Finite Group Databasebeta

← #471×D127 #30648425

Subgroups — #471×D127 #30648425

Order 16256.

468 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#137×D127C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D127C211yesmaximal, characteristic
3C2×SD16×D127C211yesmaximal, characteristic, complemented
4#131×D127C211yesmaximal, characteristic, complemented
5#22550142C211yesmaximal, characteristic, complemented
6#22550141C211yesmaximal, characteristic, complemented
7#22550140C211yesmaximal, characteristic, complemented
8#22550139C211yesmaximal, characteristic, complemented
9#22550138C211yesmaximal, characteristic, complemented
10#22550137C211yesmaximal, characteristic, complemented
11#22550136C211yesmaximal, characteristic, complemented
12#108×D127C211yesmaximal, characteristic, complemented
13#101×D127C211yesmaximal, characteristic, complemented
14#100×D127C211yesmaximal, characteristic, complemented
15#471×C127C211yesmaximal, characteristic, complemented
16#41×D127C2211yescharacteristic
17C2×Q8×D127C2211yescharacteristic
18C2×D4×D127C2211yescharacteristic
19#16861242C2211yescharacteristic, complemented
20#16861241—21no—
21#16861240C2211yescharacteristic, complemented
22#16861239C2211yescharacteristic, complemented
23#16861238—21no—
24#16861237C2211yescharacteristic
25#16861236—21no—
26#16861235C2211yescharacteristic
27#16861234C2211yescharacteristic
28#16861233—21no—
29Q16×D127—21no—
30C2×#14523237C2211yescharacteristic, complemented
31C2×#14523236C2211yescharacteristic, complemented
32C2×#14523235C2211yescharacteristic, complemented
33SD16×D127—21no—
34SD16×D127—21no—
35SD16×D127—21no—
36SD16×D127—21no—
37SD16×D127—21no—
38SD16×D127—21no—
39D8×D127—21no—
40#16861217—21no—
41#16861217C2211yescharacteristic, complemented
42#16861216C2211yescharacteristic, complemented
43#16861215—21no—
44#16861215C2211yescharacteristic, complemented
45M16×D127—21no—
46M16×D127C2211yescharacteristic
47M16×D127C2211yescharacteristic
48M16×D127C2211yescharacteristic
49C2×C8×D127—21no—
50C2×C8×D127C2211yescharacteristic
51#16861145C2211yescharacteristic, complemented
52#16861144C2211yescharacteristic, complemented
53#16861144C2211yescharacteristic, complemented
54#16861128C2211yescharacteristic, complemented
55#16861127C2211yescharacteristic, complemented
56#16861127C2211yescharacteristic, complemented
57#16861126C2211yescharacteristic, 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#14523249D411yescharacteristic
69#14523248—21no—
70#14523248C2311yescharacteristic
71#14523247—21no—
72#14523247D411yescharacteristic
73C2×Dic254—21no—
74C2×Dic254C2311yescharacteristic
75Q8×D127—21no—
76Q8×D127—21no—
77Q8×D127—21no—
78Q8×D127—21no—
79Q8×D127D411yescharacteristic
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
91Dic508—21no—
92#14523238—21no—
93#14523238—21no—
94#14523237—21no—
95#14523237—21no—
96#14523237—21no—
97#14523237—21no—
98#14523236—21no—
99#14523236—21no—
100#14523236—21no—
101#14523236—21no—
102#14523235—21no—
103#14523235—21no—
104#14523235—21no—
105#14523235—21no—
106D1016—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—
118C2×C127⋊C8—21no—
119C2×C127⋊C8C2311yescharacteristic
120C8×D127—21no—
121C8×D127—21no—
122C8×D127—21no—
123C8×D127—21no—
124C8×D127—21no—
125C8×D127D411yescharacteristic
126C8×D127D411yescharacteristic
127C8×D127D411yescharacteristic
128C8×D127D411yescharacteristic
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—41no—
