Finite Group Databasebeta

← #27153536

Subgroups — #27153536

Order 12416.

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#20261540C212yesmaximal, complemented
2#20261537C212yesmaximal, complemented
3#20261112C211yesmaximal, characteristic, complemented
4#20261006C212yesmaximal, complemented
5#20261005C212yesmaximal, complemented
6#20260904C214yesmaximal, complemented
7#20260853C211yesmaximal, characteristic, complemented
8#443×C97C211yesmaximal, characteristic, complemented
9#15690856C2214yescomplemented
10#15690838—24no—
11#15690838C2214yescomplemented
12#15690837—24no—
13#15690835C2214yescomplemented
14#15690833—22no—
15#15690832—22no—
16#15690832C2211yescharacteristic
17C2×D776—22no—
18C2×D776C2212yescomplemented
19C2×D776C2212yescomplemented
20D8×D97—22no—
21#15690816C2212yescomplemented
22#15690807C2212yescomplemented
23#15690746C2211yescharacteristic, complemented
24#15690738C2212yescomplemented
25#15690726C2214yescomplemented
26#135×C97C2211yescharacteristic
27#131×C97C2212yescomplemented
28#108×C97C2211yescharacteristic, complemented
29#104×C97C2212yescomplemented
30C4×C8×C97C2211yescharacteristic, complemented
31#13095512—24no—
32#13095511—24no—
33#13095511C2312yes—
34C2×D388—24no—
35C2×D388—42no—
36C2×D388C2314yes—
37D4×D97—24no—
38D4×D97—44no—
39Dic388D411yescharacteristic, complemented
40#13095500—24no—
41#13095500D412yescomplemented
42#13095498—24no—
43D776—24no—
44D776—22no—
45D776—22no—
46D776—22no—
47D776—22no—
48D776D411yescharacteristic, complemented
49#13095496—24no—
50#13095495C2312yes—
51#13095494—22no—
52C8×D97—22no—
53#41×C97C2312yes—
54Q16×C97D411yescharacteristic, complemented
55SD16×C97D412yescomplemented
56D8×C97D411yescharacteristic, complemented
57M16×C97—22no—
58M16×C97C2312yes—
59C2×C8×C97—22no—
60C2×C8×C97C2311yescharacteristic
61C2×C8×C97C2311yescharacteristic
62C42×C97C2311yescharacteristic
63C22×D97—24no—
64C22×D97—44no—
65Q8×C97C2×D412yes—
66D4×C97—22no—
67D4×C97C2×D412yes—
68C2×C4×C97—21no—
69C2×C4×C97—22no—
70C2×C4×C97C2411yescharacteristic
71C22⋊D97—22no—
72C22⋊D97—44no—
73D388—24no—
74D388—24no—
75D388—44no—
76D388—84no—
77D388C2×D412yes—
78C4×D97—22no—
79C4×D97—44no—
80Dic194C2×D412yes—
81C8×C97—22no—
82C8×C97C2×D411yescharacteristic
83C8×C97C2×D411yescharacteristic
84C8×C97C2×D411yescharacteristic
85C8×C97C2×D411yescharacteristic
86#11715895—22no—
87C22×C97—22no—
88C22×C97H(5,𝔽2)11yescharacteristic
89C2×D97—22no—
90C2×D97—44no—
91C2×D97—84no—
92C2×D97—164no—
93C4×C97—22no—
94C4×C97—22no—
95C4×C97C22×D411yescharacteristic
96C4×C97C22×D411yescharacteristic
97Dic97—22no—
98C2×C97—21no—
99C2×C97—42no—
100C2×C97D4211yescharacteristic
101D97—42no—
102D97—84no—
103#3292—971nomaximal, Sylow
104C97#329211yescharacteristic, complemented, Sylow
105#576—972no—
106#576—972no—
107#493—971no—
108#472—972no—
109#472—972no—
110#453—974no—
111#443—971no—
112#443D9711yescharacteristic, complemented
113H(5,𝔽2)—974no—
114AGL(1,ℤ/8)—974no—
115AGL(1,ℤ/8)—974no—
116AGL(1,ℤ/8)—1944no—
117AGL(1,ℤ/8)—1944no—
118#135—971no—
119#135—1942no—
