Finite Group Databasebeta

← #30650528

Subgroups — #30650528

Order 16256.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550681C211yesmaximal, characteristic, complemented
2#22550680C212yesmaximal, complemented
3#22550677C211yesmaximal, characteristic, complemented
4#22550251C211yesmaximal, characteristic, complemented
5#22550147C211yesmaximal, characteristic, complemented
6#22550145C212yesmaximal, complemented
7#22550046C212yesmaximal, complemented
8#22550042C212yesmaximal, complemented
9#22549992C211yesmaximal, characteristic, complemented
10#22549991C211yesmaximal, characteristic, complemented
11#472×C127C211yesmaximal, characteristic, complemented
12#16861255C2212yescomplemented
13#16861254C2212yescomplemented
14#16861237—22no—
15#16861237C2212yescomplemented
16#16861237C2212yescomplemented
17#16861236—22no—
18#16861235C2212yescomplemented
19#16861233—22no—
20#16861233—22no—
21#16861231—21no—
22#16861231C2211yescharacteristic
23#16861230—22no—
24#16861229C2211yescharacteristic
25#16861228—21no—
26C2×D1016—21no—
27C2×D1016C2211yescharacteristic, complemented
28C2×#14523233C2212yescomplemented
29D8×D127—21no—
30D8×D127—22no—
31#16861217C2212yescomplemented
32#16861215C2211yescharacteristic, complemented
33#16861205C2212yescomplemented
34#16861144C2211yescharacteristic, complemented
35#16861136C2211yescharacteristic, complemented
36#16861135C2212yescomplemented
37#16861135C2211yescharacteristic, complemented
38#16861125C2212yescomplemented
39C8×Dic127C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C127C2212yescomplemented
41C2×D8×C127C2211yescharacteristic, complemented
42#131×C127C2211yescharacteristic, complemented
43#108×C127C2211yescharacteristic, complemented
44#100×C127C2212yescomplemented
45#14523249—22no—
46#14523248—22no—
47#14523248C2311yescharacteristic
48#14523248C2312yes—
49#14523247—22no—
50#14523247—22no—
51C2×D508—22no—
52C2×D508C2312yes—
53C2×C22⋊D127—42no—
54C2×C22⋊D127—42no—
55D4×D127—22no—
56D4×D127—22no—
57D4×D127—22no—
58D4×D127—42no—
59Dic508D411yescharacteristic, complemented
60#14523238D411yescharacteristic, complemented
61#14523237—22no—
62#14523237—22no—
63#14523237D412yescomplemented
64#14523236—22no—
65#14523236D411yescharacteristic, complemented
66#14523235D411yescharacteristic, complemented
67D1016—22no—
68D1016—22no—
69D1016—21no—
70D1016—21no—
71D1016D411yescharacteristic, complemented
72#14523233—22no—
73#14523233—22no—
74#14523233—22no—
75#14523233D411yescharacteristic, complemented
76#14523232C2311yescharacteristic
77#14523231—22no—
78#14523231—21no—
79C2×C127⋊C8C2311yescharacteristic
80C8×D127—22no—
81C8×D127—21no—
82C4×Dic127C2311yescharacteristic
83#41×C127C2311yescharacteristic
84C2×D4×C127C2312yes—
85SD16×C127—22no—
86D8×C127—22no—
87D8×C127—21no—
88D8×C127—21no—
89M16×C127—21no—
90M16×C127C2312yes—
91M16×C127C2311yescharacteristic
92C2×C8×C127—21no—
93C2×C8×C127C2311yescharacteristic
94C22×D127—22no—
95C22×D127—42no—
96C22×D127—42no—
97Dic254C2×D412yes—
98Dic254C2×D411yescharacteristic
99D508—22no—
100D508—22no—
101D508—42no—
102D508C2×D412yes—
103D508C2×D411yescharacteristic
104C22⋊D127—22no—
105C22⋊D127—21no—
106C22⋊D127—42no—
107C22⋊D127—42no—
108C22⋊D127—42no—
109C22⋊D127—84no—
110C2×Dic127—21no—
111C2×Dic127—42no—
112C4×D127—21no—
113C4×D127—22no—
114C4×D127—42no—
115C127⋊C8—21no—
116C127⋊C8C2×D411yescharacteristic
117C127⋊C8C2×D411yescharacteristic
118C23×C127—22no—
119Q8×C127C2×D411yescharacteristic
120D4×C127—21no—
121D4×C127—22no—
122D4×C127—22no—
123D4×C127C2×D411yescharacteristic
124C2×C4×C127—21no—
125C2×C4×C127C2411yescharacteristic
126C8×C127—21no—
127C8×C127—22no—
128C8×C127C2×D411yescharacteristic
129C8×C127C2×D411yescharacteristic
130C22×C127—21no—
131C22×C127—42no—
132C22×C127—82no—
133C22×C127H(5,𝔽2)11yescharacteristic
134C2×D127—21no—
135C2×D127—22no—
136C2×D127—42no—
137C2×D127—82no—
138C2×D127—162no—
139C2×D127—162no—
140C4×C127—21no—
141C4×C127C22×D411yescharacteristic
142C4×C127C22×D411yescharacteristic
143Dic127—22no—
144Dic127—22no—
145Dic127—21no—
146C2×C127—21no—
147C2×C127—41no—
148C2×C127—82no—
149C2×C127D4211yescharacteristic
150D127—41no—
151D127—42no—
152D127—82no—
153#3292—1271nomaximal, Sylow
154C127#329211yescharacteristic, complemented, Sylow
