Finite Group Databasebeta

← #30650497

Subgroups — #30650497

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#22550688C211yesmaximal, characteristic, complemented
2#22550683C211yesmaximal, characteristic, complemented
3#22550681C211yesmaximal, characteristic, complemented
4#22550678C211yesmaximal, characteristic, complemented
5#22550265C211yesmaximal, characteristic, complemented
6#22550147C211yesmaximal, characteristic, complemented
7#22550146C211yesmaximal, characteristic, complemented
8#22550140C211yesmaximal, characteristic, complemented
9#22550139C211yesmaximal, characteristic, complemented
10#22550055C211yesmaximal, characteristic, complemented
11#22550050C211yesmaximal, characteristic, complemented
12#22550045C211yesmaximal, characteristic, complemented
13#22550045C211yesmaximal, characteristic, complemented
14#22549999C211yesmaximal, characteristic, complemented
15#444×C127C211yesmaximal, characteristic, complemented
16#16861256C2211yescharacteristic, complemented
17#16861255C2211yescharacteristic
18#16861255C2211yescharacteristic, complemented
19#16861255C2211yescharacteristic, complemented
20#16861242—21no—
21#16861242C2211yescharacteristic, complemented
22#16861241—21no—
23#16861239C2211yescharacteristic, complemented
24#16861237—21no—
25#16861237—21no—
26#16861237—21no—
27#16861237C2211yescharacteristic
28#16861237C2211yescharacteristic, complemented
29#16861237C2211yescharacteristic, complemented
30#16861236—21no—
31#16861236—21no—
32#16861234C2211yescharacteristic
33#16861234C2211yescharacteristic, complemented
34#16861234C2211yescharacteristic, complemented
35#16861233—21no—
36#16861232—21no—
37#16861231—21no—
38#16861231—21no—
39#16861231C2211yescharacteristic
40#16861230—21no—
41C2×#14523237—21no—
42C2×#14523237C2211yescharacteristic, complemented
43C2×#14523237C2211yescharacteristic, complemented
44SD16×D127—21no—
45C2×D1016—21no—
46C2×D1016C2211yescharacteristic, complemented
47C2×D1016C2211yescharacteristic, complemented
48D8×D127—21no—
49#16861215C2211yescharacteristic, complemented
50#16861215C2211yescharacteristic, complemented
51#16861206C2211yescharacteristic, complemented
52#16861197C2211yescharacteristic, complemented
53#16861145C2211yescharacteristic, complemented
54#16861137C2211yescharacteristic, complemented
55#16861137C2211yescharacteristic, complemented
56#16861127C2211yescharacteristic, complemented
57#16861125C2211yescharacteristic, complemented
58#16861125C2211yescharacteristic, complemented
59#16861125C2211yescharacteristic, complemented
60#135×C127C2211yescharacteristic
61#131×C127C2211yescharacteristic, complemented
62#131×C127C2211yescharacteristic, complemented
63#108×C127C2211yescharacteristic, complemented
64#104×C127C2211yescharacteristic, complemented
65#104×C127C2211yescharacteristic, complemented
66#97×C127C2211yescharacteristic, complemented
67#14523249—21no—
68#14523249—21no—
69#14523249—21no—
70#14523248—21no—
71#14523248—21no—
72#14523248—21no—
73#14523248—21no—
74#14523248C2311yescharacteristic
75#14523248C2311yescharacteristic
76#14523247—21no—
77#14523247—41no—
78C2×Dic254—21no—
79C2×Dic254C2311yescharacteristic
80Q8×D127—21no—
81C2×D508—21no—
82C2×D508—21no—
83C2×D508—21no—
84C2×D508—41no—
85C2×D508C2311yescharacteristic
86C2×D508C2311yescharacteristic
87C2×D508C2311yescharacteristic
88D4×D127—21no—
89D4×D127—21no—
90D4×D127—21no—
91D4×D127—41no—
92D4×D127—41no—
93D4×D127—41no—
94Dic508—21no—
95Dic508D411yescharacteristic, complemented
96#14523238—21no—
97#14523237—21no—
98#14523237—21no—
99#14523237—21no—
100#14523237—21no—
101#14523237—21no—
102#14523237—21no—
103#14523237—21no—
104#14523237—21no—
105#14523237D411yescharacteristic, complemented
106#14523237D411yescharacteristic, complemented
107#14523236—21no—
108#14523235—21no—
109#14523235—21no—
110#14523235—21no—
111D1016—21no—
112D1016—21no—
113D1016—21no—
114D1016—21no—
115D1016—21no—
116D1016—21no—
117D1016—21no—
118D1016D411yescharacteristic, complemented
