Finite Group Databasebeta

← #30650216

Subgroups — #30650216

Order 16256.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16861236C211yesmaximal, characteristic, complemented
2C2×D8×D127C211yesmaximal, characteristic, complemented
3#22550519C211yesmaximal, characteristic, complemented
4D4×D508C211yesmaximal, characteristic, complemented
5#22550266C211yesmaximal, characteristic, complemented
6#22550128C211yesmaximal, characteristic, complemented
7#22550108C211yesmaximal, characteristic, complemented
8#22550072C211yesmaximal, characteristic, complemented
9#22550025C211yesmaximal, characteristic, complemented
10#22550024C211yesmaximal, characteristic, complemented
11#22550011C211yesmaximal, characteristic, complemented
12#22550008C211yesmaximal, characteristic, complemented
13#22549986C211yesmaximal, characteristic, complemented
14#22549946C211yesmaximal, characteristic, complemented
15#463×C127C211yesmaximal, characteristic, complemented
16C2×#14523249C2211yescharacteristic, complemented
17C22×D508—21no—
18C2×D4×D127C2211yescharacteristic
19C2×D4×D127C2211yescharacteristic, complemented
20C2×D4×D127C2211yescharacteristic, complemented
21#16861236—24no—
22C2×#14523235C2211yescharacteristic, complemented
23C2×#14523235C2211yescharacteristic, complemented
24C2×D1016C2211yescharacteristic, complemented
25C2×D1016C2211yescharacteristic, complemented
26C2×#14523233C2211yescharacteristic
27C2×#14523233C2211yescharacteristic, complemented
28C2×#14523233C2211yescharacteristic, complemented
29D8×D127—24no—
30C2×#14523231C2211yescharacteristic, complemented
31C2×C8×D127C2211yescharacteristic, complemented
32#16861206C2211yescharacteristic, complemented
33#16861191—21no—
34#16861185C2211yescharacteristic, complemented
35#16861184C2211yescharacteristic
36#16861184C2211yescharacteristic, complemented
37#16861183—21no—
38#16861182—21no—
39#16861180C2211yescharacteristic, complemented
40#16861179—21no—
41#16861179—21no—
42#16861179—21no—
43C4×D508C2211yescharacteristic
44C4×D508C2211yescharacteristic, complemented
45C22⋊C4×D127—21no—
46#16861140C2211yescharacteristic, complemented
47#16861131C2211yescharacteristic, complemented
48#16861131C2211yescharacteristic, complemented
49#16861130C2211yescharacteristic, complemented
50#16861129C2211yescharacteristic, complemented
51(C2×D127)⋊C8C2211yescharacteristic, complemented
52(C2×D127)⋊C8C2211yescharacteristic, complemented
53#16861120C2211yescharacteristic, complemented
54C2×SD16×C127C2211yescharacteristic, complemented
55C2×D8×C127C2211yescharacteristic
56#124×C127C2211yescharacteristic, complemented
57C4×D4×C127C2211yescharacteristic, complemented
58C4⋊C8×C127C2211yescharacteristic, complemented
59#103×C127C2211yescharacteristic, complemented
60#102×C127C2211yescharacteristic, complemented
61C23×D127—21no—
62C23×D127—21no—
63C23×D127—21no—
64#14523249—22no—
65#14523249—41no—
66C2×D508—21no—
67C2×D508—21no—
68C2×D508—21no—
69C2×D508—21no—
70C2×D508—21no—
71C2×D508—44no—
72C2×D508D411yescharacteristic
73C2×D508C2311yescharacteristic
74C2×D508C2311yescharacteristic
