Finite Group Databasebeta

← #30650513

Subgroups — #30650513

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#22550688C211yesmaximal, characteristic, complemented
3#22550682C211yesmaximal, characteristic, complemented
4#22550680C211yesmaximal, characteristic, complemented
5#22550257C211yesmaximal, characteristic, complemented
6#22550150C211yesmaximal, characteristic, complemented
7#22550145C211yesmaximal, characteristic, complemented
8#22550140C211yesmaximal, characteristic, complemented
9#22550056C211yesmaximal, characteristic, complemented
10#22550052C211yesmaximal, characteristic, complemented
11#22550051C211yesmaximal, characteristic, complemented
12#22550047C211yesmaximal, characteristic, complemented
13#22549992C211yesmaximal, characteristic, complemented
14#22549991C211yesmaximal, characteristic, complemented
15#471×C127C211yesmaximal, characteristic, complemented
16#16861257C2211yescharacteristic, complemented
17#16861256C2211yescharacteristic, complemented
18#16861255C2211yescharacteristic
19#16861254C2211yescharacteristic
20#16861242—21no—
21#16861242C2211yescharacteristic
22#16861242C2211yescharacteristic, complemented
23#16861241—21no—
24#16861240C2211yescharacteristic
25#16861238—21no—
26#16861238—21no—
27#16861237—21no—
28#16861237C2211yescharacteristic
29#16861237C2211yescharacteristic, complemented
30#16861236—21no—
31#16861236—21no—
32#16861235C2211yescharacteristic, complemented
33#16861233—21no—
34#16861231—21no—
35#16861231C2211yescharacteristic
36#16861230—21no—
37#16861230—21no—
38#16861230—21no—
39#16861229C2211yescharacteristic
40C2×#14523238C2211yescharacteristic, complemented
41Q16×D127—21no—
42C2×#14523237—21no—
43C2×#14523237C2211yescharacteristic, complemented
44SD16×D127—21no—
45C2×#14523233C2211yescharacteristic
46D8×D127—21no—
47#16861217C2211yescharacteristic
48#16861217C2211yescharacteristic, complemented
49#16861215C2211yescharacteristic, complemented
50#16861198C2211yescharacteristic, complemented
51#16861196C2211yescharacteristic, complemented
52#16861144C2211yescharacteristic, complemented
53#16861136C2211yescharacteristic, complemented
54#16861135C2211yescharacteristic
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—
88D4×D127—21no—
89D4×D127—21no—
90D4×D127—21no—
91D4×D127—41no—
92Dic508—21no—
93Dic508—21no—
94Dic508D411yescharacteristic
95#14523238—21no—
96#14523238—21no—
97#14523238—21no—
98#14523238D411yescharacteristic
99#14523237—21no—
100#14523237—21no—
101#14523237—21no—
102#14523237—21no—
103#14523237—21no—
104#14523237—21no—
105#14523237D411yescharacteristic
106#14523237D411yescharacteristic
107#14523236—21no—
108#14523236D411yescharacteristic
109#14523235—21no—
110#14523235D411yescharacteristic
111D1016—21no—
112D1016—21no—
113D1016D411yescharacteristic
114#14523233—21no—
115#14523233—21no—
116#14523233—21no—
117#14523233D411yescharacteristic
118#14523232C2311yescharacteristic
119#14523231—21no—
120#14523231—21no—
121#14523231—21no—
122C2×C127⋊C8C2311yescharacteristic
123C8×D127—21no—
124C8×D127—21no—
125C8×D127—21no—
126C22⋊Dic127—41no—
127(C2×D127)⋊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—
168C2×Dic127—21no—
169C2×Dic127—21no—
170C2×Dic127—41no—
171C2×Dic127—41no—
172C4×D127—21no—
173C4×D127—21no—
174C4×D127—21no—
175C4×D127—41no—
176C4×D127—41no—
177C4×D127—41no—
178C4×D127—41no—
179C127⋊C8—21no—
180C127⋊C8C2×D411yescharacteristic
181C127⋊C8C2×D411yescharacteristic
182C23×C127—21no—
183Q8×C127—21no—
184Q8×C127—21no—
185Q8×C127C2×D411yescharacteristic
186D4×C127—21no—
187D4×C127—21no—
188D4×C127—21no—
189D4×C127C2×D411yescharacteristic
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127C2411yescharacteristic
193C8×C127—21no—
194C8×C127—21no—
195C8×C127—21no—
196C8×C127C2×D411yescharacteristic
197C8×C127C2×D411yescharacteristic
198C22×C127—21no—
199C22×C127—41no—
200C22×C127—81no—
201C22×C127H(5,𝔽2)11yescharacteristic
202C2×D127—21no—
203C2×D127—21no—
204C2×D127—21no—
205C2×D127—41no—
206C2×D127—81no—
207C2×D127—161no—
208C2×D127—161no—
209C4×C127—21no—
210C4×C127—41no—
211C4×C127C22×D411yescharacteristic
212C4×C127C22×D411yescharacteristic
213Dic127—21no—
214Dic127—21no—
215Dic127—21no—
216Dic127—41no—
217Dic127—41no—
218C2×C127—21no—
219C2×C127—41no—
220C2×C127—81no—
221C2×C127D4211yescharacteristic
222D127—41no—
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#471D12711yescharacteristic, complemented
237#455—1271no—
238#455—1271no—
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)—1271no—
247#137—1271no—
248#137—1271no—
249#137—1271no—
