Finite Group Databasebeta

← #30650509

Subgroups — #30650509

Order 16256.

454 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550691C211yesmaximal, characteristic, complemented
2#22550686C211yesmaximal, characteristic, complemented
3#22550684C211yesmaximal, characteristic, complemented
4#22550680C211yesmaximal, characteristic, complemented
5#22550258C211yesmaximal, characteristic, complemented
6#22550150C211yesmaximal, characteristic, complemented
7#22550144C211yesmaximal, characteristic, complemented
8#22550142C211yesmaximal, characteristic, complemented
9#22550137C211yesmaximal, characteristic, complemented
10#22550055C211yesmaximal, characteristic, complemented
11#22550053C211yesmaximal, characteristic, complemented
12#22550050C211yesmaximal, characteristic, complemented
13#22549990C211yesmaximal, characteristic, complemented
14#22549990C211yesmaximal, characteristic, complemented
15#454×C127C211yesmaximal, characteristic, complemented
16#16861256C2211yescharacteristic, complemented
17#16861255C2211yescharacteristic
18#16861254C2211yescharacteristic
19#16861242C2211yescharacteristic
20#16861240—21no—
21#16861240—21no—
22#16861240C2211yescharacteristic
23#16861240C2211yescharacteristic, complemented
24#16861239—21no—
25#16861239C2211yescharacteristic, complemented
26#16861238—21no—
27#16861237C2211yescharacteristic
28#16861236—21no—
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×#14523238—21no—
40C2×#14523238C2211yescharacteristic, complemented
41C2×#14523236C2211yescharacteristic, complemented
42C2×#14523235—21no—
43C2×#14523235C2211yescharacteristic, complemented
44SD16×D127—21no—
45C2×#14523233C2211yescharacteristic
46D8×D127—21no—
47#16861217C2211yescharacteristic
48#16861217C2211yescharacteristic, complemented
49#16861216C2211yescharacteristic, complemented
50#16861216C2211yescharacteristic, complemented
51#16861197C2211yescharacteristic, complemented
52#16861144C2211yescharacteristic, complemented
53#16861144C2211yescharacteristic, complemented
54#16861137C2211yescharacteristic
55#16861137C2211yescharacteristic, complemented
56#16861127C2211yescharacteristic, complemented
57#16861126C2211yescharacteristic, complemented
58#16861125C2211yescharacteristic, complemented
59C4×C127⋊C8C2211yescharacteristic, complemented
602−1+4×C127C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C127C2211yescharacteristic
62AGL(1,ℤ/8)×C127C2211yescharacteristic, complemented
63#124×C127C2211yescharacteristic, complemented
64#104×C127C2211yescharacteristic, complemented
65#104×C127C2211yescharacteristic, complemented
66#101×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#14523238—21no—
89#14523238—21no—
90#14523238—21no—
91#14523238—21no—
92#14523238D411yescharacteristic
93#14523238D411yescharacteristic
94#14523237—21no—
95#14523237—21no—
96#14523236—21no—
97#14523236—21no—
98#14523236—21no—
99#14523236D411yescharacteristic
100#14523236D411yescharacteristic
101#14523235—21no—
102#14523235—21no—
103#14523235—21no—
104#14523235—21no—
105#14523235—21no—
106#14523235D411yescharacteristic
107#14523235D411yescharacteristic
108D1016—21no—
109#14523233—21no—
110#14523233—21no—
111#14523233—21no—
112#14523233D411yescharacteristic
113#14523233D411yescharacteristic
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×C127—41no—
130#41×C127C2311yescharacteristic
131#41×C127C2311yescharacteristic
132C2×Q8×C127—21no—
133C2×Q8×C127—21no—
134C2×Q8×C127C2311yescharacteristic
135C2×D4×C127C2311yescharacteristic