75C2×D508C2311yescharacteristic
76C2×C22⋊D127—21no—
77C2×C22⋊D127—21no—
78C2×C22⋊D127—21no—
79C2×C22⋊D127—21no—
80D4×D127—22no—
81D4×D127—22no—
82D4×D127—22no—
83D4×D127—41no—
84D4×D127—41no—
85D4×D127—41no—
86C2×C4×D127—21no—
87C2×C4×D127—21no—
88C2×C4×D127C2311yescharacteristic
89C2×C4×D127C2311yescharacteristic
90#14523235—22no—
91#14523235—41no—
92D1016—22no—
93D1016—22no—
94#14523233—22no—
95#14523233—41no—
96#14523233D414yescomplemented
97#14523231—22no—
98C2×C127⋊C8C2311yescharacteristic
99C2×C127⋊C8C2311yescharacteristic
100C8×D127—22no—
101C4⋊Dic127D411yescharacteristic
102C4⋊Dic127C2311yescharacteristic
103Dic127⋊C4—21no—
104C22⋊Dic127—21no—
105(C2×D127)⋊C4—21no—
106(C2×D127)⋊C4—21no—
107(C2×D127)⋊C4—21no—
108(C2×D127)⋊C4—21no—
109(C2×D127)⋊C4D411yescharacteristic
110(C2×D127)⋊C4D411yescharacteristic
111C2×Q8×C127C2311yescharacteristic
112C2×D4×C127C2311yescharacteristic
113C2×D4×C127C2311yescharacteristic
114C22×C4×C127—21no—
115SD16×C127—22no—
116D8×C127—22no—
117C2×C8×C127C2311yescharacteristic
118C2×C8×C127C2311yescharacteristic
119C4⋊C4×C127C2311yescharacteristic
120C22⋊C4×C127—21no—
121C22⋊C4×C127—21no—
122C22⋊C4×C127—21no—
123C42×C127C2311yescharacteristic
124C22×D127—21no—
125C22×D127—21no—
126C22×D127—21no—
127C22×D127—42no—
128C22×D127—42no—
129C22×D127—42no—
130C22×D127—42no—
131C22×D127—42no—
132C22×D127—42no—
133C22×D127—81no—
134C22×D127—81no—
135C22×D127—81no—
136C22×D127C2×D411yescharacteristic
137C22×D127C2×D411yescharacteristic
138D508—22no—
139D508—21no—
140D508—21no—
141D508—21no—
142D508—21no—
143D508—21no—
144D508—42no—
145D508—42no—
146D508—42no—
147D508—42no—
148D508—81no—
149D508C2×D412yescomplemented
150C22⋊D127—42no—
151C22⋊D127—42no—
152C22⋊D127—42no—
153C22⋊D127—81no—
154C2×Dic127—21no—
155C2×Dic127C2×D411yescharacteristic
156C2×Dic127C2×D411yescharacteristic
157C4×D127—21no—
158C4×D127—21no—
159C4×D127—21no—
160C4×D127—21no—
161C4×D127—42no—
162C4×D127—81no—
163C127⋊C8—21no—
164C127⋊C8C2×D412yescomplemented
165C23×C127—21no—
166C23×C127—21no—
167Q8×C127—21no—
168Q8×C127—21no—
169D4×C127—21no—
170D4×C127—21no—
171D4×C127—21no—
172D4×C127C2×D412yes—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—42no—
176C2×C4×C127C2×D411yescharacteristic
177C2×C4×C127C2×D411yescharacteristic
178C2×C4×C127C2411yescharacteristic
179C8×C127—21no—
180C8×C127—21no—
181C22×C127—22no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—41no—
185C22×C127—41no—
186C22×C127—41no—
187C22×C127C22×D411yescharacteristic
188C2×D127—21no—
189C2×D127—21no—
190C2×D127—21no—
191C2×D127—22no—
192C2×D127—21no—
193C2×D127—21no—
194C2×D127—41no—
195C2×D127—41no—
196C2×D127—41no—
197C2×D127—41no—
198C2×D127—41no—
199C2×D127—41no—
200C2×D127—41no—
201C2×D127—41no—
202C2×D127—84no—
203C2×D127—84no—
204C2×D127—84no—
205C2×D127AGL(1,ℤ/8)12yescomplemented
206C4×C127—21no—
207C4×C127—21no—
208C4×C127—41no—
209C4×C127—41no—
210C4×C127C22×D411yescharacteristic
211C4×C127H(5,𝔽2)11yescharacteristic
212Dic127—21no—
213Dic127—21no—
214Dic127—41no—
215C2×C127—42no—
216C2×C127—81no—
217C2×C127D4211yescharacteristic