250#137—2541no—
251#137—2541no—
252#137—2541no—
253#137—2541no—
254#137C2×D12711yescharacteristic, complemented
255AGL(1,ℤ/8)—1271no—
256AGL(1,ℤ/8)—1271no—
257AGL(1,ℤ/8)—1271no—
258AGL(1,ℤ/8)—2541no—
259AGL(1,ℤ/8)—2541no—
260AGL(1,ℤ/8)—2541no—
261AGL(1,ℤ/8)—2541no—
262AGL(1,ℤ/8)C2×D12711yescharacteristic
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—2541no—
273C2×SD16—2541no—
274C2×SD16C2×D12711yescharacteristic, complemented
275C2×D8—1271no—
276C2×D8—2541no—
277#131—1271no—
278#131—1271no—
279#131—1271no—
280#131C2×D12711yescharacteristic, complemented
281#124—1271no—
282#124—1271no—
283#108—1271no—
284#108C2×D12711yescharacteristic, complemented
285#104—1271no—
286#104—1271no—
287#104—1271no—
288#104—1271no—
289#101—1271no—
290#101C2×D12711yescharacteristic, complemented
291#100—1271no—
292#100C2×D12711yescharacteristic, complemented
293C4×C8—1271no—
294#41—1271no—
295#41—1271no—
296#41—1271no—
297#41—5081no—
298#41—5081no—
299#41—2541no—
300#41—2541no—
301#41—2541no—
302#41—2541no—
303#41—2541no—
304#41—2541no—
305#41—2541no—
306#41—2541no—
307#41C22×D12711yescharacteristic
308C2×Q8—1271no—
309C2×Q8—2541no—
310C2×Q8—2541no—
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—21no—
323Q16—1271no—
324Q16—1271no—
325Q16—2541no—
326Q16—2541no—
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—
346D8—21no—
347D8—1271no—
348D8—1271no—
349D8—2541no—
350D8—2541no—
351D8—2541no—
352D8—2541no—
353D8—2541no—
354M16—21no—
355M16—1271no—
356M16—2541no—
357M16—2541no—
358M16—2541no—
359M16C22×D12711yescharacteristic
360M16C22×D12711yescharacteristic
361M16C22×D12711yescharacteristic
362C2×C8—21no—
363C2×C8—1271no—
364C2×C8—2541no—
365C2×C8—2541no—
366C2×C8—2541no—
367C2×C8C22×D12711yescharacteristic
368C22⋊C4—5081no—
369C22⋊C4—5081no—
370C42—1271no—
371C23—21no—
372C23—5081no—
373C23—5081no—
374C23—2541no—
375Q8—21no—
376Q8—21no—
377Q8—1271no—
378Q8—1271no—
379Q8—1271no—
380Q8—5081no—
381Q8—5081no—
382Q8—2541no—
383Q8—2541no—
384Q8D4×D12711yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—1271no—
389D4—1271no—
390D4—1271no—
391D4—5081no—
392D4—5081no—
393D4—5081no—
394D4—5081no—
395D4—5081no—
396D4—5081no—
397D4—2541no—
398D4—2541no—
399D4—2541no—
400D4—2541no—
401D4—2541no—
402D4D4×D12711yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—5081no—
406C2×C4—5081no—
407C2×C4—5081no—
408C2×C4—5081no—
409C2×C4—5081no—
410C2×C4—5081no—
411C2×C4—2541no—
412C2×C4—2541no—
413C2×C4—2541no—
414C2×C4—2541no—
415C2×C4—2541no—
416C2×C4C23×D12711yescharacteristic
417C8—21no—
418C8—21no—
419C8—21no—
420C8—1271no—
421C8—1271no—
422C8—2541no—
423C8D4×D12711yescharacteristic
424C8D4×D12711yescharacteristic
425C22—21no—
426C22—41no—
427C22—81no—
428C22—20321no—
429C22—20321no—
430C22—10161no—
431C22—5081no—
432C22—2541no—
433C22—2541no—
434C22—2541no—
435C22#1686125511yescharacteristic
436C4—21no—
437C4—41no—
438C4—5081no—
439C4—5081no—
440C4—2541no—
441C4—2541no—
442C4—2541no—
443C4C2×D4×D12711yescharacteristic
444C4C2×D4×D12711yescharacteristic
445C2—21no—
446C2—41no—
447C2—81no—
448C2—10161no—
449C2—5081no—
450C2—5081no—
451C2—5081no—
452C2#2255051011yescharacteristic
subgroupG/H
centerC2#22550510
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#471×C127C2
socleC2×C127D42
hypercenter#471D127
hypocenterC127#3290
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. #22550692C2semidirect
2. #22550688C2semidirect
3. #22550682C2semidirect
4. #22550680C2semidirect
5. #22550257C2semidirect
6. #22550150C2semidirect
7. #22550145C2semidirect
8. #22550140C2semidirect
9. #22550056C2semidirect
10. #22550052C2semidirect
11. #22550051C2semidirect
12. #22550047C2semidirect
13. #22549992C2semidirect
14. #22549991C2semidirect
15. #471×C127C2semidirect
16. #16861257C22semidirect
17. #16861256C22semidirect
22. #16861242C22semidirect
29. #16861237C22semidirect
32. #16861235C22semidirect
40. C2×#14523238C22semidirect
43. C2×#14523237C22semidirect
48. #16861217C22semidirect
49. #16861215C22semidirect
50. #16861198C22semidirect
51. #16861196C22semidirect
52. #16861144C22semidirect
53. #16861136C22semidirect
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
227. C127#3290semidirect
236. #471D127semidirect
254. #137C2×D127semidirect
274. C2×SD16C2×D127semidirect
280. #131C2×D127semidirect
284. #108C2×D127semidirect
290. #101C2×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
#225506882
#225506822
#225506802
#225502572
#225501502
#225501452
#225501402
#225500562
#225500522
#225500512
#225500472
#225499922
#225499912
#471×C1272
#3290127