136SD16×C127—21no—
137SD16×C127—21no—
138D8×C127—21no—
139D8×C127—21no—
140M16×C127C2311yescharacteristic
141M16×C127C2311yescharacteristic
142C22⋊C4×C127—41no—
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—
177Q8×C127—21no—
178Q8×C127—21no—
179Q8×C127—41no—
180Q8×C127C2×D411yescharacteristic
181Q8×C127C2×D411yescharacteristic
182D4×C127—21no—
183D4×C127—21no—
184D4×C127—21no—
185D4×C127—21no—
186D4×C127—41no—
187D4×C127C2×D411yescharacteristic
188D4×C127C2×D411yescharacteristic
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127—21no—
193C2×C4×C127—41no—
194C2×C4×C127—41no—
195C2×C4×C127C2411yescharacteristic
196C8×C127—21no—
197C8×C127—21no—
198C22×C127—21no—
199C22×C127—21no—
200C22×C127—41no—
201C22×C127—81no—
202C22×C127H(5,𝔽2)11yescharacteristic
203C2×D127—21no—
204C2×D127—21no—
205C2×D127—41no—
206C2×D127—81no—
207C2×D127—161no—
208C2×D127—161no—
209C4×C127—21no—
210C4×C127—21no—
211C4×C127—41no—
212C4×C127—41no—
213C4×C127C22×D411yescharacteristic
214C4×C127C22×D411yescharacteristic
215Dic127—21no—
216Dic127—21no—
217Dic127—41no—
218C2×C127—21no—
219C2×C127—41no—
220C2×C127—41no—
221C2×C127—81no—
222C2×C127D4211yescharacteristic
223D127—41no—
224D127—41no—
225D127—81no—
226#3290—1271nomaximal, Sylow
227C127#329011yescharacteristic, complemented, Sylow
228#578—1271no—
229#577—1271no—
230#577—1271no—
231#576—1271no—
232#495—1271no—
233#473—1271no—
234#472—1271no—
235#471—1271no—
236#471—1271no—
237#455—1271no—
238#455—1271no—
239#454—1271no—
240#454D12711yescharacteristic, complemented
241#443—1271no—
242#443—1271no—
2432−1+4—1271no—
2442−1+4C2×D12711yescharacteristic, complemented
245H(5,𝔽2)—1271no—
246H(5,𝔽2)—1271no—
247#137—1271no—
248#137—1271no—
249#137—1271no—
250#137—1271no—
251#137—2541no—
252#137—2541no—
253#137—2541no—
254#137—2541no—
255AGL(1,ℤ/8)—1271no—
256AGL(1,ℤ/8)—1271no—
257AGL(1,ℤ/8)—2541no—
258AGL(1,ℤ/8)—2541no—
259AGL(1,ℤ/8)—2541no—
260AGL(1,ℤ/8)—2541no—
261AGL(1,ℤ/8)C2×D12711yescharacteristic
262AGL(1,ℤ/8)C2×D12711yescharacteristic, complemented
263#135—1271no—
264#135—1271no—
265#135—2541no—
266#135—2541no—
267#135—2541no—
268#135—2541no—
269C2×Q16—1271no—
270C2×Q16—2541no—
271C2×SD16—1271no—
272C2×SD16—1271no—
273C2×SD16—2541no—
274C2×SD16—2541no—
275C2×D8—1271no—
276C2×D8—2541no—
277#131—1271no—
278#131—1271no—
279#131—1271no—
280#131—1271no—
281#124—1271no—
282#124C2×D12711yescharacteristic, complemented
283#108—1271no—
284#108—1271no—
285#104—1271no—
286#104—1271no—
287#104C2×D12711yescharacteristic, complemented
288#104C2×D12711yescharacteristic, complemented
289#101—1271no—
290#101C2×D12711yescharacteristic, complemented
291#100—1271no—
292#100—1271no—
293C4×C8—1271no—
294#41—21no—
295#41—21no—
296#41—41no—
297#41—1271no—
298#41—1271no—
299#41—5081no—
300#41—2541no—
301#41—2541no—
302#41—2541no—
303#41—2541no—
304#41—2541no—
305#41—2541no—
306#41C22×D12711yescharacteristic
307#41C22×D12711yescharacteristic
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—1271no—
311C2×Q8—2541no—
312C2×Q8—2541no—
313C2×Q8C22×D12711yescharacteristic
314C2×D4—1271no—
315C2×D4—5081no—
316C2×D4—5081no—
317C2×D4—2541no—
318C2×D4—2541no—
319C2×D4—2541no—
320C2×D4—2541no—
321C2×D4C22×D12711yescharacteristic
322Q16—1271no—
323Q16—1271no—
324Q16—2541no—
325Q16—2541no—
326Q16—2541no—
327Q16—2541no—
328Q16—2541no—
329Q16—2541no—
330SD16—21no—
331SD16—21no—
332SD16—1271no—
333SD16—1271no—
334SD16—1271no—
335SD16—1271no—
336SD16—2541no—
337SD16—2541no—