218C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C127#57611yescharacteristic
220D127—22no—
221D127—41no—
222D127—42no—
223D127—81no—
224D127—81no—
225#3264—1271nomaximal, Sylow
226C127#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1271no—
228C22×D8—1271no—
229#546—1271no—
230D42—1271no—
231#496—1271no—
232#469—1271no—
233#465—1271no—
234#463D12711yescharacteristic, complemented
235#459—1271no—
236#450—1271no—
237#449—1271no—
238#447—1271no—
239#447—1271no—
240#442—1271no—
241#435—1271no—
242C2×#41—1271no—
243C22×D4—1271no—
244C22×D4—1271no—
245C22×D4—1271no—
246C22×D4—2541no—
247AGL(1,ℤ/8)—2544no—
248C2×SD16—1271no—
249C2×SD16—1271no—
250C2×SD16C2×D12711yescharacteristic, complemented
251C2×D8—1271no—
252C2×D8—1271no—
253C2×D8—1271no—
254C2×D8—1271no—
255C2×D8—1271no—
256C2×D8—2544no—
257C2×D8C2×D12711yescharacteristic
258C2×M16—1271no—
259C22×C8—1271no—
260#127—1271no—
261#124C2×D12711yescharacteristic, complemented
262#123—2541no—
263C22⋊Q8—1271no—
264#121—1271no—
265#121—1271no—
266#121—1271no—
267#121—2541no—
268#121—2541no—
269#120—2541no—
270#120—2541no—
271#120—2541no—
272C4×D4—1271no—
273C4×D4—1271no—
274C4×D4C2×D12711yescharacteristic, complemented
275C2×C22⋊C4—2541no—
276#106—1271no—
277C4⋊C8C2×D12711yescharacteristic, complemented
278#103C2×D12711yescharacteristic, complemented
279#102—1271no—
280#102—1271no—
281#102—1271no—
282#102—1271no—
283#102C2×D12711yescharacteristic, complemented
284C22⋊C8—1271no—
285C22⋊C8—1271no—
286#97—1271no—
287C24—2541no—
288C24—2541no—
289C24—2541no—
290#41—5081no—
291#41—2542no—
292C2×Q8C22×D12711yescharacteristic
293C2×D4—1271no—
294C2×D4—1271no—
295C2×D4—1271no—
296C2×D4—1271no—
297C2×D4—5084no—
298C2×D4—5081no—
299C2×D4—5081no—
300C2×D4—5081no—
301C2×D4—2541no—
302C2×D4—2541no—
303C2×D4—2541no—
304C2×D4—2541no—
305C2×D4—2541no—
306C2×D4—2541no—
307C2×D4—2542no—
308C2×D4—2541no—
309C2×D4—2542no—
310C2×D4—2542no—
311C2×D4—2541no—
312C2×D4—2541no—
313C2×D4C22×D12711yescharacteristic
314C2×D4C22×D12711yescharacteristic
315C22×C4—21no—
316C22×C4—1271no—
317C22×C4—1271no—
318C22×C4—2541no—
319C22×C4—2541no—
320SD16—22no—
321SD16—5081no—
322SD16—2542no—
323D8—22no—
324D8—1274no—
325D8—5081no—
326D8—2542no—
327D8—2542no—
328D8—2542no—
329M16—2542no—
330C2×C8—1271no—
331C2×C8—1271no—
332C2×C8—2542no—
333C2×C8C22×D12711yescharacteristic
334C2×C8C22×D12711yescharacteristic
335C4⋊C4—1271no—
336C4⋊C4—1271no—
337C4⋊C4—2541no—
338C4⋊C4C22×D12711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—1271no—
343C22⋊C4—1271no—
344C22⋊C4—2541no—
345C22⋊C4—2541no—
346C22⋊C4—2541no—
347C22⋊C4—2541no—
348C22⋊C4—2541no—
349C42C22×D12711yescharacteristic
350C23—21no—
351C23—21no—
352C23—1271no—
353C23—1271no—
354C23—10161no—
355C23—10161no—
356C23—10161no—
357C23—5082no—
358C23—5082no—
359C23—5082no—
360C23—5082no—
361C23—5082no—
362C23—5082no—
363C23—2541no—
364C23—2541no—
365C23—2541no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—1272no—
372D4—10161no—
373D4—10161no—
374D4—5082no—
375D4—5082no—
376D4—5082no—
377D4—5082no—
378D4—5082no—
