Finite Group Databasebeta

← #30650493

Subgroups — #30650493

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#22550686C211yesmaximal, characteristic, complemented
3#22550680C211yesmaximal, characteristic, complemented
4#22550677C211yesmaximal, characteristic, complemented
5#22550264C211yesmaximal, characteristic, complemented
6#22550145C211yesmaximal, characteristic, complemented
7#22550140C211yesmaximal, characteristic, complemented
8#22550138C211yesmaximal, characteristic, complemented
9#22550056C211yesmaximal, characteristic, complemented
10#22550047C211yesmaximal, characteristic, complemented
11#22550046C211yesmaximal, characteristic, complemented
12#22550042C211yesmaximal, characteristic, complemented
13#22549997C211yesmaximal, characteristic, complemented
14#22549996C211yesmaximal, characteristic, complemented
15#472×C127C211yesmaximal, characteristic, complemented
16#16861256C2211yescharacteristic, complemented
17#16861255C2211yescharacteristic, complemented
18#16861254C2211yescharacteristic
19#16861254C2211yescharacteristic, complemented
20#16861242—21no—
21#16861242C2211yescharacteristic, complemented
22#16861242C2211yescharacteristic, complemented
23#16861238—21no—
24#16861237—21no—
25#16861237C2211yescharacteristic
26#16861237C2211yescharacteristic, complemented
27#16861236—21no—
28#16861235C2211yescharacteristic
29#16861235C2211yescharacteristic, complemented
30#16861233—21no—
31#16861233—21no—
32#16861233—21no—
33#16861233—21no—
34#16861231—21no—
35#16861231C2211yescharacteristic
36#16861230—21no—
37#16861229C2211yescharacteristic
38#16861228—21no—
39#16861228—21no—
40C2×#14523237—21no—
41C2×#14523237C2211yescharacteristic, complemented
42C2×#14523236C2211yescharacteristic, complemented
43SD16×D127—21no—
44C2×#14523233C2211yescharacteristic, complemented
45D8×D127—21no—
46D8×D127—21no—
47#16861217C2211yescharacteristic, complemented
48#16861217C2211yescharacteristic, complemented
49#16861215C2211yescharacteristic, complemented
50#16861205C2211yescharacteristic, complemented
51#16861196C2211yescharacteristic, complemented
52#16861144C2211yescharacteristic, complemented
53#16861136C2211yescharacteristic, complemented
54#16861135C2211yescharacteristic, complemented
55#16861135C2211yescharacteristic, complemented
56#16861135C2211yescharacteristic, complemented
57#16861127C2211yescharacteristic, complemented
58#16861125C2211yescharacteristic, complemented
59#16861119C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C127C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C127C2211yescharacteristic, complemented
62C2×D8×C127C2211yescharacteristic, complemented
63#131×C127C2211yescharacteristic, complemented
64#108×C127C2211yescharacteristic, complemented
65#100×C127C2211yescharacteristic, complemented
66#100×C127C2211yescharacteristic, complemented
67#14523249—21no—
68#14523248—21no—
69#14523248—21no—
70#14523248C2311yescharacteristic
71#14523248C2311yescharacteristic
72#14523248C2311yescharacteristic
73#14523247—21no—
74#14523247—21no—
75#14523247—21no—
76#14523247—21no—
77#14523247—21no—
78#14523247—41no—
79C2×Dic254—21no—
80C2×Dic254C2311yescharacteristic
81Q8×D127—21no—
82C2×D508—21no—
83C2×D508C2311yescharacteristic
84C2×C22⋊D127—41no—
85C2×C22⋊D127—41no—
86C2×C22⋊D127—41no—
87D4×D127—21no—
88D4×D127—21no—
89D4×D127—21no—
90D4×D127—21no—
91D4×D127—21no—
92D4×D127—41no—
93Dic508—21no—
94Dic508—21no—
95Dic508D411yescharacteristic, complemented
96#14523238D411yescharacteristic, complemented
97#14523237—21no—
98#14523237—21no—
99#14523237—21no—
100#14523237—21no—
101#14523237—21no—
102#14523237—21no—
103#14523237D411yescharacteristic, complemented
104#14523237D411yescharacteristic, complemented
105#14523236—21no—
106#14523236—21no—
107#14523236—21no—
108#14523236—21no—
109#14523236D411yescharacteristic, complemented
