Finite Group Databasebeta

← #30650522

Subgroups — #30650522

Order 16256.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 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#22550685C211yesmaximal, characteristic, complemented
4#22550682C211yesmaximal, characteristic, complemented
5#22550247C211yesmaximal, characteristic, complemented
6#22550142C211yesmaximal, characteristic, complemented
7#22550140C211yesmaximal, characteristic, complemented
8#22550138C211yesmaximal, characteristic, complemented
9#22550061C211yesmaximal, characteristic, complemented
10#22550057C211yesmaximal, characteristic, complemented
11#22550046C211yesmaximal, characteristic, complemented
12#22550042C211yesmaximal, characteristic, complemented
13#22549992C211yesmaximal, characteristic, complemented
14#22549991C211yesmaximal, characteristic, complemented
15#471×C127C211yesmaximal, characteristic, complemented
16#16861257C2211yescharacteristic
17#16861256C2211yescharacteristic
18#16861255C2211yescharacteristic, complemented
19#16861254C2211yescharacteristic, complemented
20#16861242—21no—
21#16861242C2211yescharacteristic, complemented
22#16861242C2211yescharacteristic, complemented
23#16861241—21no—
24#16861240C2211yescharacteristic, complemented
25#16861238—21no—
26#16861238—21no—
27#16861237—21no—
28#16861237C2211yescharacteristic
29#16861237C2211yescharacteristic
30#16861236—21no—
31#16861236—21no—
32#16861235C2211yescharacteristic
33#16861233—21no—
34#16861232—21no—
35#16861231—21no—
36#16861231C2211yescharacteristic
37#16861230—21no—
38#16861229C2211yescharacteristic
39#16861228—21no—
40C2×#14523237—21no—
41C2×#14523237C2211yescharacteristic, complemented
42C2×#14523236C2211yescharacteristic, complemented
43C2×#14523235C2211yescharacteristic, complemented
44SD16×D127—21no—
45SD16×D127—21no—
46SD16×D127—21no—
47#16861217C2211yescharacteristic, complemented
48#16861217C2211yescharacteristic, complemented
49#16861215C2211yescharacteristic, complemented
50#16861207C2211yescharacteristic, complemented
51#16861205C2211yescharacteristic
52#16861144C2211yescharacteristic, complemented
53#16861136C2211yescharacteristic, complemented
54#16861135C2211yescharacteristic, complemented
55#16861135C2211yescharacteristic, complemented
56#16861135C2211yescharacteristic, complemented
57#16861127C2211yescharacteristic, complemented
58#16861125C2211yescharacteristic, complemented
59C8×Dic127C2211yescharacteristic, complemented
60#137×C127C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C127C2211yescharacteristic
62C2×SD16×C127C2211yescharacteristic, complemented
63#131×C127C2211yescharacteristic, complemented
64#108×C127C2211yescharacteristic, complemented
65#101×C127C2211yescharacteristic, complemented
66#100×C127C2211yescharacteristic, complemented
67#14523249—21no—
68#14523249—21no—
69#14523249—21no—
70#14523248—21no—
71#14523248—21no—
72#14523248—41no—
73#14523248C2311yescharacteristic
74#14523248C2311yescharacteristic
75#14523248C2311yescharacteristic
76#14523247—21no—
77#14523247—21no—
78#14523247—21no—
79#14523247—41no—
80C2×Dic254—21no—
81C2×Dic254C2311yescharacteristic
82Q8×D127—21no—
83Q8×D127—21no—
84Q8×D127—21no—
85C2×D508—21no—
86C2×D508C2311yescharacteristic
87C2×C22⋊D127—41no—
88C2×C22⋊D127—41no—
89D4×D127—21no—
90D4×D127—21no—
91D4×D127—21no—
92D4×D127—41no—
93Dic508—21no—
94Dic508—21no—
95Dic508D411yescharacteristic, complemented
96#14523238—21no—
97#14523238D411yescharacteristic, complemented
98#14523237—21no—
99#14523237—21no—
100#14523237—21no—
101#14523237—21no—
102#14523237—21no—
103#14523237—21no—
104#14523237D411yescharacteristic, complemented
105#14523237D411yescharacteristic, complemented
106#14523236—21no—
107#14523236—21no—
108#14523236—21no—
109#14523236D411yescharacteristic, complemented
110#14523235—21no—
111#14523235—21no—
112#14523235—21no—
113#14523235D411yescharacteristic, complemented
114D1016—21no—
115D1016—21no—
