Finite Group Databasebeta

← #30650527

Subgroups — #30650527

Order 16256.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550679C212yesmaximal, complemented
2#22550678C211yesmaximal, characteristic, complemented
3#22550251C211yesmaximal, characteristic, complemented
4#22550146C211yesmaximal, characteristic, complemented
5#22550144C212yesmaximal, complemented
6#22550143C211yesmaximal, characteristic, complemented
7#22550044C212yesmaximal, complemented
8#22550042C212yesmaximal, complemented
9#22549993C211yesmaximal, characteristic, complemented
10#22549991C211yesmaximal, characteristic, complemented
11#576×C127C211yesmaximal, characteristic, complemented
12#16861255C2212yescomplemented
13#16861236—22no—
14#16861235—22no—
15#16861235C2212yescomplemented
16#16861234—22no—
17#16861234C2212yescomplemented
18#16861234C2212yescomplemented
19#16861232—21no—
20#16861229—22no—
21#16861229C2211yescharacteristic
22C2×D1016C2211yescharacteristic, complemented
23C2×#14523233—22no—
24C2×#14523233C2212yescomplemented
25D8×D127—21no—
26#16861216C2212yescomplemented
27#16861215C2211yescharacteristic, complemented
28#16861205C2212yescomplemented
29#16861145C2211yescharacteristic, complemented
30#16861144C2211yescharacteristic, complemented
31#16861136C2212yescomplemented
32#16861136C2211yescharacteristic, complemented
33#16861135C2211yescharacteristic, complemented
34#16861126C2212yescomplemented
35#16861125C2212yescomplemented
36C8×Dic127C2211yescharacteristic, complemented
37H(5,𝔽2)×C127C2212yescomplemented
38AGL(1,ℤ/8)×C127—22no—
39AGL(1,ℤ/8)×C127C2212yescomplemented
40#135×C127—21no—
41#135×C127C2211yescharacteristic
42C2×D8×C127—21no—
43C2×D8×C127C2211yescharacteristic, complemented
44#131×C127C2211yescharacteristic, complemented
45#14523249—22no—
46#14523248—22no—
47#14523248C2311yescharacteristic
48C2×D508—22no—
49C2×D508C2312yes—
50C2×C22⋊D127—42no—
51D4×D127—22no—
52D4×D127—42no—
53#14523238D411yescharacteristic, complemented
54#14523236—22no—
55#14523236D411yescharacteristic, complemented
56#14523235—22no—
57#14523235—22no—
58#14523235D411yescharacteristic, complemented
59D1016—21no—
60D1016—21no—
61#14523233—22no—
62#14523233—22no—
63#14523233—22no—
64#14523233—22no—
65#14523233—22no—
66#14523233D411yescharacteristic, complemented
67#14523232—22no—
68#14523232C2311yescharacteristic
69#14523232C2312yes—
70#14523231—21no—
71C2×C127⋊C8—22no—
72C2×C127⋊C8C2311yescharacteristic
73C8×D127—21no—
74C4×Dic127C2311yescharacteristic
75#41×C127—22no—
76#41×C127—22no—
77#41×C127C2312yes—
78#41×C127C2311yescharacteristic
79C2×D4×C127—22no—
80C2×D4×C127—22no—
81C2×D4×C127—42no—
82C2×D4×C127C2312yes—
83Q16×C127D411yescharacteristic, complemented
84SD16×C127—22no—
85SD16×C127D412yescomplemented
86D8×C127—22no—
87D8×C127—21no—
88D8×C127—21no—
89D8×C127D411yescharacteristic, complemented
90M16×C127—21no—
91M16×C127C2311yescharacteristic
92C2×C8×C127—21no—
93C2×C8×C127C2311yescharacteristic
94C22×D127—22no—
95C22×D127—42no—
96Dic254C2×D411yescharacteristic
97D508—22no—
98D508—22no—
99D508—42no—
100D508C2×D411yescharacteristic
101C22⋊D127—21no—
102C22⋊D127—42no—
103C22⋊D127—84no—
104C2×Dic127—21no—
105C4×D127—21no—
106C4×D127—42no—
107C127⋊C8—21no—
108C127⋊C8—22no—
109C127⋊C8C2×D411yescharacteristic
110C127⋊C8C2×D411yescharacteristic
111C23×C127—22no—
112C23×C127—42no—
113Q8×C127C2×D411yescharacteristic
114Q8×C127C2×D412yes—
115D4×C127—21no—
116D4×C127—22no—
117D4×C127—22no—
118D4×C127—22no—
119D4×C127—42no—
120D4×C127—42no—
121D4×C127C2×D411yescharacteristic
122D4×C127C2×D412yes—
123C2×C4×C127—21no—
124C2×C4×C127—22no—
125C2×C4×C127—42no—
126C2×C4×C127C2411yescharacteristic
127C8×C127—21no—
128C8×C127C2×D411yescharacteristic
129C8×C127C2×D411yescharacteristic
130C22×C127—21no—
131C22×C127—22no—
132C22×C127—42no—
133C22×C127—82no—
134C22×C127—162no—
135C22×C127H(5,𝔽2)11yescharacteristic
136C2×D127—21no—
137C2×D127—42no—
138C2×D127—82no—
139C2×D127—162no—
140C4×C127—21no—
141C4×C127—22no—
142C4×C127C22×D411yescharacteristic
143C4×C127C22×D411yescharacteristic
144Dic127—22no—
145Dic127—21no—
146C2×C127—21no—
147C2×C127—42no—
148C2×C127—41no—
149C2×C127—82no—
150C2×C127D4211yescharacteristic
151D127—41no—
152D127—82no—
153#3292—1271nomaximal, Sylow
154C127#329211yescharacteristic, complemented, Sylow
155#576—1272no—
156#576—1271no—