120#135—1942no—
121#135C2×D9711yescharacteristic
122C2×D8—972no—
123C2×D8—972no—
124C2×D8—1942no—
125C2×D8—1942no—
126#131—972no—
127#131C2×D9712yescomplemented
128#127—972no—
129#108—971no—
130#108C2×D9711yescharacteristic, complemented
131#104—972no—
132#104C2×D9712yescomplemented
133#100—974no—
134C4×C8C2×D9711yescharacteristic, complemented
135#41—972no—
136#41—1944no—
137#41—1944no—
138#41C22×D9712yes—
139C2×D4—974no—
140C2×D4—3882no—
141C2×D4—3884no—
142C2×D4—1944no—
143C2×D4—1944no—
144Q16—971no—
145Q16D38811yescharacteristic, complemented
146SD16—972no—
147SD16—1944no—
148SD16—1944no—
149SD16D38812yescomplemented
150D8—971no—
151D8—1944no—
152D8—1944no—
153D8—1942no—
154D8—1942no—
155D8—1942no—
156D8—1942no—
157D8D38811yescharacteristic, complemented
158M16—22no—
159M16—972no—
160M16—1942no—
161M16C22×D9712yes—
162C2×C8—22no—
163C2×C8—1942no—
164C2×C8C22×D9711yescharacteristic
165C2×C8C22×D9711yescharacteristic
166C42C22×D9711yescharacteristic
167C23—3884no—
168C23—1944no—
169Q8—972no—
170Q8C2×D38812yes—
171D4—22no—
172D4—7764no—
173D4—972no—
174D4—3884no—
175D4—3884no—
176D4—1942no—
177D4—1944no—
178D4—1944no—
179D4C2×D38812yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—3884no—
183C2×C4—1942no—
184C2×C4C23×D9711yescharacteristic
185C8—22no—
186C8—1942no—
187C8D4×D9711yescharacteristic
188C8D4×D9711yescharacteristic
189C8C2×D38811yescharacteristic
190C8C2×D38811yescharacteristic
191C22—22no—
192C22—7764no—
193C22—15524no—
194C22—3884no—
195C22—1942no—
196C22#1569085611yescharacteristic
197C4—22no—
198C4—22no—
199C4—1942no—
200C4C2×D4×D9711yescharacteristic
201C4C22×D38811yescharacteristic
202C2—21no—
203C2—42no—
204C2—7764no—
205C2—3882no—
206C2D4×D38811yescharacteristic
subgroupG/H
centerC2D4×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#443×C97C2
socleC2×C97D42
hypercenter#443D97
hypocenterC97#3292
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #20261540C2semidirect
2. #20261537C2semidirect
3. #20261112C2semidirect
4. #20261006C2semidirect
5. #20261005C2semidirect
6. #20260904C2semidirect
7. #20260853C2semidirect
8. #443×C97C2semidirect
9. #15690856C22semidirect
11. #15690838C22semidirect
13. #15690835C22semidirect
18. C2×D776C22semidirect
19. C2×D776C22semidirect
21. #15690816C22semidirect
22. #15690807C22semidirect
23. #15690746C22semidirect
24. #15690738C22semidirect
25. #15690726C22semidirect
27. #131×C97C22semidirect
28. #108×C97C22semidirect
29. #104×C97C22semidirect
30. C4×C8×C97C22semidirect
39. Dic388D4semidirect
41. #13095500D4semidirect
48. D776D4semidirect
54. Q16×C97D4semidirect
55. SD16×C97D4semidirect
56. D8×C97D4semidirect
104. C97#3292semidirect
112. #443D97semidirect
127. #131C2×D97semidirect
130. #108C2×D97semidirect
132. #104C2×D97semidirect
134. C4×C8C2×D97semidirect
145. Q16D388semidirect
149. SD16D388semidirect
157. D8D388semidirect

Maximal subgroups

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

subgroupindex
#202615402
#202615372
#202611122
#202610062
#202610052
#202609042
#202608532
#443×C972
#329297