119#14523233—21no—
120#14523233—21no—
121#14523233—21no—
122#14523232C2311yescharacteristic
123#14523232C2311yescharacteristic
124#14523231—21no—
125#14523231—21no—
126C8×D127—21no—
127C8×D127—21no—
128(C2×D127)⋊C4—41no—
129#41×C127C2311yescharacteristic
130#41×C127C2311yescharacteristic
131Q16×C127D411yescharacteristic, complemented
132SD16×C127D411yescharacteristic, complemented
133SD16×C127D411yescharacteristic, complemented
134D8×C127D411yescharacteristic, complemented
135M16×C127—21no—
136M16×C127—21no—
137M16×C127C2311yescharacteristic
138M16×C127C2311yescharacteristic
139C2×C8×C127—21no—
140C2×C8×C127—21no—
141C2×C8×C127C2311yescharacteristic
142C2×C8×C127C2311yescharacteristic
143C42×C127C2311yescharacteristic
144C22×D127—21no—
145C22×D127—21no—
146C22×D127—21no—
147C22×D127—41no—
148C22×D127—41no—
149C22×D127—41no—
150Dic254—21no—
151Dic254—21no—
152Dic254—41no—
153Dic254C2×D411yescharacteristic
154Dic254C2×D411yescharacteristic
155D508—21no—
156D508—21no—
157D508—21no—
158D508—21no—
159D508—21no—
160D508—21no—
161D508—41no—
162D508—41no—
163D508—41no—
164D508—82no—
165D508C2×D411yescharacteristic
166D508C2×D411yescharacteristic
167C22⋊D127—21no—
168C22⋊D127—21no—
169C22⋊D127—41no—
170C22⋊D127—41no—
171C22⋊D127—41no—
172C22⋊D127—41no—
173C2×Dic127—21no—
174C2×Dic127—41no—
175C4×D127—21no—
176C4×D127—21no—
177C4×D127—41no—
178C4×D127—41no—
179C4×D127—41no—
180C4×D127—41no—
181C127⋊C8—21no—
182C127⋊C8—21no—
183Q8×C127C2×D411yescharacteristic
184Q8×C127C2×D411yescharacteristic
185D4×C127—21no—
186D4×C127—21no—
187D4×C127C2×D411yescharacteristic
188D4×C127C2×D411yescharacteristic
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127C2411yescharacteristic
193C8×C127—21no—
194C8×C127—21no—
195C8×C127C2×D411yescharacteristic
196C8×C127C2×D411yescharacteristic
197C8×C127C2×D411yescharacteristic
198C8×C127C2×D411yescharacteristic
199C22×C127—21no—
200C22×C127—21no—
201C22×C127H(5,𝔽2)11yescharacteristic
202C2×D127—21no—
203C2×D127—21no—
204C2×D127—41no—
205C2×D127—41no—
206C2×D127—41no—
207C2×D127—81no—
208C2×D127—81no—
209C2×D127—81no—
210C2×D127—161no—
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×C127D4211yescharacteristic
225D127—41no—
226D127—41no—
227D127—81no—
228D127—81no—
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#453—1271no—
245#444—1271no—
246#444D12711yescharacteristic, complemented
2472−1+4—1271no—
248H(5,𝔽2)—1271no—
249H(5,𝔽2)—1271no—
250H(5,𝔽2)—1271no—
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)—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)—2541no—
267#135—1271no—
268#135—2541no—
269#135—2541no—
270#135—2541no—
271#135—2541no—
272#135C2×D12711yescharacteristic
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#131C2×D12711yescharacteristic, complemented
284#131C2×D12711yescharacteristic, complemented
285#127—1271no—
286#124—1271no—
287#108—1271no—
288#108C2×D12711yescharacteristic, complemented
289#104—1271no—
290#104—1271no—
291#104C2×D12711yescharacteristic, complemented
292#104C2×D12711yescharacteristic, complemented
293#101—1271no—
294#100—1271no—
295#100—1271no—
296#100—1271no—
297#97C2×D12711yescharacteristic, complemented
298#41—1271no—
299#41—1271no—
300#41—5081no—
301#41—2541no—
302#41—2541no—
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—1271no—
315C2×D4—1271no—
316C2×D4—1271no—
317C2×D4—5081no—
318C2×D4—5081no—
319C2×D4—5081no—
320C2×D4—5081no—
321C2×D4—2541no—
322C2×D4—2541no—
323C2×D4—2541no—
324C2×D4—2541no—
325C2×D4—2541no—
326C2×D4—2541no—
327Q16—1271no—
328Q16—2541no—
329Q16—2541no—
330Q16D50811yescharacteristic, complemented
331SD16—1271no—
332SD16—1271no—
333SD16—2541no—
334SD16—2541no—
335SD16—2541no—
336SD16—2541no—
337SD16—2541no—
338SD16—2541no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342SD16—2541no—
343SD16—2541no—
344SD16—2541no—
345SD16D50811yescharacteristic, complemented
346SD16D50811yescharacteristic, complemented
347D8—1271no—
348D8—2541no—
349D8—2541no—
350D8—2541no—