116D1016D411yescharacteristic, complemented
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—
127Dic127⋊C4—41no—
128C4×Dic127C2311yescharacteristic
129#41×C127C2311yescharacteristic
130C2×Q8×C127C2311yescharacteristic
131C2×D4×C127C2311yescharacteristic
132Q16×C127—21no—
133SD16×C127—21no—
134SD16×C127—21no—
135SD16×C127—21no—
136SD16×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—
147Dic254—21no—
148Dic254—21no—
149Dic254—41no—
150Dic254—41no—
151Dic254C2×D411yescharacteristic
152Dic254C2×D411yescharacteristic
153Dic254C2×D411yescharacteristic
154D508—21no—
155D508—21no—
156D508—41no—
157D508—41no—
158D508C2×D411yescharacteristic
159D508C2×D411yescharacteristic
160D508C2×D411yescharacteristic
161C22⋊D127—21no—
162C22⋊D127—21no—
163C22⋊D127—21no—
164C22⋊D127—41no—
165C22⋊D127—41no—
166C22⋊D127—41no—
167C22⋊D127—41no—
168C22⋊D127—82no—
169C2×Dic127—21no—
170C2×Dic127—21no—
171C2×Dic127—41no—
172C2×Dic127—41no—
173C4×D127—21no—
174C4×D127—21no—
175C4×D127—21no—
176C4×D127—41no—
177C4×D127—41no—
178C4×D127—41no—
179C4×D127—41no—
180C127⋊C8—21no—
181C127⋊C8C2×D411yescharacteristic
182C127⋊C8C2×D411yescharacteristic
183C23×C127—21no—
184Q8×C127—21no—
185Q8×C127—21no—
186Q8×C127C2×D411yescharacteristic
187D4×C127—21no—
188D4×C127—21no—
189D4×C127—21no—
190D4×C127C2×D411yescharacteristic
191C2×C4×C127—21no—
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—81no—
202C22×C127H(5,𝔽2)11yescharacteristic
203C2×D127—21no—
204C2×D127—21no—
205C2×D127—21no—
206C2×D127—41no—
207C2×D127—81no—
208C2×D127—161no—
209C2×D127—161no—
210C4×C127—21no—
211C4×C127—41no—
212C4×C127C22×D411yescharacteristic
213C4×C127C22×D411yescharacteristic
214Dic127—22no—
215Dic127—21no—
216Dic127—21no—
217Dic127—21no—
218Dic127—41no—
219C2×C127—21no—
220C2×C127—41no—
221C2×C127—81no—
222C2×C127D4211yescharacteristic
223D127—41no—
224D127—41no—
225D127—41no—
226D127—81no—
227#3291—1271nomaximal, Sylow
228C127#329111yescharacteristic, complemented, Sylow
229#577—1271no—
230#577—1271no—
231#577—1271no—
232#577—1271no—
233#492—1271no—
234#471—1271no—
235#471—1271no—
236#471—1271no—
237#471D12711yescharacteristic, complemented
238#456—1271no—
239#456—1271no—
240#453—1271no—
241#453—1271no—
242#443—1271no—
243#443—1271no—
2442−1+4—1271no—
2452−1+4—1271no—
246H(5,𝔽2)—1271no—
247H(5,𝔽2)—1271no—
248#137—1271no—
249#137—1271no—
250#137—1271no—
251#137—2541no—
252#137—2541no—
253#137—2541no—
254#137—2541no—
255#137C2×D12711yescharacteristic, complemented
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)C2×D12711yescharacteristic
264#135—1271no—
265#135—1271no—
266#135—2541no—
267#135—2541no—
268#135—2541no—
269#135—2541no—
270C2×SD16—1271no—
271C2×SD16—1271no—
272C2×SD16—1271no—
273C2×SD16—2541no—
274C2×SD16—2541no—
275C2×SD16—2541no—
276C2×SD16—2541no—
277C2×SD16C2×D12711yescharacteristic, complemented
278#131—1271no—
279#131—1271no—
280#131—1271no—
281#131C2×D12711yescharacteristic, complemented
282C4⋊Q8—1271no—
283#127—1271no—
284#108—1271no—
285#108C2×D12711yescharacteristic, complemented
286#104—1271no—
287#104—1271no—
288#104—1271no—
289#104—1271no—
290#101—1271no—
291#101C2×D12711yescharacteristic, complemented
292#100—1271no—
293#100C2×D12711yescharacteristic, complemented
294C4×C8—1271no—
295#41—1271no—
296#41—1271no—
297#41—1271no—
298#41—5081no—
299#41—5081no—
300#41—2541no—
301#41—2541no—
302#41—2541no—
303#41—2541no—
304#41—2541no—
305#41—2541no—
306#41—2541no—
307#41—2541no—
308#41C22×D12711yescharacteristic
309C2×Q8—1271no—
310C2×Q8—2541no—
311C2×Q8—2541no—
312C2×Q8—2541no—
313C2×Q8—2541no—
314C2×Q8C22×D12711yescharacteristic
315C2×D4—1271no—
316C2×D4—5081no—
317C2×D4—5081no—
318C2×D4—5081no—
319C2×D4—2541no—
320C2×D4—2541no—
321C2×D4—2541no—
322C2×D4—2541no—
323C2×D4C22×D12711yescharacteristic
324Q16—21no—
325Q16—1271no—
326Q16—1271no—
327Q16—2541no—
328Q16—2541no—
329Q16—2541no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—1271no—
335SD16—1271no—