379D4—5082no—
380D4—5082no—
381D4—2542no—
382D4—2541no—
383D4—2541no—
384D4—2541no—
385D4—2541no—
386D4—2541no—
387D4D4×D12712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—1271no—
392C2×C4—1271no—
393C2×C4—10161no—
394C2×C4—5082no—
395C2×C4—2541no—
396C2×C4—2541no—
397C2×C4—2541no—
398C2×C4—2541no—
399C2×C4—2541no—
400C2×C4D4×D12711yescharacteristic
401C2×C4D4×D12711yescharacteristic
402C2×C4C23×D12711yescharacteristic
403C8—21no—
404C8—21no—
405C8—1272no—
406C8—2541no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—1272no—
414C22—10164no—
415C22—10164no—
416C22—10164no—
417C22—5081no—
418C22—5081no—
419C22—5081no—
420C22—5081no—
421C22—5081no—
422C22—5081no—
423C22—5081no—
424C22—5081no—
425C22—2541no—
426C22—2541no—
427C22—2542no—
428C22—2541no—
429C22—2541no—
430C22—2541no—
431C22C2×D4×D12711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—5081no—
437C4—2541no—
438C4—2541no—
439C4C2×D4×D12711yescharacteristic
440C4#1686125511yescharacteristic
441C2—42no—
442C2—81no—
443C2—10161no—
444C2—10161no—
445C2—5081no—
446C2—5082no—
447C2—2542no—
448C2#2255051011yescharacteristic
449C2AGL(1,ℤ/8)×D12711yescharacteristic
450C2#2255067811yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#463×C127C2
socleC22×C127C22×D4
hypercenter#463D127
hypocenterC127#3264
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×D8×D127C2semidirect
3. #22550519C2semidirect
4. D4×D508C2semidirect
5. #22550266C2semidirect
6. #22550128C2semidirect
7. #22550108C2semidirect
8. #22550072C2semidirect
9. #22550025C2semidirect
10. #22550024C2semidirect
11. #22550011C2semidirect
12. #22550008C2semidirect
13. #22549986C2semidirect
14. #22549946C2semidirect
15. #463×C127C2semidirect
16. C2×#14523249C22semidirect
19. C2×D4×D127C22semidirect
20. C2×D4×D127C22semidirect
22. C2×#14523235C22semidirect
23. C2×#14523235C22semidirect
24. C2×D1016C22semidirect
25. C2×D1016C22semidirect
27. C2×#14523233C22semidirect
28. C2×#14523233C22semidirect
30. C2×#14523231C22semidirect
31. C2×C8×D127C22semidirect
32. #16861206C22semidirect
34. #16861185C22semidirect
36. #16861184C22semidirect
39. #16861180C22semidirect
44. C4×D508C22semidirect
46. #16861140C22semidirect
47. #16861131C22semidirect
48. #16861131C22semidirect
49. #16861130C22semidirect
50. #16861129C22semidirect
51. (C2×D127)⋊C8C22semidirect
52. (C2×D127)⋊C8C22semidirect
53. #16861120C22semidirect
54. C2×SD16×C127C22semidirect
56. #124×C127C22semidirect
57. C4×D4×C127C22semidirect
58. C4⋊C8×C127C22semidirect
59. #103×C127C22semidirect
60. #102×C127C22semidirect
96. #14523233D4semidirect
149. D508C2×D4semidirect
164. C127⋊C8C2×D4semidirect
205. C2×D127AGL(1,ℤ/8)semidirect
226. C127#3264semidirect
234. #463D127semidirect
250. C2×SD16C2×D127semidirect
261. #124C2×D127semidirect
274. C4×D4C2×D127semidirect
277. C4⋊C8C2×D127semidirect
278. #103C2×D127semidirect
283. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C2×D8×D1272
#225505192
D4×D5082
#225502662
#225501282
#225501082
#225500722
#225500252
#225500242
#225500112
#225500082
#225499862
#225499462
#463×C1272
#3264127