Finite Group Databasebeta

← #30650744

Subgroups — #30650744

Order 16256.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550560C211yesmaximal, characteristic, complemented
2D4×C22⋊D127C211yesmaximal, characteristic
3#22550239C211yesmaximal, characteristic, complemented
4#22550201C211yesmaximal, characteristic, complemented
5#22550188C211yesmaximal, characteristic, complemented
6#22550178C211yesmaximal, characteristic, complemented
7#22550124C211yesmaximal, characteristic, complemented
8#22550102C211yesmaximal, characteristic, complemented
9#22550074C211yesmaximal, characteristic, complemented
10#22550025C211yesmaximal, characteristic, complemented
11#22549977C211yesmaximal, characteristic, complemented
12#22549950C211yesmaximal, characteristic, complemented
13#22549881C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#450×C127C211yesmaximal, characteristic, complemented
16C22×C22⋊D127—21no—
17C2×D4×D127C2211yescharacteristic
18C2×#14523235C2211yescharacteristic, complemented
19C2×#14523232C2211yescharacteristic, complemented
20C2×C8×D127C2211yescharacteristic, complemented
21#16861205C2211yescharacteristic
22#16861202—21no—
23#16861200C2211yescharacteristic, complemented
24#16861199—21no—
25#16861191—21no—
26#16861185C2211yescharacteristic, complemented
27#16861183C2211yescharacteristic
28#16861181—21no—
29#16861180C2211yescharacteristic
30#16861178—21no—
31#16861177—21no—
32Q8×Dic127C2211yescharacteristic, complemented
33D4×Dic127C2211yescharacteristic
34C4×C22⋊D127C2211yescharacteristic
35#16861168C2211yescharacteristic, complemented
36#16861167—21no—
37C4⋊C4×D127C2211yescharacteristic, complemented
38#16861143C2211yescharacteristic, complemented
39#16861142C2211yescharacteristic, complemented
40#16861141C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42#16861133C2211yescharacteristic, complemented
43#16861131C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic, complemented
45#16861130C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic, complemented
48#16861130C2211yescharacteristic, complemented
49#16861129C2211yescharacteristic, complemented
50(C2×D127)⋊C8C2211yescharacteristic, complemented
51#16861119C2211yescharacteristic, complemented
52C22×D4×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic
54C2×SD16×C127C2211yescharacteristic, complemented
55C22⋊Q8×C127C2211yescharacteristic, complemented
56#102×C127C2211yescharacteristic
57#102×C127C2211yescharacteristic, complemented
58C22⋊C8×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60C2×D508C2311yescharacteristic
61C2×C22⋊D127—21no—
62C2×C22⋊D127—21no—
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127—44no—
67C2×C22⋊D127D411yescharacteristic
68D4×D127—22no—
69D4×D127—41no—
70C22×Dic127—21no—
71C2×C4×D127—21no—
72C2×C4×D127C2311yescharacteristic
73#14523235—41no—
74#14523232—41no—
75C2×C127⋊C8C2311yescharacteristic
76C2×C127⋊C8C2311yescharacteristic
77C8×D127—41no—
78C4⋊Dic127—21no—
79C4⋊Dic127—21no—
80C4⋊Dic127C2311yescharacteristic
81C4⋊Dic127C2311yescharacteristic
82C4⋊Dic127C2311yescharacteristic
83Dic127⋊C4—21no—
84Dic127⋊C4—21no—
85Dic127⋊C4—21no—
86Dic127⋊C4D411yescharacteristic
87C22⋊Dic127—21no—
88C22⋊Dic127—21no—
89C22⋊Dic127—21no—
90C22⋊Dic127D411yescharacteristic
91(C2×D127)⋊C4—21no—
92(C2×D127)⋊C4—21no—
93(C2×D127)⋊C4D411yescharacteristic
94C4×Dic127—21no—
95C4×Dic127—21no—
96C4×Dic127C2311yescharacteristic
97C24×C127—21no—
98C2×Q8×C127C2311yescharacteristic
99C2×D4×C127—22no—
100C2×D4×C127—21no—
101C2×D4×C127—41no—