147C22×D127—41no—
148C22×D127—82no—
149C22×D127#4111yescharacteristic
150Dic254—21no—
151Dic254—21no—
152Dic254—21no—
153Dic254—41no—
154Dic254—41no—
155Dic254C2×D411yescharacteristic
156D508—21no—
157D508—21no—
158D508—21no—
159D508C2×D411yescharacteristic
160C22⋊D127—21no—
161C22⋊D127—21no—
162C22⋊D127—21no—
163C22⋊D127—41no—
164C22⋊D127—41no—
165C2×Dic127—21no—
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—22no—
203C2×D127—21no—
204C2×D127—42no—
205C2×D127—41no—
206C2×D127—82no—
207C2×D127—82no—
208C2×D127—81no—
209C2×D127#12111yescharacteristic
210C2×D127#12111yescharacteristic
211C4×C127—21no—
212C4×C127—41no—
213C4×C127C22×D411yescharacteristic
214C4×C127C22×D411yescharacteristic
215Dic127—21no—
216Dic127—41no—
217Dic127#12111yescharacteristic
218Dic127#12111yescharacteristic
219C2×C127—21no—
220C2×C127—41no—
221C2×C127—81no—
222C2×C127C2×#12111yescharacteristic
223D127—21no—
224D127—41no—
225D127—81no—
226D127#47112yesdirect product factor
227C2×#471—1271nomaximal, Sylow
228C127C2×#47111yescharacteristic, complemented, Sylow
229C2×#137—1271no—
230C2×AGL(1,ℤ/8)—1271no—
231C22×SD16—1271no—
232C2×#131—1271no—
233#471—1271no—
234#471—1271no—
235#471—1271no—
236#471—1271no—
237#471—1271no—
238#471—1271no—
239#471—1271no—
240#471D12711yescharacteristic, direct product factor
241C2×#108—1271no—
242C2×#101—1271no—
243C2×#100—1271no—
244C2×#41—1271no—
245C22×Q8—1271no—
246C22×D4—1271no—
247#137—1271no—
248#137—1271no—
249#137—1271no—
250#137—2541no—
251#137—2541no—
252#137C2×D12711yescharacteristic, complemented
253AGL(1,ℤ/8)—1271no—
254AGL(1,ℤ/8)—1271no—
255AGL(1,ℤ/8)—1271no—
256AGL(1,ℤ/8)—2541no—
257AGL(1,ℤ/8)—2541no—
258AGL(1,ℤ/8)C2×D12711yescharacteristic
259C2×Q16—2541no—
260C2×SD16—1271no—
261C2×SD16—1271no—
262C2×SD16—1271no—
263C2×SD16—2541no—
264C2×SD16—2541no—
265C2×SD16—2541no—
266C2×SD16—2541no—
267C2×SD16—2541no—
268C2×SD16—2541no—
269C2×SD16C2×D12711yescharacteristic, complemented
270C2×D8—2541no—
271#131—1271no—
272#131—1271no—
273#131—1271no—
274#131—2541no—
275#131—2541no—
276#131C2×D12711yescharacteristic, complemented
277C2×M16—1271no—
278C2×M16—1271no—
279C2×M16—1271no—
280C2×M16—2541no—
281C22×C8—1271no—
282C22×C8—2541no—
283#108—1271no—
284#108—1271no—
285#108—1271no—
286#108C2×D12711yescharacteristic, complemented
287#101—1271no—
288#101—1271no—
289#101—1271no—
290#101C2×D12711yescharacteristic, complemented
291#100—1271no—
292#100—1271no—
293#100—1271no—
294#100C2×D12711yescharacteristic, complemented
295C24—2541no—
296#41—1271no—
297#41—1271no—
298#41—1271no—
299#41—2541no—
300#41—2541no—
301#41C22×D12711yescharacteristic
302C2×Q8—1271no—
303C2×Q8—1271no—
304C2×Q8—2541no—
305C2×Q8—2541no—
306C2×Q8—2541no—
307C2×Q8—2541no—
308C2×Q8—2541no—
309C2×Q8C22×D12711yescharacteristic
310C2×D4—1271no—
311C2×D4—1271no—
312C2×D4—2541no—
313C2×D4—2541no—
314C2×D4—2541no—
315C2×D4—2541no—
316C2×D4—2541no—
317C2×D4—2541no—
318C2×D4C22×D12711yescharacteristic
319C22×C4—1271no—
320C22×C4—2541no—
321C22×C4—2541no—
322Q16—21no—
323Q16—2541no—
324Q16—2541no—
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—
338SD16—2541no—
339SD16—2541no—
340SD16—2541no—
341SD16—2541no—
342D8—21no—
343D8—2541no—
344D8—2541no—
345D8—2541no—
346M16—21no—
347M16—1271no—
348M16—1271no—
349M16—1271no—
350M16—2541no—
351M16—2541no—
352M16—2541no—
353M16—2541no—
354M16—2541no—
355M16—2541no—
356M16C22×D12711yescharacteristic
