Finite Group Databasebeta

← #30650514

Subgroups — #30650514

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#22550692C211yesmaximal, characteristic, complemented
2#22550687C211yesmaximal, characteristic, complemented
3#22550685C211yesmaximal, characteristic, complemented
4#22550679C211yesmaximal, characteristic, complemented
5#22550258C211yesmaximal, characteristic, complemented
6#22550149C211yesmaximal, characteristic, complemented
7#22550145C211yesmaximal, characteristic, complemented
8#22550141C211yesmaximal, characteristic, complemented
9#22550138C211yesmaximal, characteristic, complemented
10#22550055C211yesmaximal, characteristic, complemented
11#22550050C211yesmaximal, characteristic, complemented
12#22550048C211yesmaximal, characteristic, complemented
13#22549990C211yesmaximal, characteristic, complemented
14#22549990C211yesmaximal, characteristic, complemented
15#455×C127C211yesmaximal, characteristic, complemented
16#16861257C2211yescharacteristic, complemented
17#16861256C2211yescharacteristic, complemented
18#16861255C2211yescharacteristic
19#16861242C2211yescharacteristic, complemented
20#16861241—21no—
21#16861241—21no—
22#16861239—21no—
23#16861239C2211yescharacteristic
24#16861238—21no—
25#16861237C2211yescharacteristic, complemented
26#16861236—21no—
27#16861235—21no—
28#16861235—21no—
29#16861235C2211yescharacteristic
30#16861235C2211yescharacteristic, complemented
31#16861234—21no—
32#16861234C2211yescharacteristic
33#16861232—21no—
34#16861230—21no—
35#16861229—21no—
36#16861229—21no—
37#16861229C2211yescharacteristic
38#16861229C2211yescharacteristic
39C2×#14523238C2211yescharacteristic, complemented
40Q16×D127—21no—
41C2×#14523236—21no—
42C2×#14523236C2211yescharacteristic, complemented
43C2×#14523235C2211yescharacteristic, complemented
44SD16×D127—21no—
45C2×#14523233—21no—
46C2×#14523233C2211yescharacteristic
47#16861217C2211yescharacteristic, complemented
48#16861217C2211yescharacteristic, complemented
49#16861216C2211yescharacteristic
50#16861216C2211yescharacteristic, complemented
51#16861197C2211yescharacteristic, complemented
52#16861144C2211yescharacteristic, complemented
53#16861144C2211yescharacteristic, complemented
54#16861137C2211yescharacteristic, complemented
55#16861137C2211yescharacteristic, complemented
56#16861128C2211yescharacteristic, complemented
57#16861127C2211yescharacteristic, complemented
58#16861125C2211yescharacteristic, complemented
59C4×C127⋊C8C2211yescharacteristic, complemented
60H(5,𝔽2)×C127C2211yescharacteristic
61#137×C127C2211yescharacteristic
62#137×C127C2211yescharacteristic, complemented
63#124×C127C2211yescharacteristic, complemented
64#104×C127C2211yescharacteristic
65#104×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#14523247—21no—
76#14523247—41no—
77C2×Dic254—21no—
78C2×Dic254C2311yescharacteristic
79Q8×D127—21no—
80Q8×D127—21no—
81Q8×D127—21no—
82C2×D508—21no—
83C2×D508C2311yescharacteristic
84D4×D127—21no—
85D4×D127—41no—
86Dic508—21no—
87#14523238—21no—
88#14523238—21no—
89#14523238—21no—
90#14523238D411yescharacteristic
91#14523238D411yescharacteristic
92#14523237—21no—
93#14523237—21no—
94#14523236—21no—
95#14523236—21no—
96#14523236—21no—
97#14523236—21no—
98#14523236—21no—
99#14523236D411yescharacteristic
100#14523236D411yescharacteristic
101#14523235—21no—
102#14523235—21no—
103#14523235—21no—
104#14523235D411yescharacteristic
105#14523235D411yescharacteristic
106D1016—21no—
107#14523233—21no—
108#14523233—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×C127C2311yescharacteristic
130#41×C127C2311yescharacteristic
131C2×Q8×C127C2311yescharacteristic
132C2×D4×C127—21no—
133C2×D4×C127—21no—
134C2×D4×C127—41no—
135C2×D4×C127C2311yescharacteristic
136Q16×C127—21no—
137Q16×C127—21no—
138SD16×C127—21no—
139SD16×C127—21no—
140M16×C127C2311yescharacteristic
141M16×C127C2311yescharacteristic
142C22⋊C4×C127—41no—
143C42×C127C2311yescharacteristic
144C22×D127—21no—
145C22×D127—41no—
146Dic254—21no—
147Dic254—21no—
148Dic254—41no—
149Dic254—41no—
150Dic254C2×D411yescharacteristic
151Dic254C2×D411yescharacteristic
152D508—21no—
153D508—21no—
154D508—41no—
155D508—41no—
