—
405D4—2541no—
406D4—2541no—
407D4—2541no—
408D4D4×D12711yescharacteristic
409D4D4×D12711yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—5081no—
415C2×C4—5081no—
416C2×C4—5081no—
417C2×C4—5081no—
418C2×C4—2541no—
419C2×C4—2541no—
420C2×C4—2541no—
421C2×C4C23×D12711yescharacteristic
422C8—21no—
423C8—21no—
424C8—1271no—
425C8—1271no—
426C8—1271no—
427C8—1271no—
428C8—2541no—
429C8—2541no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—20321no—
438C22—20321no—
439C22—10161no—
440C22—5081no—
441C22—2541no—
442C22—2541no—
443C22#1686125411yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—5081no—
448C4—2541no—
449C4—2541no—
450C4C2×D4×D12711yescharacteristic
451C4C2×D4×D12711yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—10161no—
458C2—5081no—
459C2—5081no—
460C2#2255050811yescharacteristic
subgroupG/H
centerC2#22550508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#453×C127C2
socleC2×C127D42
hypercenter#453D127
hypocenterC127#3288
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22550687C2semidirect
2. #22550686C2semidirect
3. #22550680C2semidirect
4. #22550679C2semidirect
5. #22550263C2semidirect
6. #22550145C2semidirect
7. #22550144C2semidirect
8. #22550138C2semidirect
9. #22550137C2semidirect
10. #22550055C2semidirect
11. #22550050C2semidirect
12. #22550043C2semidirect
13. #22549995C2semidirect
14. #22549995C2semidirect
15. #453×C127C2semidirect
16. #16861256C22semidirect
18. #16861254C22semidirect
19. #16861242C22semidirect
21. #16861239C22semidirect
23. #16861237C22semidirect
28. #16861235C22semidirect
30. #16861234C22semidirect
40. C2×#14523236C22semidirect
41. C2×#14523236C22semidirect
44. C2×#14523233C22semidirect
45. C2×#14523233C22semidirect
47. #16861217C22semidirect
48. #16861217C22semidirect
49. #16861216C22semidirect
50. #16861216C22semidirect
51. #16861197C22semidirect
52. #16861144C22semidirect
53. #16861144C22semidirect
54. #16861137C22semidirect
55. #16861137C22semidirect
56. #16861127C22semidirect
57. #16861126C22semidirect
58. #16861125C22semidirect
59. #16861120C22semidirect
60. H(5,𝔽2)×C127C22semidirect
62. AGL(1,ℤ/8)×C127C22semidirect
63. #127×C127C22semidirect
64. #104×C127C22semidirect
65. #104×C127C22semidirect
66. #100×C127C22semidirect
88. #14523238D4semidirect
89. #14523238D4semidirect
99. #14523236D4semidirect
100. #14523236D4semidirect
102. #14523235D4semidirect
103. #14523235D4semidirect
112. #14523233D4semidirect
113. #14523233D4semidirect
231. C127#3288semidirect
244. #453D127semidirect
250. H(5,𝔽2)C2×D127semidirect
266. AGL(1,ℤ/8)C2×D127semidirect
285. #127C2×D127semidirect
291. #104C2×D127semidirect
292. #104C2×D127semidirect
296. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506872
#225506862
#225506802
#225506792
#225502632
#225501452
#225501442
#225501382
#225501372
#225500552
#225500502
#225500432
#225499952
#453×C1272
#3288127