110#14523235D411yescharacteristic, complemented
111D1016—21no—
112D1016—21no—
113D1016D411yescharacteristic, complemented
114#14523233—21no—
115#14523233—21no—
116#14523233—21no—
117#14523233—21no—
118#14523233D411yescharacteristic, complemented
119#14523232C2311yescharacteristic
120#14523231—21no—
121#14523231—21no—
122#14523231—21no—
123C2×C127⋊C8C2311yescharacteristic
124C8×D127—21no—
125C8×D127—21no—
126C8×D127—21no—
127C22⋊Dic127—41no—
128C4×Dic127C2311yescharacteristic
129#41×C127C2311yescharacteristic
130C2×D4×C127C2311yescharacteristic
131C2×D4×C127C2311yescharacteristic
132SD16×C127—21no—
133SD16×C127—21no—
134D8×C127—21no—
135D8×C127—21no—
136D8×C127—21no—
137D8×C127—21no—
138M16×C127—21no—
139M16×C127C2311yescharacteristic
140M16×C127C2311yescharacteristic
141M16×C127C2311yescharacteristic
142C2×C8×C127—21no—
143C2×C8×C127C2311yescharacteristic
144C22×D127—21no—
145C22×D127—41no—
146C22×D127—41no—
147C22×D127—41no—
148Dic254—21no—
149Dic254—21no—
150Dic254—41no—
151Dic254C2×D411yescharacteristic
152Dic254C2×D411yescharacteristic
153Dic254C2×D411yescharacteristic
154D508—21no—
155D508—21no—
156D508—41no—
157D508C2×D411yescharacteristic
158D508C2×D411yescharacteristic
159D508C2×D411yescharacteristic
160C22⋊D127—21no—
161C22⋊D127—21no—
162C22⋊D127—21no—
163C22⋊D127—41no—
164C22⋊D127—41no—
165C22⋊D127—41no—
166C22⋊D127—41no—
167C22⋊D127—41no—
168C22⋊D127—41no—
169C22⋊D127—82no—
170C2×Dic127—21no—
171C2×Dic127—21no—
172C2×Dic127—41no—
173C2×Dic127—41no—
174C2×Dic127—41no—
175C4×D127—21no—
176C4×D127—21no—
177C4×D127—21no—
178C4×D127—41no—
179C4×D127—41no—
180C127⋊C8—21no—
181C127⋊C8C2×D411yescharacteristic
182C127⋊C8C2×D411yescharacteristic
183C23×C127—21no—
184C23×C127—21no—
185Q8×C127C2×D411yescharacteristic
186D4×C127—21no—
187D4×C127—21no—
188D4×C127—21no—
189D4×C127—21no—
190D4×C127—21no—
191D4×C127C2×D411yescharacteristic
192C2×C4×C127—21no—
193C2×C4×C127C2411yescharacteristic
194C8×C127—21no—
195C8×C127—21no—
196C8×C127—21no—
197C8×C127C2×D411yescharacteristic
198C8×C127C2×D411yescharacteristic
199C22×C127—21no—
200C22×C127—41no—
201C22×C127—41no—
202C22×C127—81no—
203C22×C127—81no—
204C22×C127H(5,𝔽2)11yescharacteristic
205C2×D127—21no—
206C2×D127—21no—
207C2×D127—21no—
208C2×D127—41no—
209C2×D127—81no—
210C2×D127—161no—
211C2×D127—161no—
212C2×D127—161no—
213C4×C127—21no—
214C4×C127C22×D411yescharacteristic
215C4×C127C22×D411yescharacteristic
216Dic127—21no—
217Dic127—21no—
218Dic127—21no—
219Dic127—41no—
220Dic127—41no—
221C2×C127—21no—
222C2×C127—41no—
223C2×C127—81no—
224C2×C127—81no—
225C2×C127D4211yescharacteristic
226D127—41no—
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#472D12711yescharacteristic, complemented
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+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)—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, complemented
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—2541no—
279C2×D8—2541no—
280C2×D8C2×D12711yescharacteristic, complemented
281#131—1271no—
282#131—1271no—
283#131—1271no—
284#131C2×D12711yescharacteristic, complemented
285#127—1271no—
286#124—1271no—
287#108—1271no—
288#108C2×D12711yescharacteristic, complemented
289#104—1271no—
290#104—1271no—
291#104—1271no—
292#104—1271no—
293#101—1271no—
294#100—1271no—
295#100C2×D12711yescharacteristic, complemented
296#100C2×D12711yescharacteristic, complemented
297#97—1271no—
298#41—1271no—
299#41—1271no—
300#41—1271no—
301#41—5081no—
302#41—2541no—
303#41—2541no—
304#41—2541no—
305#41—2541no—
306#41—2541no—
307#41—2541no—
308#41—2541no—
309#41—2541no—
310#41C22×D12711yescharacteristic
311C2×Q8—1271no—
312C2×Q8—2541no—
313C2×Q8—2541no—
314C2×D4—1271no—
315C2×D4—5081no—
316C2×D4—5081no—
317C2×D4—5081no—
318C2×D4—5081no—
319C2×D4—2541no—
320C2×D4—2541no—