157#576D12711yescharacteristic, complemented
158#493—1271no—
159#472—1272no—
160#472—1271no—
161#472—1271no—
162#453—1272no—
163#453—1272no—
164#443—1271no—
165#443—1271no—
166H(5,𝔽2)—1272no—
167H(5,𝔽2)C2×D12712yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—1272no—
170AGL(1,ℤ/8)—1272no—
171AGL(1,ℤ/8)—1272no—
172AGL(1,ℤ/8)—2542no—
173AGL(1,ℤ/8)—2542no—
174AGL(1,ℤ/8)—2542no—
175AGL(1,ℤ/8)C2×D12712yescomplemented
176#135—21no—
177#135—1271no—
178#135—2542no—
179#135—2541no—
180#135C2×D12711yescharacteristic
181C2×D8—21no—
182C2×D8—1272no—
183C2×D8—1271no—
184C2×D8—2542no—
185C2×D8—2541no—
186C2×D8C2×D12711yescharacteristic, complemented
187#131—1272no—
188#131—1271no—
189#131C2×D12711yescharacteristic, complemented
190#127—1272no—
191#108—1271no—
192#108—1271no—
193#104—1271no—
194#104—1272no—
195#104—1271no—
196#100—1272no—
197#100—1272no—
198C4×C8—1271no—
199#41—22no—
200#41—22no—
201#41—1271no—
202#41—2542no—
203#41—2542no—
204#41C22×D12711yescharacteristic
205#41C22×D12712yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—1272no—
210C2×D4—5082no—
211C2×D4—5082no—
212C2×D4—2542no—
213C2×D4—2542no—
214C2×D4C22×D12712yes—
215Q16—1271no—
216Q16C22⋊D12711yescharacteristic, complemented
217SD16—22no—
218SD16—1271no—
219SD16—1271no—
220SD16—2542no—
221SD16—2542no—
222SD16—2542no—
223SD16C22⋊D12712yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—1271no—
228D8—2542no—
229D8—2542no—
230D8—2542no—
231D8—2542no—
232D8—2542no—
233D8—2541no—
234D8—2541no—
235D8C22⋊D12711yescharacteristic, complemented
236M16—21no—
237M16—1272no—
238M16—1271no—
239M16—2542no—
240M16—2541no—
241M16C22×D12711yescharacteristic
242C2×C8—21no—
243C2×C8—1271no—
244C2×C8—2542no—
245C2×C8—2541no—
246C2×C8C22×D12711yescharacteristic
247C42—1271no—
248C23—22no—
249C23—42no—
250C23—5082no—
251C23—2542no—
252Q8—1271no—
253Q8D4×D12711yescharacteristic
254Q8C2×C22⋊D12712yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—1271no—
262D4—10164no—
263D4—5082no—
264D4—5082no—
265D4—2541no—
266D4—2542no—
267D4—2542no—
268D4D4×D12711yescharacteristic
269D4C2×C22⋊D12712yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—5082no—
274C2×C4—2541no—
275C2×C4—2541no—
276C2×C4C23×D12711yescharacteristic
277C8—21no—
278C8—1271no—
279C8—1271no—
280C8—2542no—
281C8—2541no—
282C8C2×C22⋊D12711yescharacteristic
283C8C2×C22⋊D12711yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—20322no—
290C22—10162no—
291C22—5082no—
292C22—2541no—
293C22#1686125411yescharacteristic
294C4—21no—
295C4—22no—
296C4—2542no—
297C4—2541no—
298C4C2×D4×D12711yescharacteristic
299C4C22×C22⋊D12711yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—10162no—
305C2—5081no—
306C2D4×C22⋊D12711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#576×C127C2
socleC2×C127D42
hypercenter#576D127
hypocenterC127#3292
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #22550679C2semidirect
2. #22550678C2semidirect
3. #22550251C2semidirect
4. #22550146C2semidirect
5. #22550144C2semidirect
6. #22550143C2semidirect
7. #22550044C2semidirect
8. #22550042C2semidirect
9. #22549993C2semidirect
10. #22549991C2semidirect
11. #576×C127C2semidirect
12. #16861255C22semidirect
15. #16861235C22semidirect
17. #16861234C22semidirect
18. #16861234C22semidirect
22. C2×D1016C22semidirect
24. C2×#14523233C22semidirect
26. #16861216C22semidirect
27. #16861215C22semidirect
28. #16861205C22semidirect
29. #16861145C22semidirect
30. #16861144C22semidirect
31. #16861136C22semidirect
32. #16861136C22semidirect
33. #16861135C22semidirect
34. #16861126C22semidirect
35. #16861125C22semidirect
36. C8×Dic127C22semidirect
37. H(5,𝔽2)×C127C22semidirect
39. AGL(1,ℤ/8)×C127C22semidirect
43. C2×D8×C127C22semidirect
44. #131×C127C22semidirect
53. #14523238D4semidirect
55. #14523236D4semidirect
58. #14523235D4semidirect
66. #14523233D4semidirect
83. Q16×C127D4semidirect
85. SD16×C127D4semidirect
89. D8×C127D4semidirect
154. C127#3292semidirect
157. #576D127semidirect
167. H(5,𝔽2)C2×D127semidirect
175. AGL(1,ℤ/8)C2×D127semidirect
186. C2×D8C2×D127semidirect
189. #131C2×D127semidirect
216. Q16C22⋊D127semidirect
223. SD16C22⋊D127semidirect
235. D8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506792
#225506782
#225502512
#225501462
#225501442
#225501432
#225500442
#225500422
#225499932
#225499912
#576×C1272
#3292127