Finite Group Databasebeta

← #30648453

Subgroups — #30648453

Order 16256.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550675C211yesmaximal, characteristic, complemented
2C2×#16861236C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D127C211yesmaximal, characteristic, complemented
4#22550641C211yesmaximal, characteristic, complemented
5#22550147C211yesmaximal, characteristic, complemented
6#22550146C211yesmaximal, characteristic, complemented
7#22550145C211yesmaximal, characteristic, complemented
8#22550144C211yesmaximal, characteristic, complemented
9#22550142C211yesmaximal, characteristic, complemented
10#22550141C211yesmaximal, characteristic, complemented
11#22550136C211yesmaximal, characteristic, complemented
12#22549923C211yesmaximal, characteristic, complemented
13#22549859C211yesmaximal, characteristic, complemented
14#100×D127C211yesmaximal, characteristic, complemented
15#471×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic
17#41×D127C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19#16861241—21no—
20#16861240C2211yescharacteristic, complemented
21#16861237C2211yescharacteristic
22#16861237C2211yescharacteristic, complemented
23#16861236—21no—
24#16861236—21no—
25#16861236—21no—
26#16861236—21no—
27#16861236—21no—
28#16861236—21no—
29#16861235C2211yescharacteristic
30#16861234C2211yescharacteristic
31#16861234C2211yescharacteristic, complemented
32#16861233—21no—
33#16861232—21no—
34#16861230—21no—
35C2×#14523235C2211yescharacteristic, complemented
36SD16×D127—21no—
37C2×D1016C2211yescharacteristic, complemented
38C2×#14523233C2211yescharacteristic, complemented
39D8×D127—21no—
40#16861217—21no—
41#16861217C2211yescharacteristic, complemented
42#16861216C2211yescharacteristic, complemented
43#16861215—21no—
44#16861215C2211yescharacteristic, complemented
45C2×#14523231—21no—
46C2×#14523231C2211yescharacteristic
47M16×D127—21no—
48M16×D127C2211yescharacteristic
49M16×D127C2211yescharacteristic
50M16×D127C2211yescharacteristic
51#16861145C2211yescharacteristic, complemented
52#16861144C2211yescharacteristic, complemented
53#16861144C2211yescharacteristic, complemented
54#16861128C2211yescharacteristic, complemented
55#16861126C2211yescharacteristic, complemented
56#16861125C2211yescharacteristic, complemented
57#16861125C2211yescharacteristic, complemented
58#16861125C2211yescharacteristic, complemented
59#16861125C2211yescharacteristic, 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
67C23×D127—21no—
68#14523249—21no—
69#14523249—21no—
70#14523249—21no—
71#14523249—21no—
72#14523249D411yescharacteristic
73#14523248—21no—
74#14523248C2311yescharacteristic
75#14523247—21no—
76#14523247D411yescharacteristic
77Q8×D127D411yescharacteristic
78C2×D508—21no—
79C2×D508C2311yescharacteristic
80C2×D508C2311yescharacteristic
81C2×C22⋊D127—21no—
82D4×D127—21no—
83D4×D127—21no—
84D4×D127—21no—
85D4×D127—21no—
86D4×D127—21no—
87D4×D127D411yescharacteristic
88C2×C4×D127—21no—
89C2×C4×D127—21no—
90C2×C4×D127C2311yescharacteristic
91#14523238—21no—
92#14523237—21no—
93#14523237—21no—
94#14523236—21no—
95#14523235—21no—
96#14523235—21no—
97#14523235—21no—
98#14523235—21no—
99#14523235—21no—
100D1016—21no—
101D1016—21no—
102D1016—21no—
103D1016—21no—
104#14523233—21no—
105#14523233—21no—
106#14523233—21no—
107#14523233—21no—
108#14523233—21no—
109#14523232—21no—
110#14523232C2311yescharacteristic
111#14523232C2311yescharacteristic
112#14523232C2311yescharacteristic
113#14523231—21no—
114#14523231—21no—
115#14523231—21no—
116#14523231—21no—
117#14523231—21no—
118#14523231D411yescharacteristic
119#14523231D411yescharacteristic
120#14523231D411yescharacteristic
