Finite Group Databasebeta

← #30650684

Subgroups — #30650684

Order 16256.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550526C211yesmaximal, characteristic, complemented
2#22550514C211yesmaximal, characteristic, complemented
3#22550237C211yesmaximal, characteristic, complemented
4#22550205C211yesmaximal, characteristic, complemented
5#22550195C211yesmaximal, characteristic, complemented
6#22550164C211yesmaximal, characteristic, complemented
7#22550127C211yesmaximal, characteristic, complemented
8#22550102C211yesmaximal, characteristic, complemented
9#22550093C211yesmaximal, characteristic, complemented
10#22550008C211yesmaximal, characteristic, complemented
11#22549988C211yesmaximal, characteristic, complemented
12#22549950C211yesmaximal, characteristic, complemented
13#22549877C211yesmaximal, characteristic, complemented
14#102×D127C211yesmaximal, characteristic, complemented
15#449×C127C211yesmaximal, characteristic, complemented
16C2×#14523247—21no—
17C2×D4×D127C2211yescharacteristic
18C2×#14523233C2211yescharacteristic
19C2×#14523232C2211yescharacteristic, complemented
20C2×C8×D127C2211yescharacteristic
21#16861208C2211yescharacteristic, complemented
22#16861198C2211yescharacteristic
23#16861192—21no—
24#16861191—21no—
25#16861190—21no—
26#16861186—21no—
27#16861185C2211yescharacteristic
28#16861183C2211yescharacteristic
29#16861181—21no—
30#16861180C2211yescharacteristic
31#16861177—21no—
32D4×Dic127C2211yescharacteristic
33D4×Dic127C2211yescharacteristic, complemented
34C4×C22⋊D127C2211yescharacteristic
35C2×C4⋊Dic127C2211yescharacteristic, complemented
36C4⋊C4×D127C2211yescharacteristic
37C2×(C2×D127)⋊C4—21no—
38#16861141C2211yescharacteristic, complemented
39#16861141C2211yescharacteristic, complemented
40#16861140C2211yescharacteristic, complemented
41Dic127⋊C8C2211yescharacteristic, complemented
42#16861133C2211yescharacteristic, complemented
43#16861133C2211yescharacteristic, complemented
44#16861131C2211yescharacteristic
45#16861131C2211yescharacteristic
46#16861130C2211yescharacteristic
47#16861130C2211yescharacteristic
48#16861130C2211yescharacteristic, complemented
49#16861129C2211yescharacteristic
50(C2×D127)⋊C8C2211yescharacteristic
51#16861119C2211yescharacteristic, complemented
52C2×#41×C127C2211yescharacteristic
53C2×SD16×C127C2211yescharacteristic, complemented
54C2×D8×C127C2211yescharacteristic
55#121×C127C2211yescharacteristic, complemented
56#103×C127C2211yescharacteristic, complemented
57#102×C127C2211yescharacteristic
58C22⋊C8×C127C2211yescharacteristic, complemented
59C23×D127—21no—
60#14523247—44no—
61C2×Dic254—21no—
62C2×D508C2311yescharacteristic
63C2×C22⋊D127—21no—
64C2×C22⋊D127—21no—
65C2×C22⋊D127—21no—
66C2×C22⋊D127D411yescharacteristic
67D4×D127—22no—
68D4×D127—41no—
69C22×Dic127—21no—
70C22×Dic127—21no—
71C22×Dic127—21no—
72C2×C4×D127—21no—
73C2×C4×D127C2311yescharacteristic
74#14523233—41no—
75#14523232—41no—
76C2×C127⋊C8C2311yescharacteristic
77C2×C127⋊C8C2311yescharacteristic
78C8×D127—41no—
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)⋊C4—21no—
94(C2×D127)⋊C4—21no—
95(C2×D127)⋊C4D411yescharacteristic
96C4×Dic127C2311yescharacteristic
97#41×C127—22no—
98#41×C127—41no—
99C2×Q8×C127C2311yescharacteristic
100C2×D4×C127—21no—
101C2×D4×C127—21no—
102C2×D4×C127C2311yescharacteristic
