Finite Group Databasebeta

← #30650327

Subgroups — #30650327

Order 16256.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861237C211yesmaximal, characteristic, complemented
2C2×#16861236C211yesmaximal, characteristic, complemented
3#22550528C211yesmaximal, characteristic, complemented
4#22550510C211yesmaximal, characteristic, complemented
5#22550247C211yesmaximal, characteristic, complemented
6#22550123C211yesmaximal, characteristic, complemented
7#22550080C211yesmaximal, characteristic, complemented
8#22550070C211yesmaximal, characteristic, complemented
9#22550028C211yesmaximal, characteristic, complemented
10#22550026C211yesmaximal, characteristic, complemented
11#22550024C211yesmaximal, characteristic, complemented
12#22550019C211yesmaximal, characteristic, complemented
13#22549958C211yesmaximal, characteristic, complemented
14#22549952C211yesmaximal, characteristic, complemented
15#468×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic, complemented
17C2×#14523248C2211yescharacteristic, complemented
18C22×D508C2211yescharacteristic
19C2×D4×D127—21no—
20C2×D4×D127C2211yescharacteristic
21#16861237—24no—
22#16861236—24no—
23C2×#14523237C2211yescharacteristic
24C2×#14523237C2211yescharacteristic, complemented
25C2×#14523236C2211yescharacteristic, complemented
26C2×#14523235C2211yescharacteristic
27C2×#14523235C2211yescharacteristic, complemented
28C2×D1016C2211yescharacteristic
29C2×D1016C2211yescharacteristic, complemented
30C2×#14523233C2211yescharacteristic, complemented
31C2×#14523231C2211yescharacteristic, complemented
32#16861207C2211yescharacteristic, complemented
33#16861205C2211yescharacteristic
34#16861194—21no—
35#16861191—21no—
36#16861189C2211yescharacteristic, complemented
37#16861184C2211yescharacteristic
38#16861182—21no—
39#16861182—21no—
40#16861179—21no—
41#16861177—21no—
42#16861171C2211yescharacteristic
43#16861171C2211yescharacteristic, complemented
44C4×C22⋊D127C2211yescharacteristic
45C4⋊C4×D127—21no—
46Dic127⋊C8C2211yescharacteristic, complemented
47#16861132C2211yescharacteristic, complemented
48#16861129C2211yescharacteristic
49#16861129C2211yescharacteristic, complemented
50#16861129C2211yescharacteristic, complemented
51C22⋊(C127⋊C8)C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53C8×Dic127C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C2×M16×C127C2211yescharacteristic, complemented
56C22⋊Q8×C127C2211yescharacteristic, complemented
57#121×C127C2211yescharacteristic
58#107×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63#14523249—22no—
64#14523249—41no—
65#14523248—22no—
66#14523248—41no—
67C2×Dic254C2311yescharacteristic
68C2×D508—22no—
69C2×D508—21no—
70C2×D508—21no—
71C2×D508—21no—
72C2×D508—41no—
73C2×D508C2311yescharacteristic
74C2×D508C2311yescharacteristic
75C2×D508C2311yescharacteristic
76C2×C22⋊D127—21no—
77C2×C22⋊D127—21no—
78C2×C22⋊D127—21no—
79C2×C22⋊D127—21no—
80C2×C22⋊D127D411yescharacteristic
81D4×D127—22no—
82D4×D127—41no—
83D4×D127—44no—
84C2×C4×D127—21no—
85C2×C4×D127—21no—
86C2×C4×D127—21no—
87C2×C4×D127—21no—
88C2×C4×D127C2311yescharacteristic
89#14523237—22no—
90#14523237—22no—
91#14523236—41no—
92#14523235—22no—
93#14523235—41no—
94D1016—22no—
95D1016D414yescomplemented
96#14523233—22no—
97#14523231—22no—
98C2×C127⋊C8C2311yescharacteristic
99C2×C127⋊C8C2311yescharacteristic
100C4⋊Dic127—21no—
101Dic127⋊C4—21no—
102Dic127⋊C4—21no—
103Dic127⋊C4D411yescharacteristic
104C22⋊Dic127D411yescharacteristic
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4—21no—
109(C2×D127)⋊C4D411yescharacteristic
110C4×Dic127C2311yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127—21no—
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127C2311yescharacteristic
115SD16×C127—22no—
116M16×C127—22no—
117C2×C8×C127C2311yescharacteristic
118C2×C8×C127C2311yescharacteristic
119C4⋊C4×C127—21no—