121#14523231D411yescharacteristic
122C2×C127⋊C8—21no—
123C2×C127⋊C8C2311yescharacteristic
124C8×D127—21no—
125C8×D127—21no—
126C8×D127—21no—
127C8×D127—21no—
128C8×D127—21no—
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—21no—
146C22×D127—21no—
147C22×D127—41no—
148C22×D127—41no—
149C22×D127—82no—
150C22×D127#4111yescharacteristic
151Dic254—21no—
152Dic254C2×D411yescharacteristic
153D508—21no—
154D508—21no—
155D508—21no—
156D508—21no—
157D508—21no—
158D508—41no—
159D508—41no—
160D508C2×D411yescharacteristic
161C22⋊D127—21no—
162C22⋊D127—21no—
163C22⋊D127—21no—
164C22⋊D127—41no—
165C22⋊D127—41no—
166C2×Dic127—21no—
167C2×Dic127#4111yescharacteristic
168C4×D127—21no—
169C4×D127—21no—
170C4×D127—21no—
171C4×D127—41no—
172C4×D127—41no—
173C4×D127C2×D411yescharacteristic
174C4×D127C2×D411yescharacteristic
175C4×D127C2×D411yescharacteristic
176C4×D127C2×D411yescharacteristic
177C127⋊C8—21no—
178C127⋊C8—21no—
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×C127C2×#4111yescharacteristic
202C2×D127—21no—
203C2×D127—41no—
204C2×D127—41no—
205C2×D127—41no—
206C2×D127—81no—
207C2×D127—82no—
208C2×D127—82no—
209C2×D127—81no—
210C2×D127—81no—
211C2×D127#12111yescharacteristic
212C2×D127#12111yescharacteristic
213C4×C127—21no—
214C4×C127—41no—
215C4×C127C22×D411yescharacteristic
216C4×C127C22×D411yescharacteristic
217Dic127—21no—
218Dic127#12111yescharacteristic
219Dic127#12111yescharacteristic
220C2×C127—21no—
221C2×C127—41no—
222C2×C127—81no—
223C2×C127C2×#12111yescharacteristic
224D127—21no—
225D127—21no—
226D127—41no—
227D127—81no—
228D127—81no—
229#3068—1271nomaximal, Sylow
230C127#306811yescharacteristic, complemented, Sylow
231#575—1271no—
232C2×AGL(1,ℤ/8)—1271no—
233C2×AGL(1,ℤ/8)—1271no—
234#568—1271no—
235#472—1271no—
236#472—1271no—
237#472—1271no—
238#472—1271no—
239#471—1271no—
240#471—1271no—
241#471—1271no—
242#471D12711yescharacteristic, complemented
243#430—1271no—
244#413—1271no—
245C2×#100—1271no—
246C2×#41—1271no—
247C2×#41—1271no—
248C22×D4—1271no—
249#137—1271no—
250#137—2541no—
251#137C2×D12711yescharacteristic, complemented
252AGL(1,ℤ/8)—1271no—
253AGL(1,ℤ/8)—1271no—
254AGL(1,ℤ/8)—1271no—
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)—2541no—
262AGL(1,ℤ/8)—2541no—
263AGL(1,ℤ/8)—2541no—
264AGL(1,ℤ/8)C2×D12711yescharacteristic
265#135—2541no—
266#135—2541no—
267C2×SD16—1271no—
268C2×SD16—2541no—
269C2×SD16C2×D12711yescharacteristic, complemented
270C2×D8—1271no—
271C2×D8—1271no—
272C2×D8—2541no—
273#131—1271no—
274#131—1271no—
275#131—1271no—
276#131—2541no—
277#131—2541no—
278#131C2×D12711yescharacteristic, complemented
279C2×M16—1271no—
280C2×M16—1271no—
281C2×M16—1271no—
282C2×M16—1271no—
283C2×M16—2541no—
284C2×M16—2541no—
285#108—1271no—
286#108—1271no—
287#108—1271no—
288#108C2×D12711yescharacteristic, complemented
289#101—1271no—
290#101C2×D12711yescharacteristic, complemented
291#100—1271no—
292#100—1271no—
293#100—1271no—
294#100—1271no—
295#100—1271no—
296#100C2×D12711yescharacteristic, complemented
297C24—2541no—
298#41—1271no—
299#41—1271no—
300#41—1271no—
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×Q8C22×D12711yescharacteristic
310C2×D4—1271no—
311C2×D4—1271no—
312C2×D4—1271no—
313C2×D4—2541no—
314C2×D4—2541no—
315C2×D4—2541no—
316C2×D4—2541no—
317C2×D4—2541no—
318C2×D4—2541no—
319C2×D4—2541no—
320C2×D4C22×D12711yescharacteristic
321C22×C4—1271no—
322C22×C4—2541no—
323C22×C4—2541no—
324Q16—21no—
325Q16—2541no—
326SD16—21no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—2541no—
331SD16—2541no—
332SD16—2541no—
333SD16—2541no—
334SD16—2541no—
335SD16—2541no—
336SD16—2541no—
337SD16—2541no—
338D8—21no—
339D8—2541no—
340D8—2541no—