103C2×D4×C127C2311yescharacteristic
104C22×C4×C127—21no—
105C22×C4×C127C2311yescharacteristic
106SD16×C127—41no—
107D8×C127—22no—
108C2×C8×C127C2311yescharacteristic
109C2×C8×C127C2311yescharacteristic
110C4⋊C4×C127C2311yescharacteristic
111C22⋊C4×C127—21no—
112C22×D127—21no—
113C22×D127—42no—
114C22×D127—42no—
115C22×D127—81no—
116C22×D127C2×D411yescharacteristic
117Dic254—42no—
118D508—21no—
119D508—21no—
120C22⋊D127—22no—
121C22⋊D127—42no—
122C22⋊D127—42no—
123C22⋊D127—81no—
124C2×Dic127—21no—
125C2×Dic127—21no—
126C2×Dic127—21no—
127C2×Dic127—42no—
128C2×Dic127—42no—
129C2×Dic127—81no—
130C2×Dic127C2×D411yescharacteristic
131C2×Dic127C2×D411yescharacteristic
132C2×Dic127C2×D411yescharacteristic
133C4×D127—21no—
134C4×D127—21no—
135C4×D127—42no—
136C127⋊C8—21no—
137C127⋊C8—21no—
138C23×C127—21no—
139C23×C127—21no—
140C23×C127C2×D411yescharacteristic
141Q8×C127—21no—
142Q8×C127—21no—
143D4×C127—21no—
144D4×C127—21no—
145D4×C127—21no—
146D4×C127—42no—
147D4×C127—42no—
148D4×C127#4112yescomplemented
149C2×C4×C127—21no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—42no—
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×C127C22×D411yescharacteristic
168C2×D127—21no—
169C2×D127—21no—
170C2×D127—41no—
171C2×D127—41no—
172C2×D127—41no—
173C2×D127—84no—
174C2×D127AGL(1,ℤ/8)12yes—
175C4×C127—21no—
176C4×C127—41no—
177C4×C127—41no—
178C4×C127C2×#4111yescharacteristic
179C4×C127H(5,𝔽2)11yescharacteristic
180Dic127—21no—
181Dic127—21no—
182Dic127—21no—
183Dic127—41no—
184Dic127—41no—
185Dic127—41no—
186C2×C127—42no—
187C2×C127—41no—
188C2×C127—81no—
189C2×C127#54611yescharacteristic
190C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C127#57711yescharacteristic
192D127—22no—
193D127—81no—
194#3306—1271nomaximal, Sylow
195C127#330611yescharacteristic, complemented, Sylow
196#547—1271no—
197#546—1271no—
198#489—1271no—
199#483—1271no—
200#481—1271no—
201#476—1271no—
202#469—1271no—
203#465—1271no—
204#463—1271no—
205#449D12711yescharacteristic, complemented
206#447—1271no—
207#442—1271no—
208#435—1271no—
209#417—1271no—
210C2×#102—1271no—
211C2×#41—2541no—
212C2×#41C2×D12711yescharacteristic
213C22×D4—1271no—
214C2×SD16C2×D12711yescharacteristic, complemented
215C2×D8—1271no—
216C2×D8C2×D12711yescharacteristic
217C2×M16—1271no—
218C22×C8—1271no—
219C4⋊Q8—1271no—
220#124—1271no—
221#123—2541no—
222#123—2541no—
223#123—2541no—
224C22⋊Q8—1271no—
225C22⋊Q8—2541no—
226#121—1271no—
227#121—1271no—
228#121—2541no—
229#121C2×D12711yescharacteristic, complemented
230#120—2541no—
231C4×D4—1271no—
232C4×D4—1271no—
233C4×D4—1271no—
234C2×C4⋊C4—1271no—
235C2×C4⋊C4—1271no—
236C2×C22⋊C4—2541no—
237#106—1271no—
238#106—1271no—
239#106—1271no—
240C4⋊C8—1271no—
241#103—1271no—
242#103—1271no—
243#103C2×D12711yescharacteristic, complemented
244#102—1271no—
245#102—1271no—
246#102—1271no—
247#102—1271no—
248#102—1271no—
249#102—1271no—
250#102C2×D12711yescharacteristic
251C22⋊C8—1271no—
252C22⋊C8C2×D12711yescharacteristic, complemented
253#97—1271no—
254C24—2541no—
255#41—22no—
256#41—41no—
257#41—5084no—
258C2×Q8—2541no—
259C2×Q8C22×D12711yescharacteristic
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—1271no—
263C2×D4—1271no—
264C2×D4—5081no—
265C2×D4—2541no—
266C2×D4—2541no—
267C2×D4—2541no—
268C2×D4—2542no—
269C2×D4C22×D12711yescharacteristic
270C2×D4C22×D12711yescharacteristic