102C2×D4×C127C2311yescharacteristic
103C2×D4×C127C2311yescharacteristic
104C22×C4×C127C2311yescharacteristic
105SD16×C127—22no—
106SD16×C127—41no—
107C2×C8×C127C2311yescharacteristic
108C2×C8×C127C2311yescharacteristic
109C4⋊C4×C127—21no—
110C4⋊C4×C127C2311yescharacteristic
111C22⋊C4×C127—21no—
112C22×D127—21no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127—81no—
116C22×D127C2×D411yescharacteristic
117D508—21no—
118D508—21no—
119C22⋊D127—22no—
120C22⋊D127—42no—
121C22⋊D127—42no—
122C22⋊D127—42no—
123C22⋊D127—42no—
124C22⋊D127—81no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—21no—
128C2×Dic127—42no—
129C2×Dic127C2×D411yescharacteristic
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C4×D127—21no—
133C4×D127—21no—
134C4×D127—42no—
135C127⋊C8—21no—
136C127⋊C8—21no—
137C23×C127—21no—
138C23×C127—21no—
139C23×C127—42no—
140C23×C127—42no—
141C23×C127—81no—
142C23×C127C2×D411yescharacteristic
143Q8×C127—21no—
144Q8×C127—21no—
145D4×C127—21no—
146D4×C127—21no—
147D4×C127—21no—
148D4×C127—42no—
149D4×C127#4112yes—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127C2×D411yescharacteristic
155C2×C4×C127C2411yescharacteristic
156C8×C127—21no—
157C8×C127—21no—
158C22×C127—21no—
159C22×C127—22no—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—41no—
163C22×C127—41no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—84no—
168C22×C127C22×D411yescharacteristic
169C2×D127—21no—
170C2×D127—21no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—41no—
174C2×D127—84no—
175C2×D127AGL(1,ℤ/8)12yes—
176C4×C127—21no—
177C4×C127—41no—
178C4×C127—41no—
179C4×C127C2×#4111yescharacteristic
180C4×C127H(5,𝔽2)11yescharacteristic
181Dic127—21no—
182Dic127—21no—
183Dic127—21no—
184Dic127—41no—
185Dic127—41no—
186Dic127—41no—
187C2×C127—42no—
188C2×C127—41no—
189C2×C127—81no—
190C2×C127#54611yescharacteristic
191C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C127#57711yescharacteristic
193D127—22no—
194D127—81no—
195#3310—1271nomaximal, Sylow
196C127#331011yescharacteristic, complemented, Sylow
197#552—1271no—
198D42—1271no—
199#490—1271no—
200#482—1271no—
201#480—1271no—
202#478—1271no—
203#468—1271no—
204#465—1271no—
205#460—1271no—
206#450—1271no—
207#450D12711yescharacteristic, complemented
208#440—1271no—
209#435—1271no—
210#418—1271no—
211C2×#102—1271no—
212C22×D4—1271no—
213C22×D4—2541no—
214C22×D4C2×D12711yescharacteristic
215C2×SD16—1271no—
216C2×SD16C2×D12711yescharacteristic
217C2×SD16C2×D12711yescharacteristic, complemented
218C2×M16—1271no—
219C22×C8—1271no—
220#127—1271no—
221#126—2541no—
222#125—1271no—
223#125—2541no—
224#123—2541no—
225C22⋊Q8—1271no—
226C22⋊Q8C2×D12711yescharacteristic, complemented
227#121—1271no—
228#121—1271no—
229#121—2541no—
230#120—2541no—
231#120—2541no—
232C4×Q8—1271no—
233C4×D4—1271no—
234C4×D4—1271no—
235#117—1271no—
236#117—2541no—
237C2×C4⋊C4—1271no—
238#107—1271no—
239#107—1271no—
240#106—1271no—
241C4⋊C8—1271no—
242#103—1271no—
243#102—1271no—
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102C2×D12711yescharacteristic
251#102C2×D12711yescharacteristic, complemented
252C22⋊C8—1271no—
253C22⋊C8C2×D12711yescharacteristic, complemented
254#97—1271no—
255C24—21no—
256C24—2541no—
257C2×Q8C22×D12711yescharacteristic
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—1271no—
262C2×D4—1271no—
263C2×D4—5084no—
264C2×D4—5081no—
265C2×D4—2541no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4—2541no—
269C2×D4—2541no—
270C2×D4—2542no—
271C2×D4C22×D12711yescharacteristic