341D8—2541no—
342D8—2541no—
343D8—2541no—
344D8—2541no—
345D8—2541no—
346D8—2541no—
347D8—2541no—
348M16—21no—
349M16—1271no—
350M16—1271no—
351M16—1271no—
352M16—1271no—
353M16—1271no—
354M16—1271no—
355M16—1271no—
356M16—2541no—
357M16—2541no—
358M16—2541no—
359M16—2541no—
360M16—2541no—
361M16—2541no—
362M16C22×D12711yescharacteristic
363M16C22×D12711yescharacteristic
364M16C22×D12711yescharacteristic
365C2×C8—21no—
366C2×C8—1271no—
367C2×C8—2541no—
368C2×C8—2541no—
369C2×C8—2541no—
370C2×C8—2541no—
371C2×C8—2541no—
372C2×C8—2541no—
373C2×C8C22×D12711yescharacteristic
374C23—21no—
375C23—1271no—
376C23—10162no—
377C23—5081no—
378C23—5081no—
379C23—2541no—
380C23—2541no—
381C23—2541no—
382Q8—21no—
383Q8—21no—
384Q8—1271no—
385Q8—2541no—
386Q8D4×D12711yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—1271no—
391D4—5081no—
392D4—5081no—
393D4—5081no—
394D4—5081no—
395D4—2541no—
396D4—2541no—
397D4—2541no—
398D4—2541no—
399D4—2541no—
400D4—2541no—
401D4—2541no—
402D4—2541no—
403D4D4×D12711yescharacteristic
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—1271no—
407C2×C4—1271no—
408C2×C4—1271no—
409C2×C4—1271no—
410C2×C4—1271no—
411C2×C4—5081no—
412C2×C4—5081no—
413C2×C4—2541no—
414C2×C4—2541no—
415C2×C4—2541no—
416C2×C4—2541no—
417C2×C4C23×D12711yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—1271no—
422C8—1271no—
423C8—2541no—
424C8—2541no—
425C8—2541no—
426C8D4×D12711yescharacteristic
427C8D4×D12711yescharacteristic
428C22—21no—
429C22—41no—
430C22—81no—
431C22—1271no—
432C22—1271no—
433C22—10161no—
434C22—10161no—
435C22—10161no—
436C22—10162no—
437C22—10162no—
438C22—5081no—
439C22—5081no—
440C22—5081no—
441C22—2541no—
442C22#41×D12711yescharacteristic
443C4—21no—
444C4—41no—
445C4—1271no—
446C4—1271no—
447C4—2541no—
448C4C2×D4×D12711yescharacteristic
449C4C2×D4×D12711yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—10161no—
454C2—10161no—
455C2—5081no—
456C2—2541no—
457C2—2541no—
458C2#121×D12711yescharacteristic
subgroupG/H
centerC2#121×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#471×C127C2
socleC2×C127C2×#121
hypercenter#471D127
hypocenterC127#3068
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. #22550675C2semidirect
2. C2×#16861236C2semidirect
3. AGL(1,ℤ/8)×D127C2semidirect
4. #22550641C2semidirect
5. #22550147C2semidirect
6. #22550146C2semidirect
7. #22550145C2semidirect
8. #22550144C2semidirect
9. #22550142C2semidirect
10. #22550141C2semidirect
11. #22550136C2semidirect
12. #22549923C2semidirect
13. #22549859C2semidirect
14. #100×D127C2semidirect
15. #471×C127C2semidirect
20. #16861240C22semidirect
22. #16861237C22semidirect
31. #16861234C22semidirect
35. C2×#14523235C22semidirect
37. C2×D1016C22semidirect
38. C2×#14523233C22semidirect
41. #16861217C22semidirect
42. #16861216C22semidirect
44. #16861215C22semidirect
51. #16861145C22semidirect
52. #16861144C22semidirect
53. #16861144C22semidirect
54. #16861128C22semidirect
55. #16861126C22semidirect
56. #16861125C22semidirect
57. #16861125C22semidirect
58. #16861125C22semidirect
59. #16861125C22semidirect
60. #137×C127C22semidirect
62. C2×SD16×C127C22semidirect
63. #131×C127C22semidirect
64. #108×C127C22semidirect
65. #101×C127C22semidirect
66. #100×C127C22semidirect
230. C127#3068semidirect
242. #471D127semidirect
251. #137C2×D127semidirect
269. C2×SD16C2×D127semidirect
278. #131C2×D127semidirect
288. #108C2×D127semidirect
290. #101C2×D127semidirect
296. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506752
C2×#168612362
AGL(1,ℤ/8)×D1272
#225506412
#225501472
#225501462
#225501452
#225501442
#225501422
#225501412
#225501362
#225499232
#225498592
#100×D1272
#471×C1272
#3068127