156D508C2×D411yescharacteristic
157D508C2×D411yescharacteristic
158C22⋊D127—21no—
159C22⋊D127—21no—
160C22⋊D127—41no—
161C22⋊D127—41no—
162C2×Dic127—21no—
163C2×Dic127—41no—
164C4×D127—21no—
165C4×D127—21no—
166C4×D127—41no—
167C4×D127—41no—
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—41no—
178Q8×C127—21no—
179Q8×C127—21no—
180Q8×C127C2×D411yescharacteristic
181Q8×C127C2×D411yescharacteristic
182D4×C127—21no—
183D4×C127—21no—
184D4×C127—21no—
185D4×C127—21no—
186D4×C127—41no—
187D4×C127—41no—
188D4×C127C2×D411yescharacteristic
189D4×C127C2×D411yescharacteristic
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127—21no—
193C2×C4×C127—21no—
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×C127—161no—
203C22×C127H(5,𝔽2)11yescharacteristic
204C2×D127—21no—
205C2×D127—21no—
206C2×D127—41no—
207C2×D127—81no—
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#455D12711yescharacteristic, complemented
239#454—1271no—
240#454—1271no—
241#443—1271no—
242#443—1271no—
2432−1+4—1271no—
2442−1+4—1271no—
245H(5,𝔽2)—1271no—
246H(5,𝔽2)C2×D12711yescharacteristic
247#137—1271no—
248#137—1271no—
249#137—2541no—
250#137—2541no—
251#137—2541no—
252#137—2541no—
253#137C2×D12711yescharacteristic
254#137C2×D12711yescharacteristic, complemented
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—
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
288#104C2×D12711yescharacteristic, complemented
289#101—1271no—
290#101—1271no—
291#100—1271no—
292#100C2×D12711yescharacteristic, complemented
293C4×C8—1271no—
294#41—21no—
295#41—21no—
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#41C22×D12711yescharacteristic
307#41C22×D12711yescharacteristic
308C2×Q8—1271no—
309C2×Q8—2541no—
310C2×Q8—2541no—
311C2×Q8—2541no—
312C2×Q8—2541no—
313C2×Q8C22×D12711yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—1271no—
318C2×D4—5081no—
319C2×D4—2541no—
320C2×D4—2541no—
321C2×D4C22×D12711yescharacteristic
322Q16—21no—
323Q16—21no—
324Q16—1271no—
325Q16—1271no—
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—1271no—
347D8—1271no—
348D8—2541no—
349D8—2541no—
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—41no—
373C23—5081no—
374C23—2541no—
375Q8—21no—
376Q8—21no—
377Q8—1271no—
378Q8—1271no—
379Q8—5081no—
380Q8—5081no—
381Q8—2541no—
382Q8—2541no—
383Q8D4×D12711yescharacteristic
384Q8D4×D12711yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—21no—
389D4—41no—
390D4—41no—
391D4—1271no—
392D4—1271no—
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—5081no—
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—161no—
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#455×C127C2
socleC2×C127D42
hypercenter#455D127
hypocenterC127#3290
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #22550692C2semidirect
2. #22550687C2semidirect
3. #22550685C2semidirect
4. #22550679C2semidirect
5. #22550258C2semidirect
6. #22550149C2semidirect
7. #22550145C2semidirect
8. #22550141C2semidirect
9. #22550138C2semidirect
10. #22550055C2semidirect
11. #22550050C2semidirect
12. #22550048C2semidirect
13. #22549990C2semidirect
14. #22549990C2semidirect
15. #455×C127C2semidirect
16. #16861257C22semidirect
17. #16861256C22semidirect
19. #16861242C22semidirect
25. #16861237C22semidirect
30. #16861235C22semidirect
39. C2×#14523238C22semidirect
42. C2×#14523236C22semidirect
43. C2×#14523235C22semidirect
47. #16861217C22semidirect
48. #16861217C22semidirect
50. #16861216C22semidirect
51. #16861197C22semidirect
52. #16861144C22semidirect
53. #16861144C22semidirect
54. #16861137C22semidirect
55. #16861137C22semidirect
56. #16861128C22semidirect
57. #16861127C22semidirect
58. #16861125C22semidirect
59. C4×C127⋊C8C22semidirect
62. #137×C127C22semidirect
63. #124×C127C22semidirect
65. #104×C127C22semidirect
66. #100×C127C22semidirect
227. C127#3290semidirect
238. #455D127semidirect
254. #137C2×D127semidirect
282. #124C2×D127semidirect
288. #104C2×D127semidirect
292. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506922
#225506872
#225506852
#225506792
#225502582
#225501492
#225501452
#225501412
#225501382
#225500552
#225500502
#225500482
#225499902
#455×C1272
#3290127