Finite Group Databasebeta

← #30650530

Subgroups — #30650530

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#22550681C212yesmaximal, complemented
2#22550678C212yesmaximal, complemented
3#22550253C211yesmaximal, characteristic, complemented
4#22550147C212yesmaximal, complemented
5#22550146C212yesmaximal, complemented
6#22550045C214yesmaximal, complemented
7#22549994C211yesmaximal, characteristic, complemented
8#443×C127C211yesmaximal, characteristic, complemented
9#16861255C2214yescomplemented
10#16861237—24no—
11#16861237C2214yescomplemented
12#16861236—24no—
13#16861234C2214yescomplemented
14#16861232—22no—
15#16861231—22no—
16#16861231C2211yescharacteristic
17C2×D1016—22no—
18C2×D1016C2212yescomplemented
19C2×D1016C2212yescomplemented
20D8×D127—22no—
21#16861215C2212yescomplemented
22#16861206C2212yescomplemented
23#16861145C2211yescharacteristic, complemented
24#16861137C2212yescomplemented
25#16861125C2214yescomplemented
26#135×C127C2211yescharacteristic
27#131×C127C2212yescomplemented
28#108×C127C2211yescharacteristic, complemented
29#104×C127C2212yescomplemented
30C4×C8×C127C2211yescharacteristic, complemented
31#14523249—24no—
32#14523248—24no—
33#14523248C2312yes—
34C2×D508—24no—
35C2×D508—42no—
36C2×D508C2314yes—
37D4×D127—24no—
38D4×D127—44no—
39Dic508D411yescharacteristic, complemented
40#14523237—24no—
41#14523237D412yescomplemented
42#14523235—24no—
43D1016—24no—
44D1016—22no—
45D1016—22no—
46D1016—22no—
47D1016—22no—
48D1016D411yescharacteristic, complemented
49#14523233—24no—
50#14523232C2312yes—
51#14523231—22no—
52C8×D127—22no—
53#41×C127C2312yes—
54Q16×C127D411yescharacteristic, complemented
55SD16×C127D412yescomplemented
56D8×C127D411yescharacteristic, complemented
57M16×C127—22no—
58M16×C127C2312yes—
59C2×C8×C127—22no—
60C2×C8×C127C2311yescharacteristic
61C2×C8×C127C2311yescharacteristic
62C42×C127C2311yescharacteristic
63C22×D127—24no—
64C22×D127—44no—
65Dic254C2×D412yes—
66D508—24no—
67D508—24no—
68D508—44no—
69D508—84no—
70D508C2×D412yes—
71C22⋊D127—22no—
72C22⋊D127—44no—
73C4×D127—22no—
74C4×D127—44no—
75C127⋊C8—22no—
76Q8×C127C2×D412yes—
77D4×C127—22no—
78D4×C127C2×D412yes—
79C2×C4×C127—21no—
80C2×C4×C127—22no—
81C2×C4×C127C2411yescharacteristic
82C8×C127—22no—
83C8×C127C2×D411yescharacteristic
84C8×C127C2×D411yescharacteristic
85C8×C127C2×D411yescharacteristic
86C8×C127C2×D411yescharacteristic
87C22×C127—22no—
88C22×C127H(5,𝔽2)11yescharacteristic
89C2×D127—22no—
90C2×D127—44no—
91C2×D127—84no—
92C2×D127—164no—
93C4×C127—22no—
94C4×C127—22no—
95C4×C127C22×D411yescharacteristic
96C4×C127C22×D411yescharacteristic
97Dic127—22no—
98C2×C127—21no—
99C2×C127—42no—
100C2×C127D4211yescharacteristic
101D127—42no—
102D127—84no—
103#3292—1271nomaximal, Sylow
104C127#329211yescharacteristic, complemented, Sylow
105#576—1272no—
106#576—1272no—
107#493—1271no—
108#472—1272no—
109#472—1272no—
110#453—1274no—
111#443—1271no—
112#443D12711yescharacteristic, complemented
113H(5,𝔽2)—1274no—
114AGL(1,ℤ/8)—1274no—
115AGL(1,ℤ/8)—1274no—
116AGL(1,ℤ/8)—2544no—
117AGL(1,ℤ/8)—2544no—
118#135—1271no—
119#135—2542no—