338SD16—2541no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342SD16—2541no—
343SD16—2541no—
344SD16—2541no—
345SD16—2541no—
346D8—21no—
347D8—21no—
348D8—1271no—
349D8—1271no—
350D8—2541no—
351D8—2541no—
352D8—2541no—
353D8—2541no—
354M16—1271no—
355M16—1271no—
356M16—2541no—
357M16—2541no—
358M16—2541no—
359M16—2541no—
360M16C22×D12711yescharacteristic
361M16C22×D12711yescharacteristic
362C2×C8—1271no—
363C2×C8—1271no—
364C2×C8—2541no—
365C2×C8—2541no—
366C2×C8—2541no—
367C2×C8—2541no—
368C22⋊C4—41no—
369C22⋊C4—5081no—
370C42C22×D12711yescharacteristic
371C23—21no—
372C23—5081no—
373C23—5081no—
374C23—2541no—
375Q8—21no—
376Q8—21no—
377Q8—41no—
378Q8—1271no—
379Q8—1271no—
380Q8—5081no—
381Q8—2541no—
382Q8—2541no—
383Q8D4×D12711yescharacteristic
384Q8D4×D12711yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—21no—
389D4—41no—
390D4—1271no—
391D4—1271no—
392D4—5081no—
393D4—5081no—
394D4—5081no—
395D4—5081no—
396D4—5081no—
397D4—2541no—
398D4—2541no—
399D4—2541no—
400D4—2541no—
401D4D4×D12711yescharacteristic
402D4D4×D12711yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—5081no—
410C2×C4—5081no—
411C2×C4—5081no—
412C2×C4—5081no—
413C2×C4—2541no—
414C2×C4—2541no—
415C2×C4—2541no—
416C2×C4C23×D12711yescharacteristic
417C8—21no—
418C8—21no—
419C8—1271no—
420C8—1271no—
421C8—1271no—
422C8—1271no—
423C8—2541no—
424C8—2541no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—81no—
429C22—20321no—
430C22—20321no—
431C22—10161no—
432C22—5081no—
433C22—2541no—
434C22—2541no—
435C22#1686125411yescharacteristic
436C4—21no—
437C4—21no—
438C4—41no—
439C4—41no—
440C4—5081no—
441C4—2541no—
442C4—2541no—
443C4C2×D4×D12711yescharacteristic
444C4C2×D4×D12711yescharacteristic
445C2—21no—
446C2—41no—
447C2—41no—
448C2—81no—
449C2—10161no—
450C2—5081no—
451C2—5081no—
452C2#2255050811yescharacteristic
subgroupG/H
centerC2#22550508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#454×C127C2
socleC2×C127D42
hypercenter#454D127
hypocenterC127#3290
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #22550691C2semidirect
2. #22550686C2semidirect
3. #22550684C2semidirect
4. #22550680C2semidirect
5. #22550258C2semidirect
6. #22550150C2semidirect
7. #22550144C2semidirect
8. #22550142C2semidirect
9. #22550137C2semidirect
10. #22550055C2semidirect
11. #22550053C2semidirect
12. #22550050C2semidirect
13. #22549990C2semidirect
14. #22549990C2semidirect
15. #454×C127C2semidirect
16. #16861256C22semidirect
23. #16861240C22semidirect
25. #16861239C22semidirect
30. #16861234C22semidirect
40. C2×#14523238C22semidirect
41. C2×#14523236C22semidirect
43. C2×#14523235C22semidirect
48. #16861217C22semidirect
49. #16861216C22semidirect
50. #16861216C22semidirect
51. #16861197C22semidirect
52. #16861144C22semidirect
53. #16861144C22semidirect
55. #16861137C22semidirect
56. #16861127C22semidirect
57. #16861126C22semidirect
58. #16861125C22semidirect
59. C4×C127⋊C8C22semidirect
60. 2−1+4×C127C22semidirect
62. AGL(1,ℤ/8)×C127C22semidirect
63. #124×C127C22semidirect
64. #104×C127C22semidirect
65. #104×C127C22semidirect
66. #101×C127C22semidirect
227. C127#3290semidirect
240. #454D127semidirect
244. 2−1+4C2×D127semidirect
262. AGL(1,ℤ/8)C2×D127semidirect
282. #124C2×D127semidirect
287. #104C2×D127semidirect
288. #104C2×D127semidirect
290. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506912
#225506862
#225506842
#225506802
#225502582
#225501502
#225501442
#225501422
#225501372
#225500552
#225500532
#225500502
#225499902
#454×C1272
#3290127