357M16C22×D12711yescharacteristic
358M16C22×D12711yescharacteristic
359C2×C8—21no—
360C2×C8—1271no—
361C2×C8—1271no—
362C2×C8—1271no—
363C2×C8—1271no—
364C2×C8—1271no—
365C2×C8—2541no—
366C2×C8—2541no—
367C2×C8—2541no—
368C2×C8—2541no—
369C2×C8—2541no—
370C2×C8—2541no—
371C2×C8C22×D12711yescharacteristic
372C23—21no—
373C23—1271no—
374C23—10162no—
375C23—5081no—
376C23—5081no—
377C23—2541no—
378C23—2541no—
379Q8—21no—
380Q8—21no—
381Q8—1271no—
382Q8—5081no—
383Q8—5081no—
384Q8—2541no—
385Q8—2541no—
386Q8—2541no—
387Q8D4×D12711yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—1271no—
392D4—5081no—
393D4—5081no—
394D4—2541no—
395D4—2541no—
396D4—2541no—
397D4—2541no—
398D4—2541no—
399D4—2541no—
400D4D4×D12711yescharacteristic
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—1271no—
404C2×C4—1271no—
405C2×C4—1271no—
406C2×C4—1271no—
407C2×C4—1271no—
408C2×C4—5081no—
409C2×C4—5081no—
410C2×C4—2541no—
411C2×C4—2541no—
412C2×C4—2541no—
413C2×C4—2541no—
414C2×C4—2541no—
415C2×C4C23×D12711yescharacteristic
416C8—21no—
417C8—21no—
418C8—21no—
419C8—1271no—
420C8—1271no—
421C8—2541no—
422C8—2541no—
423C8—2541no—
424C8D4×D12711yescharacteristic
425C8D4×D12711yescharacteristic
426C22—21no—
427C22—41no—
428C22—81no—
429C22—1271no—
430C22—1271no—
431C22—10161no—
432C22—10162no—
433C22—10162no—
434C22—5082no—
435C22—5081no—
436C22—2542no—
437C22—2541no—
438C22#41×D12711yescharacteristic
439C4—21no—
440C4—41no—
441C4—1271no—
442C4—1271no—
443C4—5081no—
444C4—2541no—
445C4C2×D4×D12711yescharacteristic
446C4C2×D4×D12711yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—1272no—
451C2—10161no—
452C2—5081no—
453C2—2541no—
454C2#121×D12711yescharacteristic
subgroupG/H
centerC2#121×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#471×C127C2
socleC2×C127C2×#121
prime radicalAGL(1,ℤ/8)×D127C2
hypercenter#471D127
hypocenterC127C2×#471
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. #137×D127C2semidirect
3. C2×SD16×D127C2semidirect
4. #131×D127C2semidirect
5. #22550142C2semidirect
6. #22550141C2semidirect
7. #22550140C2semidirect
8. #22550139C2semidirect
9. #22550138C2semidirect
10. #22550137C2semidirect
11. #22550136C2semidirect
12. #108×D127C2semidirect
13. #101×D127C2semidirect
14. #100×D127C2semidirect
15. #471×C127C2semidirect
19. #16861242C22semidirect
21. #16861240C22semidirect
22. #16861239C22semidirect
30. C2×#14523237C22semidirect
31. C2×#14523236C22semidirect
32. C2×#14523235C22semidirect
41. #16861217C22semidirect
42. #16861216C22semidirect
44. #16861215C22semidirect
51. #16861145C22semidirect
52. #16861144C22semidirect
53. #16861144C22semidirect
54. #16861128C22semidirect
55. #16861127C22semidirect
56. #16861127C22semidirect
57. #16861126C22semidirect
58. #16861125C22semidirect
59. #16861125C22semidirect
60. #137×C127C22semidirect
62. C2×SD16×C127C22semidirect
63. #131×C127C22semidirect
64. #108×C127C22semidirect
65. #101×C127C22semidirect
66. #100×C127C22semidirect
226. D127#471direct (N × H)
228. C127C2×#471semidirect
240. #471D127direct (N × H)
252. #137C2×D127semidirect
269. C2×SD16C2×D127semidirect
276. #131C2×D127semidirect
286. #108C2×D127semidirect
290. #101C2×D127semidirect
294. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#137×D1272
AGL(1,ℤ/8)×D1272
C2×SD16×D1272
#131×D1272
#225501422
#225501412
#225501402
#225501392
#225501382
#225501372
#225501362
#108×D1272
#101×D1272
#100×D1272
#471×C1272
C2×#471127