120#135—2542no—
121#135C2×D12711yescharacteristic
122C2×D8—1272no—
123C2×D8—1272no—
124C2×D8—2542no—
125C2×D8—2542no—
126#131—1272no—
127#131C2×D12712yescomplemented
128#127—1272no—
129#108—1271no—
130#108C2×D12711yescharacteristic, complemented
131#104—1272no—
132#104C2×D12712yescomplemented
133#100—1274no—
134C4×C8C2×D12711yescharacteristic, complemented
135#41—1272no—
136#41—2544no—
137#41—2544no—
138#41C22×D12712yes—
139C2×D4—1274no—
140C2×D4—5082no—
141C2×D4—5084no—
142C2×D4—2544no—
143C2×D4—2544no—
144Q16—1271no—
145Q16D50811yescharacteristic, complemented
146SD16—1272no—
147SD16—2544no—
148SD16—2544no—
149SD16D50812yescomplemented
150D8—1271no—
151D8—2544no—
152D8—2544no—
153D8—2542no—
154D8—2542no—
155D8—2542no—
156D8—2542no—
157D8D50811yescharacteristic, complemented
158M16—22no—
159M16—1272no—
160M16—2542no—
161M16C22×D12712yes—
162C2×C8—22no—
163C2×C8—2542no—
164C2×C8C22×D12711yescharacteristic
165C2×C8C22×D12711yescharacteristic
166C42C22×D12711yescharacteristic
167C23—5084no—
168C23—2544no—
169Q8—1272no—
170Q8C2×D50812yes—
171D4—22no—
172D4—1272no—
173D4—10164no—
174D4—5084no—
175D4—5084no—
176D4—2542no—
177D4—2544no—
178D4—2544no—
179D4C2×D50812yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—5084no—
183C2×C4—2542no—
184C2×C4C23×D12711yescharacteristic
185C8—22no—
186C8—2542no—
187C8D4×D12711yescharacteristic
188C8D4×D12711yescharacteristic
189C8C2×D50811yescharacteristic
190C8C2×D50811yescharacteristic
191C22—22no—
192C22—20324no—
193C22—10164no—
194C22—5084no—
195C22—2542no—
196C22#1686125511yescharacteristic
197C4—22no—
198C4—22no—
199C4—2542no—
200C4C2×D4×D12711yescharacteristic
201C4C22×D50811yescharacteristic
202C2—21no—
203C2—42no—
204C2—10164no—
205C2—5082no—
206C2D4×D50811yescharacteristic
subgroupG/H
centerC2D4×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#443×C127C2
socleC2×C127D42
hypercenter#443D127
hypocenterC127#3292
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #22550681C2semidirect
2. #22550678C2semidirect
3. #22550253C2semidirect
4. #22550147C2semidirect
5. #22550146C2semidirect
6. #22550045C2semidirect
7. #22549994C2semidirect
8. #443×C127C2semidirect
9. #16861255C22semidirect
11. #16861237C22semidirect
13. #16861234C22semidirect
18. C2×D1016C22semidirect
19. C2×D1016C22semidirect
21. #16861215C22semidirect
22. #16861206C22semidirect
23. #16861145C22semidirect
24. #16861137C22semidirect
25. #16861125C22semidirect
27. #131×C127C22semidirect
28. #108×C127C22semidirect
29. #104×C127C22semidirect
30. C4×C8×C127C22semidirect
39. Dic508D4semidirect
41. #14523237D4semidirect
48. D1016D4semidirect
54. Q16×C127D4semidirect
55. SD16×C127D4semidirect
56. D8×C127D4semidirect
104. C127#3292semidirect
112. #443D127semidirect
127. #131C2×D127semidirect
130. #108C2×D127semidirect
132. #104C2×D127semidirect
134. C4×C8C2×D127semidirect
145. Q16D508semidirect
149. SD16D508semidirect
157. D8D508semidirect

Maximal subgroups

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

subgroupindex
#225506812
#225506782
#225502532
#225501472
#225501462
#225500452
#225499942
#443×C1272
#3292127