271C22×C4—21no—
272C22×C4—1271no—
273C22×C4—2541no—
274C22×C4—2541no—
275C22×C4—2541no—
276C22×C4—2541no—
277C22×C4C22×D12711yescharacteristic
278SD16—41no—
279D8—22no—
280D8—5081no—
281M16—5081no—
282C2×C8—1271no—
283C2×C8—1271no—
284C2×C8—5081no—
285C2×C8C22×D12711yescharacteristic
286C2×C8C22×D12711yescharacteristic
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⋊C4C22×D12711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—1271no—
298C22⋊C4—1271no—
299C22⋊C4—2541no—
300C22⋊C4—2541no—
301C22⋊C4—2541no—
302C22⋊C4—2541no—
303C22⋊C4—2541no—
304C22⋊C4—2541no—
305C22⋊C4—2541no—
306C42—1271no—
307C23—21no—
308C23—21no—
309C23—1271no—
310C23—10161no—
311C23—5082no—
312C23—5082no—
313C23—2541no—
314C23D4×D12711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—5082no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—10161no—
324D4—5082no—
325D4—5082no—
326D4—2542no—
327D4—2541no—
328D4—2541no—
329D4#1452324712yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—1271no—
336C2×C4—1271no—
337C2×C4—1271no—
338C2×C4—10161no—
339C2×C4—5082no—
340C2×C4—5082no—
341C2×C4—5082no—
342C2×C4—2541no—
343C2×C4—2541no—
344C2×C4—2541no—
345C2×C4—2541no—
346C2×C4—2541no—
347C2×C4D4×D12711yescharacteristic
348C2×C4C23×D12711yescharacteristic
349C8—21no—
350C8—21no—
351C8—2541no—
352C8—2541no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—1272no—
363C22—10164no—
364C22—5081no—
365C22—5081no—
366C22—5081no—
367C22—2541no—
368C22—2541no—
369C22C2×D4×D12711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—5081no—
374C4—5081no—
375C4—5081no—
376C4—2541no—
377C4—2541no—
378C4—2541no—
379C4C2×#1452324711yescharacteristic
380C4#1686125411yescharacteristic
381C2—41no—
382C2—42no—
383C2—81no—
384C2—10161no—
385C2—2542no—
386C2#2255051311yescharacteristic
387C2AGL(1,ℤ/8)×D12711yescharacteristic
388C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#449×C127C2
socleC22×C127C22×D4
hypercenter#449D127
hypocenterC127#3306
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #22550526C2semidirect
2. #22550514C2semidirect
3. #22550237C2semidirect
4. #22550205C2semidirect
5. #22550195C2semidirect
6. #22550164C2semidirect
7. #22550127C2semidirect
8. #22550102C2semidirect
9. #22550093C2semidirect
10. #22550008C2semidirect
11. #22549988C2semidirect
12. #22549950C2semidirect
13. #22549877C2semidirect
14. #102×D127C2semidirect
15. #449×C127C2semidirect
19. C2×#14523232C22semidirect
21. #16861208C22semidirect
33. D4×Dic127C22semidirect
35. C2×C4⋊Dic127C22semidirect
38. #16861141C22semidirect
39. #16861141C22semidirect
40. #16861140C22semidirect
41. Dic127⋊C8C22semidirect
42. #16861133C22semidirect
43. #16861133C22semidirect
48. #16861130C22semidirect
51. #16861119C22semidirect
53. C2×SD16×C127C22semidirect
55. #121×C127C22semidirect
56. #103×C127C22semidirect
58. C22⋊C8×C127C22semidirect
148. D4×C127#41semidirect
195. C127#3306semidirect
205. #449D127semidirect
214. C2×SD16C2×D127semidirect
229. #121C2×D127semidirect
243. #103C2×D127semidirect
252. C22⋊C8C2×D127semidirect
329. D4#14523247semidirect

Maximal subgroups

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

subgroupindex
#225505262
#225505142
#225502372
#225502052
#225501952
#225501642
#225501272
#225501022
#225500932
#225500082
#225499882
#225499502
#225498772
#102×D1272
#449×C1272
#3306127