120C4⋊C4×C127C2311yescharacteristic
121C4⋊C4×C127C2311yescharacteristic
122C22⋊C4×C127—21no—
123C22⋊C4×C127—21no—
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127—42no—
130C22×D127—42no—
131C22×D127—81no—
132C22×D127—81no—
133C22×D127C2×D411yescharacteristic
134Dic254—21no—
135Dic254—21no—
136D508—21no—
137D508—21no—
138D508—21no—
139D508—21no—
140D508—42no—
141D508—42no—
142D508—42no—
143D508C2×D412yes—
144D508C2×D412yescomplemented
145C22⋊D127—22no—
146C22⋊D127—42no—
147C22⋊D127—42no—
148C22⋊D127—42no—
149C22⋊D127—81no—
150C2×Dic127—21no—
151C2×Dic127C2×D411yescharacteristic
152C2×Dic127C2×D411yescharacteristic
153C2×Dic127C2×D411yescharacteristic
154C4×D127—21no—
155C4×D127—21no—
156C4×D127—42no—
157C4×D127—42no—
158C4×D127—42no—
159C4×D127—42no—
160C127⋊C8—21no—
161C127⋊C8—21no—
162C23×C127—21no—
163C23×C127C2×D411yescharacteristic
164Q8×C127—21no—
165Q8×C127—21no—
166D4×C127—21no—
167D4×C127—21no—
168D4×C127—42no—
169C2×C4×C127—21no—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127C2×D411yescharacteristic
175C2×C4×C127C2411yescharacteristic
176C8×C127—21no—
177C8×C127C2×D412yescomplemented
178C22×C127—21no—
179C22×C127—21no—
180C22×C127—21no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127C22×D411yescharacteristic
185C2×D127—21no—
186C2×D127—22no—
187C2×D127—22no—
188C2×D127—21no—
189C2×D127—21no—
190C2×D127—41no—
191C2×D127—41no—
192C2×D127—41no—
193C2×D127—41no—
194C2×D127—41no—
195C2×D127—41no—
196C2×D127—41no—
197C2×D127—84no—
198C2×D127—84no—
199C4×C127—21no—
200C4×C127—41no—
201C4×C127—41no—
202C4×C127—41no—
203C4×C127C22×D411yescharacteristic
204C4×C127H(5,𝔽2)11yescharacteristic
205Dic127—21no—
206Dic127—21no—
207Dic127—41no—
208Dic127AGL(1,ℤ/8)12yescomplemented
209C2×C127—41no—
210C2×C127—81no—
211C2×C127D4211yescharacteristic
212C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C127#57711yescharacteristic
214D127—41no—
215D127—42no—
216D127—42no—
217D127—81no—
218#3272—1271nomaximal, Sylow
219C127#327211yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1271no—
221C2×AGL(1,ℤ/8)—1271no—
222#547—1271no—
223D42—1271no—
224#492—1271no—
225#468—1271no—
226#468D12711yescharacteristic, complemented
227#461—1271no—
228#459—1271no—
229#450—1271no—
230#450—1271no—
231#449—1271no—
232#449—1271no—
233C4×D8—1271no—
234#436—1271no—
235C2×#41—1271no—
236C2×#41—1271no—
237C22×D4—1271no—
238C22×D4—1271no—
239C22×D4—2541no—
240AGL(1,ℤ/8)—2544no—
241AGL(1,ℤ/8)—2544no—
242C2×SD16—1271no—
243C2×SD16—1271no—
244C2×SD16—1271no—
245C2×SD16—1271no—
246C2×SD16—1271no—
247C2×SD16C2×D12711yescharacteristic, complemented
248C2×D8—1271no—
249C2×D8—1271no—
250C2×D8—1271no—
251C2×M16—1271no—
252C2×M16C2×D12711yescharacteristic, complemented
253C4⋊Q8—1271no—
254#127—1271no—
255#123—2541no—
256#123—2541no—
257C22⋊Q8—1271no—
258C22⋊Q8C2×D12711yescharacteristic, complemented
259#121—1271no—
260#121—2541no—
261#121—2541no—
262#121C2×D12711yescharacteristic
263#120—2541no—
264#120—2541no—
265C4×D4—1271no—
266C4×D4—1271no—
267C4×D4—1271no—
268C2×C4⋊C4—2541no—
269#107C2×D12711yescharacteristic, complemented
270C4⋊C8—1271no—
271#103—1271no—
272#103C2×D12711yescharacteristic, complemented
273#102—1271no—
274#102—1271no—
275#102—1271no—
276#102C2×D12711yescharacteristic, complemented
277C22⋊C8—1271no—
278C22⋊C8—1271no—
279C4×C8—1271no—
280C24—2541no—
281C24—2541no—
282#41—5081no—
283#41—5081no—
284#41—2542no—
285#41—2542no—
286C2×Q8—1271no—
287C2×Q8C22×D12711yescharacteristic
288C2×D4—21no—
289C2×D4—1271no—
290C2×D4—1271no—
291C2×D4—1271no—
292C2×D4—1271no—
293C2×D4—5081no—
294C2×D4—5084no—
295C2×D4—5081no—
296C2×D4—2542no—
297C2×D4—2541no—
298C2×D4—2541no—
299C2×D4—2541no—
300C2×D4—2541no—
301C2×D4—2541no—
302C2×D4—2542no—
303C2×D4—2541no—
304C2×D4—2541no—
305C2×D4C22×D12711yescharacteristic
306C22×C4—1271no—
307C22×C4—2541no—
308C22×C4—2541no—
309C22×C4—2541no—
310C22×C4—2541no—