336SD16—1271no—
337SD16—1271no—
338SD16—2541no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342SD16—2541no—
343SD16—2541no—
344SD16—2541no—
345SD16—2541no—
346SD16—2541no—
347SD16—2541no—
348SD16—2541no—
349SD16—2541no—
350D8—21no—
351D8—1271no—
352D8—1271no—
353D8—2541no—
354D8—2541no—
355D8—2541no—
356M16—21no—
357M16—1271no—
358M16—2541no—
359M16—2541no—
360M16—2541no—
361M16C22×D12711yescharacteristic
362M16C22×D12711yescharacteristic
363M16C22×D12711yescharacteristic
364C2×C8—21no—
365C2×C8—1271no—
366C2×C8—2541no—
367C2×C8—2541no—
368C2×C8—2541no—
369C2×C8C22×D12711yescharacteristic
370C4⋊C4—5081no—
371C42—1271no—
372C23—21no—
373C23—5081no—
374C23—5081no—
375C23—2541no—
376Q8—21no—
377Q8—21no—
378Q8—1271no—
379Q8—1271no—
380Q8—1271no—
381Q8—5081no—
382Q8—5081no—
383Q8—2541no—
384Q8—2541no—
385Q8D4×D12711yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—1271no—
390D4—1271no—
391D4—1271no—
392D4—10162no—
393D4—5081no—
394D4—5081no—
395D4—5081no—
396D4—5081no—
397D4—5081no—
398D4—5081no—
399D4—2541no—
400D4—2541no—
401D4—2541no—
402D4—2541no—
403D4—2541no—
404D4D4×D12711yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—5081no—
408C2×C4—5081no—
409C2×C4—5081no—
410C2×C4—5081no—
411C2×C4—5081no—
412C2×C4—5081no—
413C2×C4—2541no—
414C2×C4—2541no—
415C2×C4—2541no—
416C2×C4—2541no—
417C2×C4—2541no—
418C2×C4C23×D12711yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—1271no—
423C8—1271no—
424C8—2541no—
425C8D4×D12711yescharacteristic
426C8D4×D12711yescharacteristic
427C22—21no—
428C22—41no—
429C22—81no—
430C22—20321no—
431C22—20321no—
432C22—10161no—
433C22—5081no—
434C22—2541no—
435C22—2541no—
436C22—2541no—
437C22#1686125511yescharacteristic
438C4—21no—
439C4—41no—
440C4—5081no—
441C4—2542no—
442C4—2541no—
443C4—2541no—
444C4—2541no—
445C4C2×D4×D12711yescharacteristic
446C4C2×D4×D12711yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—10161no—
451C2—5081no—
452C2—5081no—
453C2—5081no—
454C2#2255051011yescharacteristic
subgroupG/H
centerC2#22550510
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#471×C127C2
socleC2×C127D42
hypercenter#471D127
hypocenterC127#3291
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #22550688C2semidirect
2. #22550686C2semidirect
3. #22550685C2semidirect
4. #22550682C2semidirect
5. #22550247C2semidirect
6. #22550142C2semidirect
7. #22550140C2semidirect
8. #22550138C2semidirect
9. #22550061C2semidirect
10. #22550057C2semidirect
11. #22550046C2semidirect
12. #22550042C2semidirect
13. #22549992C2semidirect
14. #22549991C2semidirect
15. #471×C127C2semidirect
18. #16861255C22semidirect
19. #16861254C22semidirect
21. #16861242C22semidirect
22. #16861242C22semidirect
24. #16861240C22semidirect
41. C2×#14523237C22semidirect
42. C2×#14523236C22semidirect
43. C2×#14523235C22semidirect
47. #16861217C22semidirect
48. #16861217C22semidirect
49. #16861215C22semidirect
50. #16861207C22semidirect
52. #16861144C22semidirect
53. #16861136C22semidirect
54. #16861135C22semidirect
55. #16861135C22semidirect
56. #16861135C22semidirect
57. #16861127C22semidirect
58. #16861125C22semidirect
59. C8×Dic127C22semidirect
60. #137×C127C22semidirect
62. C2×SD16×C127C22semidirect
63. #131×C127C22semidirect
64. #108×C127C22semidirect
65. #101×C127C22semidirect
66. #100×C127C22semidirect
95. Dic508D4semidirect
97. #14523238D4semidirect
104. #14523237D4semidirect
105. #14523237D4semidirect
109. #14523236D4semidirect
113. #14523235D4semidirect
116. D1016D4semidirect
118. #14523233D4semidirect
228. C127#3291semidirect
237. #471D127semidirect
255. #137C2×D127semidirect
277. C2×SD16C2×D127semidirect
281. #131C2×D127semidirect
285. #108C2×D127semidirect
291. #101C2×D127semidirect
293. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506882
#225506862
#225506852
#225506822
#225502472
#225501422
#225501402
#225501382
#225500612
#225500572
#225500462
#225500422
#225499922
#225499912
#471×C1272
#3291127