351D8—2541no—
352D8—2541no—
353D8—2541no—
354D8—2541no—
355D8—2541no—
356D8—2541no—
357D8—2541no—
358D8D50811yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—1271no—
362M16—1271no—
363M16—2541no—
364M16—2541no—
365M16C22×D12711yescharacteristic
366M16C22×D12711yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—2541no—
370C2×C8—2541no—
371C2×C8C22×D12711yescharacteristic
372C2×C8C22×D12711yescharacteristic
373C22⋊C4—5081no—
374C42C22×D12711yescharacteristic
375C23—5081no—
376C23—5081no—
377C23—5081no—
378C23—2541no—
379C23—2541no—
380C23—2541no—
381Q8—1271no—
382Q8—1271no—
383Q8—5081no—
384Q8—2541no—
385Q8—2541no—
386Q8C2×D50811yescharacteristic
387Q8C2×D50811yescharacteristic
388D4—21no—
389D4—21no—
390D4—1271no—
391D4—1271no—
392D4—10162no—
393D4—5081no—
394D4—5081no—
395D4—5081no—
396D4—5081no—
397D4—5081no—
398D4—5081no—
399D4—5081no—
400D4—2541no—
401D4—2541no—
402D4—2541no—
403D4—2541no—
404D4—2541no—
405D4—2541no—
406D4—2541no—
407D4—2541no—
408D4C2×D50811yescharacteristic
409D4C2×D50811yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—5081no—
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—2541no—
425C8—2541no—
426C8D4×D12711yescharacteristic
427C8D4×D12711yescharacteristic
428C8C2×D50811yescharacteristic
429C8C2×D50811yescharacteristic
430C22—21no—
431C22—21no—
432C22—20321no—
433C22—20321no—
434C22—20321no—
435C22—10161no—
436C22—10161no—
437C22—10161no—
438C22—5081no—
439C22—5081no—
440C22—5081no—
441C22—2541no—
442C22—2541no—
443C22#1686125511yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—5081no—
448C4—2541no—
449C4—2541no—
450C4C2×D4×D12711yescharacteristic
451C4C22×D50811yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—10161no—
456C2—10161no—
457C2—10161no—
458C2—5081no—
459C2—5081no—
460C2D4×D50811yescharacteristic
subgroupG/H
centerC2D4×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#444×C127C2
socleC2×C127D42
hypercenter#444D127
hypocenterC127#3288
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #22550688C2semidirect
2. #22550683C2semidirect
3. #22550681C2semidirect
4. #22550678C2semidirect
5. #22550265C2semidirect
6. #22550147C2semidirect
7. #22550146C2semidirect
8. #22550140C2semidirect
9. #22550139C2semidirect
10. #22550055C2semidirect
11. #22550050C2semidirect
12. #22550045C2semidirect
13. #22550045C2semidirect
14. #22549999C2semidirect
15. #444×C127C2semidirect
16. #16861256C22semidirect
18. #16861255C22semidirect
19. #16861255C22semidirect
21. #16861242C22semidirect
23. #16861239C22semidirect
28. #16861237C22semidirect
29. #16861237C22semidirect
33. #16861234C22semidirect
34. #16861234C22semidirect
42. C2×#14523237C22semidirect
43. C2×#14523237C22semidirect
46. C2×D1016C22semidirect
47. C2×D1016C22semidirect
49. #16861215C22semidirect
50. #16861215C22semidirect
51. #16861206C22semidirect
52. #16861197C22semidirect
53. #16861145C22semidirect
54. #16861137C22semidirect
55. #16861137C22semidirect
56. #16861127C22semidirect
57. #16861125C22semidirect
58. #16861125C22semidirect
59. #16861125C22semidirect
61. #131×C127C22semidirect
62. #131×C127C22semidirect
63. #108×C127C22semidirect
64. #104×C127C22semidirect
65. #104×C127C22semidirect
66. #97×C127C22semidirect
95. Dic508D4semidirect
105. #14523237D4semidirect
106. #14523237D4semidirect
118. D1016D4semidirect
131. Q16×C127D4semidirect
132. SD16×C127D4semidirect
133. SD16×C127D4semidirect
134. D8×C127D4semidirect
231. C127#3288semidirect
246. #444D127semidirect
283. #131C2×D127semidirect
284. #131C2×D127semidirect
288. #108C2×D127semidirect
291. #104C2×D127semidirect
292. #104C2×D127semidirect
297. #97C2×D127semidirect
330. Q16D508semidirect
345. SD16D508semidirect
346. SD16D508semidirect
358. D8D508semidirect

Maximal subgroups

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

subgroupindex
#225506882
#225506832
#225506812
#225506782
#225502652
#225501472
#225501462
#225501402
#225501392
#225500552
#225500502
#225500452
#225499992
#444×C1272
#3288127