311C22×C4C22×D12711yescharacteristic
312SD16—22no—
313SD16—5081no—
314SD16—5081no—
315SD16—2542no—
316SD16—2542no—
317SD16—2542no—
318D8—1274no—
319D8—2542no—
320D8—2542no—
321M16—22no—
322M16—2542no—
323C2×C8—1271no—
324C2×C8—1271no—
325C2×C8C22×D12711yescharacteristic
326C2×C8C22×D12711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—1271no—
329C4⋊C4—2541no—
330C4⋊C4—2541no—
331C4⋊C4—2541no—
332C4⋊C4C22×D12711yescharacteristic
333C4⋊C4C22×D12711yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—1271no—
337C22⋊C4—1271no—
338C22⋊C4—2541no—
339C22⋊C4—2541no—
340C22⋊C4—2541no—
341C22⋊C4—2541no—
342C42—1271no—
343C23—21no—
344C23—1271no—
345C23—10161no—
346C23—10161no—
347C23—5082no—
348C23—5082no—
349C23—5082no—
350C23—5082no—
351C23—2541no—
352C23—2541no—
353C23—2541no—
354C23D4×D12711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2541no—
358Q8—2541no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—1272no—
363D4—1272no—
364D4—10161no—
365D4—5082no—
366D4—5082no—
367D4—5082no—
368D4—5082no—
369D4—5082no—
370D4—5082no—
371D4—2542no—
372D4—2541no—
373D4—2541no—
374D4—2541no—
375D4—2541no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—1271no—
382C2×C4—1271no—
383C2×C4—1271no—
384C2×C4—5082no—
385C2×C4—5082no—
386C2×C4—5082no—
387C2×C4—5082no—
388C2×C4—2541no—
389C2×C4—2541no—
390C2×C4—2541no—
391C2×C4D4×D12711yescharacteristic
392C2×C4C23×D12711yescharacteristic
393C8—21no—
394C8—2541no—
395C8—2541no—
396C8D4×D12712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—10164no—
404C22—10164no—
405C22—5081no—
406C22—5081no—
407C22—5081no—
408C22—5081no—
409C22—5081no—
410C22—5081no—
411C22—5081no—
412C22—2541no—
413C22—2541no—
414C22—2542no—
415C22—2541no—
416C22—2542no—
417C22C2×D4×D12711yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—1272no—
423C4—5081no—
424C4—2541no—
425C4—2541no—
426C4C2×D4×D12711yescharacteristic
427C4#1686125511yescharacteristic
428C2—41no—
429C2—81no—
430C2—10161no—
431C2—5082no—
432C2—5081no—
433C2—5082no—
434C2#2255051011yescharacteristic
435C2AGL(1,ℤ/8)×D12711yescharacteristic
436C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#468×C127C2
socleC22×C127C22×D4
hypercenter#468D127
hypocenterC127#3272
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. C2×#16861236C2semidirect
3. #22550528C2semidirect
4. #22550510C2semidirect
5. #22550247C2semidirect
6. #22550123C2semidirect
7. #22550080C2semidirect
8. #22550070C2semidirect
9. #22550028C2semidirect
10. #22550026C2semidirect
11. #22550024C2semidirect
12. #22550019C2semidirect
13. #22549958C2semidirect
14. #22549952C2semidirect
15. #468×C127C2semidirect
16. C2×#14523249C22semidirect
17. C2×#14523248C22semidirect
24. C2×#14523237C22semidirect
25. C2×#14523236C22semidirect
27. C2×#14523235C22semidirect
29. C2×D1016C22semidirect
30. C2×#14523233C22semidirect
31. C2×#14523231C22semidirect
32. #16861207C22semidirect
36. #16861189C22semidirect
43. #16861171C22semidirect
46. Dic127⋊C8C22semidirect
47. #16861132C22semidirect
49. #16861129C22semidirect
50. #16861129C22semidirect
51. C22⋊(C127⋊C8)C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. C8×Dic127C22semidirect
54. C2×SD16×C127C22semidirect
55. C2×M16×C127C22semidirect
56. C22⋊Q8×C127C22semidirect
58. #107×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
95. D1016D4semidirect
144. D508C2×D4semidirect
177. C8×C127C2×D4semidirect
208. Dic127AGL(1,ℤ/8)semidirect
219. C127#3272semidirect
226. #468D127semidirect
247. C2×SD16C2×D127semidirect
252. C2×M16C2×D127semidirect
258. C22⋊Q8C2×D127semidirect
269. #107C2×D127semidirect
272. #103C2×D127semidirect
276. #102C2×D127semidirect
396. C8D4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C2×#168612362
#225505282
#225505102
#225502472
#225501232
#225500802
#225500702
#225500282
#225500262
#225500242
#225500192
#225499582
#225499522
#468×C1272
#3272127