272C2×D4C22×D12711yescharacteristic
273C22×C4—1271no—
274C22×C4—2541no—
275C22×C4—2541no—
276C22×C4C22×D12711yescharacteristic
277SD16—22no—
278SD16—41no—
279SD16—5081no—
280M16—5081no—
281C2×C8—1271no—
282C2×C8—1271no—
283C2×C8—5081no—
284C2×C8C22×D12711yescharacteristic
285C2×C8C22×D12711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—1271no—
288C4⋊C4—1271no—
289C4⋊C4—1271no—
290C4⋊C4—1271no—
291C4⋊C4—2541no—
292C4⋊C4—2541no—
293C4⋊C4—2541no—
294C4⋊C4—2541no—
295C4⋊C4—2541no—
296C4⋊C4C22×D12711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—1271no—
299C22⋊C4—1271no—
300C22⋊C4—2541no—
301C22⋊C4—2541no—
302C22⋊C4—2541no—
303C22⋊C4—2541no—
304C22⋊C4—2541no—
305C42—1271no—
306C42—2541no—
307C42—2541no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—1271no—
314C23—10161no—
315C23—5082no—
316C23—5082no—
317C23—2541no—
318C23D4×D12711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—10161no—
326D4—5082no—
327D4—5082no—
328D4—5082no—
329D4—5082no—
330D4—2542no—
331D4—2541no—
332D4—2541no—
333D4#1452324712yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—1271no—
339C2×C4—1271no—
340C2×C4—1271no—
341C2×C4—5082no—
342C2×C4—5082no—
343C2×C4—2541no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4—2541no—
348C2×C4D4×D12711yescharacteristic
349C2×C4C23×D12711yescharacteristic
350C8—21no—
351C8—21no—
352C8—2541no—
353C8—2541no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—84no—
364C22—1272no—
365C22—10164no—
366C22—5081no—
367C22—5081no—
368C22—5081no—
369C22—2541no—
370C22—2541no—
371C22C2×D4×D12711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—5081no—
376C4—5081no—
377C4—5081no—
378C4—2541no—
379C4—2541no—
380C4—2541no—
381C4C2×#1452324711yescharacteristic
382C4#1686125411yescharacteristic
383C2—41no—
384C2—42no—
385C2—81no—
386C2—10161no—
387C2—2542no—
388C2#2255051311yescharacteristic
389C2AGL(1,ℤ/8)×D12711yescharacteristic
390C2#2255068211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#450×C127C2
socleC22×C127C22×D4
prime radicalD4×C22⋊D127C2
hypercenter#450D127
hypocenterC127#3310
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22550560C2semidirect
3. #22550239C2semidirect
4. #22550201C2semidirect
5. #22550188C2semidirect
6. #22550178C2semidirect
7. #22550124C2semidirect
8. #22550102C2semidirect
9. #22550074C2semidirect
10. #22550025C2semidirect
11. #22549977C2semidirect
12. #22549950C2semidirect
13. #22549881C2semidirect
14. #102×D127C2semidirect
15. #450×C127C2semidirect
18. C2×#14523235C22semidirect
19. C2×#14523232C22semidirect
20. C2×C8×D127C22semidirect
23. #16861200C22semidirect
26. #16861185C22semidirect
32. Q8×Dic127C22semidirect
35. #16861168C22semidirect
37. C4⋊C4×D127C22semidirect
38. #16861143C22semidirect
39. #16861142C22semidirect
40. #16861141C22semidirect
41. Dic127⋊C8C22semidirect
42. #16861133C22semidirect
43. #16861131C22semidirect
44. #16861131C22semidirect
47. #16861130C22semidirect
48. #16861130C22semidirect
49. #16861129C22semidirect
50. (C2×D127)⋊C8C22semidirect
51. #16861119C22semidirect
54. C2×SD16×C127C22semidirect
55. C22⋊Q8×C127C22semidirect
57. #102×C127C22semidirect
58. C22⋊C8×C127C22semidirect
196. C127#3310semidirect
207. #450D127semidirect
217. C2×SD16C2×D127semidirect
226. C22⋊Q8C2×D127semidirect
251. #102C2×D127semidirect
253. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505602
D4×C22⋊D1272
#225502392
#225502012
#225501882
#225501782
#225501242
#225501022
#225500742
#225500252
#225499772
#225499502
#225498812
#102×D1272
#450×C1272
#3310127