155#576—1272no—
156#576—1271no—
157#576—1271no—
158#493—1271no—
159#472—1271no—
160#472—1272no—
161#472D12711yescharacteristic, complemented
162#453—1272no—
163#453—1272no—
164#443—1271no—
165#443—1271no—
166H(5,𝔽2)—1272no—
167H(5,𝔽2)—1272no—
168AGL(1,ℤ/8)—1272no—
169AGL(1,ℤ/8)—1272no—
170AGL(1,ℤ/8)—1272no—
171AGL(1,ℤ/8)—2542no—
172AGL(1,ℤ/8)—2542no—
173AGL(1,ℤ/8)—2542no—
174AGL(1,ℤ/8)—2542no—
175AGL(1,ℤ/8)C2×D12712yescomplemented
176#135—1271no—
177#135—1271no—
178#135—2541no—
179#135—2541no—
180#135—2542no—
181C2×D8—1272no—
182C2×D8—1271no—
183C2×D8—2541no—
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#108C2×D12711yescharacteristic, complemented
193#104—1271no—
194#104—1272no—
195#104—1271no—
196#100—1272no—
197#100C2×D12712yescomplemented
198C4×C8—1271no—
199#41—1271no—
200#41—1272no—
201#41—2542no—
202#41—2542no—
203#41—2542no—
204#41—2542no—
205#41C22×D12711yescharacteristic
206C2×D4—1272no—
207C2×D4—5082no—
208C2×D4—5082no—
209C2×D4—5082no—
210C2×D4—2542no—
211C2×D4—2542no—
212C2×D4—2542no—
213C2×D4—2542no—
214C2×D4C22×D12712yes—
215Q16—1271no—
216Q16—1271no—
217SD16—22no—
218SD16—1271no—
219SD16—1271no—
220SD16—1272no—
221SD16—2542no—
222SD16—2542no—
223SD16—2542no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—1271no—
228D8—1271no—
229D8—2542no—
230D8—2542no—
231D8—2542no—
232D8—2542no—
233D8—2542no—
234D8—2541no—
235D8—2541no—
236M16—21no—
237M16—1271no—
238M16—2542no—
239M16—2541no—
240M16C22×D12712yes—
241M16C22×D12711yescharacteristic
242C2×C8—21no—
243C2×C8—1271no—
244C2×C8—2542no—
245C2×C8—2541no—
246C2×C8C22×D12711yescharacteristic
247C42—1271no—
248C23—22no—
249C23—5082no—
250C23—5082no—
251C23—2542no—
252Q8—1271no—
253Q8—1272no—
254Q8D4×D12711yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—1271no—
259D4—1272no—
260D4—10164no—
261D4—5082no—
262D4—5082no—
263D4—5082no—
264D4—5082no—
265D4—2542no—
266D4—2541no—
267D4—2542no—
268D4—2542no—
269D4D4×D12711yescharacteristic
270C2×C4—21no—
271C2×C4—5082no—
272C2×C4—5082no—
273C2×C4—2541no—
274C2×C4—2542no—
275C2×C4—2541no—
276C2×C4C23×D12711yescharacteristic
277C8—21no—
278C8—22no—
279C8—1271no—
280C8—1271no—
281C8—2541no—
282C8D4×D12711yescharacteristic
283C8D4×D12711yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—20322no—
288C22—20322no—
289C22—10162no—
290C22—5082no—
291C22—2542no—
292C22—2541no—
293C22#1686125511yescharacteristic
294C4—21no—
295C4—2542no—
296C4—2542no—
297C4—2541no—
298C4C2×D4×D12711yescharacteristic
299C4C2×D4×D12711yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—10162no—
304C2—5082no—
305C2—5081no—
306C2#2255051011yescharacteristic
subgroupG/H
centerC2#22550510
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#472×C127C2
socleC2×C127D42
hypercenter#472D127
hypocenterC127#3292
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #22550681C2semidirect
2. #22550680C2semidirect
3. #22550677C2semidirect
4. #22550251C2semidirect
5. #22550147C2semidirect
6. #22550145C2semidirect
7. #22550046C2semidirect
8. #22550042C2semidirect
9. #22549992C2semidirect
10. #22549991C2semidirect
11. #472×C127C2semidirect
12. #16861255C22semidirect
13. #16861254C22semidirect
15. #16861237C22semidirect
16. #16861237C22semidirect
18. #16861235C22semidirect
27. C2×D1016C22semidirect
28. C2×#14523233C22semidirect
31. #16861217C22semidirect
32. #16861215C22semidirect
33. #16861205C22semidirect
34. #16861144C22semidirect
35. #16861136C22semidirect
36. #16861135C22semidirect
37. #16861135C22semidirect
38. #16861125C22semidirect
39. C8×Dic127C22semidirect
40. AGL(1,ℤ/8)×C127C22semidirect
41. C2×D8×C127C22semidirect
42. #131×C127C22semidirect
43. #108×C127C22semidirect
44. #100×C127C22semidirect
59. Dic508D4semidirect
60. #14523238D4semidirect
63. #14523237D4semidirect
65. #14523236D4semidirect
66. #14523235D4semidirect
71. D1016D4semidirect
75. #14523233D4semidirect
154. C127#3292semidirect
161. #472D127semidirect
175. AGL(1,ℤ/8)C2×D127semidirect
186. C2×D8C2×D127semidirect
189. #131C2×D127semidirect
192. #108C2×D127semidirect
197. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506812
#225506802
#225506772
#225502512
#225501472
#225501452
#225500462
#225500422
#225499922
#225499912
#472×C1272
#3292127