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—21no—
350D8—21no—
351D8—1271no—
352D8—1271no—
353D8—2541no—
354D8—2541no—
355D8—2541no—
356D8—2541no—
357D8—2541no—
358D8—2541no—
359M16—21no—
360M16—1271no—
361M16—2541no—
362M16—2541no—
363M16—2541no—
364M16C22×D12711yescharacteristic
365M16C22×D12711yescharacteristic
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—21no—
377C23—5081no—
378C23—5081no—
379C23—5081no—
380C23—2541no—
381Q8—1271no—
382Q8—1271no—
383Q8—1271no—
384Q8—5081no—
385Q8—2541no—
386Q8—2541no—
387Q8D4×D12711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—1271no—
394D4—1271no—
395D4—1271no—
396D4—10162no—
397D4—5081no—
398D4—5081no—
399D4—5081no—
400D4—5081no—
401D4—5081no—
402D4—5081no—
403D4—5081no—
404D4—2541no—
405D4—2541no—
406D4—2541no—
407D4—2541no—
408D4—2541no—
409D4D4×D12711yescharacteristic
410C2×C4—21no—
411C2×C4—5081no—
412C2×C4—5081no—
413C2×C4—5081no—
414C2×C4—5081no—
415C2×C4—5081no—
416C2×C4—2541no—
417C2×C4—2541no—
418C2×C4—2541no—
419C2×C4—2541no—
420C2×C4—2541no—
421C2×C4C23×D12711yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—1271no—
426C8—1271no—
427C8—2541no—
428C8D4×D12711yescharacteristic
429C8D4×D12711yescharacteristic
430C22—21no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—81no—
435C22—20321no—
436C22—20321no—
437C22—20321no—
438C22—10161no—
439C22—5081no—
440C22—2541no—
441C22—2541no—
442C22—2541no—
443C22#1686125511yescharacteristic
444C4—21no—
445C4—5081no—
446C4—5081no—
447C4—2541no—
448C4—2541no—
449C4—2541no—
450C4C2×D4×D12711yescharacteristic
451C4C2×D4×D12711yescharacteristic
452C2—21no—
453C2—41no—
454C2—81no—
455C2—81no—
456C2—10161no—
457C2—5081no—
458C2—5081no—
459C2—5081no—
460C2#2255051011yescharacteristic
subgroupG/H
centerC2#22550510
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#472×C127C2
socleC2×C127D42
hypercenter#472D127
hypocenterC127#3288
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #22550688C2semidirect
2. #22550686C2semidirect
3. #22550680C2semidirect
4. #22550677C2semidirect
5. #22550264C2semidirect
6. #22550145C2semidirect
7. #22550140C2semidirect
8. #22550138C2semidirect
9. #22550056C2semidirect
10. #22550047C2semidirect
11. #22550046C2semidirect
12. #22550042C2semidirect
13. #22549997C2semidirect
14. #22549996C2semidirect
15. #472×C127C2semidirect
16. #16861256C22semidirect
17. #16861255C22semidirect
19. #16861254C22semidirect
21. #16861242C22semidirect
22. #16861242C22semidirect
26. #16861237C22semidirect
29. #16861235C22semidirect
41. C2×#14523237C22semidirect
42. C2×#14523236C22semidirect
44. C2×#14523233C22semidirect
47. #16861217C22semidirect
48. #16861217C22semidirect
49. #16861215C22semidirect
50. #16861205C22semidirect
51. #16861196C22semidirect
52. #16861144C22semidirect
53. #16861136C22semidirect
54. #16861135C22semidirect
55. #16861135C22semidirect
56. #16861135C22semidirect
57. #16861127C22semidirect
58. #16861125C22semidirect
59. #16861119C22semidirect
60. AGL(1,ℤ/8)×C127C22semidirect
61. AGL(1,ℤ/8)×C127C22semidirect
62. C2×D8×C127C22semidirect
63. #131×C127C22semidirect
64. #108×C127C22semidirect
65. #100×C127C22semidirect
66. #100×C127C22semidirect
95. Dic508D4semidirect
96. #14523238D4semidirect
103. #14523237D4semidirect
104. #14523237D4semidirect
109. #14523236D4semidirect
110. #14523235D4semidirect
113. D1016D4semidirect
118. #14523233D4semidirect
231. C127#3288semidirect
238. #472D127semidirect
265. AGL(1,ℤ/8)C2×D127semidirect
266. AGL(1,ℤ/8)C2×D127semidirect
280. C2×D8C2×D127semidirect
284. #131C2×D127semidirect
288. #108C2×D127semidirect
295. #100C2×D127semidirect
296. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506882
#225506862
#225506802
#225506772
#225502642
#225501452
#225501402
#225501382
#225500562
#225500472
#225500462
#225500422
#225499972
#225499962
#472×C1272
#3288127