Finite Group Databasebeta

← #27150647

Subgroups — #27150647

Order 12416.

458 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 185 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#20261535C211yesmaximal, characteristic
2#20261501C212yesmaximal, complemented
3#20260858C214yesmaximal, complemented
4#20260853C214yesmaximal, complemented
5#20260784C211yesmaximal, characteristic, complemented
6C2×#15690738C212yesmaximal, complemented
7C4×M16×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2212yes—
9#15690843C414yescomplemented
10#15690838C414yescomplemented
11#15690832C2214yes—
12#15690818—24no—
13#15690816C2218yescomplemented
14C2×#13095495C2212yes—
15M16×D97—24no—
16#15690746C2214yescomplemented
17#15690738C2214yescomplemented
18#15690738C2214yescomplemented
19C2×M16×C97C2211yescharacteristic
20C2×M16×C97C2211yescharacteristic
21C2×C42×C97C2211yescharacteristic
22#97×C97C2212yescomplemented
23C4×C8×C97C2212yescomplemented
24#13095511—22no—
25#13095511—22no—
26#13095511C2×C414yes—
27#13095511C2314yes—
28C2×Dic194C2×C412yes—
29C2×D388C2×C412yes—
30C2×C22⋊D97—22no—
31C2×C4×D97—22no—
32Dic388C2×C414yescomplemented
33#13095500C2×C418yescomplemented
34D776C2×C414yescomplemented
35#13095495—22no—
36#13095495C2314yes—
37#13095494—28no—
38C2×#11715895—22no—
39C8×D97—28no—
40C22×C4×C97—21no—
41C22×C4×C97C2311yescharacteristic
42M16×C97C2×C414yes—
43M16×C97D414yes—
44C2×C8×C97C2312yes—
45C2×C8×C97C2312yes—
46C42×C97—21no—
47C42×C97C2311yescharacteristic
48C42×C97C2311yescharacteristic
49C23×C97#4111yescharacteristic
50C22×D97—22no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—42no—
54C2×C4×C97C22×C412yes—
55C2×C4×C97C2×D412yes—
56C2×C4×C97#4111yescharacteristic
57C2×C4×C97C2411yescharacteristic
58C22⋊D97—24no—
59C22⋊D97—24no—
60C2×Dic97—22no—
61D388—22no—
62D388C22×C414yes—
63C4×D97—24no—
64C4×D97—24no—
65Dic194—22no—
66Dic194C22×C414yes—
67C8×C97C22×C414yes—
68C8×C97C2×D414yes—
69#11715895—24no—
70C22×C97—41no—
71C22×C97C4×D412yes—
72C22×C97C2×#4111yescharacteristic
73C2×D97—24no—
74C2×D97—82no—
75C4×C97—22no—
76C4×C97—41no—
77C4×C97C4×D412yes—
78C4×C97C23×C411yescharacteristic
79C4×C97C22×D411yescharacteristic
80Dic97—24no—
81C2×C97—21no—
82C2×C97—22no—
83C2×C97C2×C4×D411yescharacteristic
84D97—44no—
85#2956—971nomaximal, Sylow
86C97#295611yescharacteristic, complemented, Sylow
87#575—971no—
88#568—972no—
89#444—974no—
90#443—974no—
91#430—971no—
92C2×#104—972no—
93C4×M16D9711yescharacteristic, complemented
94C2×#41—972no—
95#137—974no—
96AGL(1,ℤ/8)—974no—
97#135—974no—
98#131—978no—
99#131—1944no—
100C2×M16—972no—
101C2×M16—1944no—
102C2×M16C2×D9711yescharacteristic
103C2×M16C2×D9711yescharacteristic
104C2×C42C2×D9711yescharacteristic
105#108—974no—
106#104—974no—
107#104—974no—
108#97C2×D9712yescomplemented
109C4×C8C2×D9712yescomplemented
110#41—974no—
111#41—974no—
112#41—1942no—
113#41—1942no—
114C2×Q8—972no—
115C2×D4—972no—
116C2×D4—1942no—
117C22×C4—21no—
118C22×C4—1942no—
119C22×C4C22×D9711yescharacteristic
120Q16—974no—
121SD16—978no—
122D8—974no—
123M16—974no—
124M16—1942no—
125M16—1948no—
126M16C4×D9714yes—
127M16D38814yes—
128C2×C8—1942no—
129C2×C8—1948no—
130C2×C8C22×D9712yes—
131C2×C8C22×D9712yes—
132C42—21no—
133C42C22×D9711yescharacteristic
134C42C22×D9711yescharacteristic
135C23—1942no—
136C23#1309551111yescharacteristic
137Q8—974no—
138Q8—1942no—
139D4—974no—
140D4—1944no—
141D4—1944no—
142D4—1942no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—1942no—
147C2×C4—1944no—
148C2×C4—1944no—
149C2×C4C2×C4×D9712yes—
150C2×C4C2×D38812yes—
151C2×C4#1309551111yescharacteristic
152C2×C4C23×D9711yescharacteristic
153C8—1944no—
154C8C2×C4×D9714yes—
155C8C2×D38814yes—
156C22—41no—
157C22—7762no—
158C22—1944no—
159C22C4×D38812yes—
160C22C2×#1309551111yescharacteristic
161C4—22no—
162C4—41no—
163C4—1944no—
164C4C4×D38812yes—
165C4C22×C4×D9711yescharacteristic
166C4C22×D38811yescharacteristic
167C2—21no—
168C2—22no—
169C2—3884no—
170C2C2×C4×D38811yescharacteristic
subgroupG/H
centerC4C22×D388
commutatorC4×C97C23×C4
FrattiniC2×C4C23×D97
FittingC4×M16×C97C2
socleC2×C97C2×C4×D4
prime radical#20261535C2
cosocleC2×C4×C97C24
hypercenterC4×M16D97
hypocenterC97#2956
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×M16 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. #20261501C2semidirect
3. #20260858C2semidirect
4. #20260853C2semidirect
5. #20260784C2semidirect
6. C2×#15690738C2semidirect
7. C4×M16×C97C2semidirect
9. #15690843C4semidirect
10. #15690838C4semidirect
13. #15690816C22semidirect
16. #15690746C22semidirect
17. #15690738C22semidirect
18. #15690738C22semidirect
22. #97×C97C22semidirect
23. C4×C8×C97C22semidirect
32. Dic388C2×C4semidirect
33. #13095500C2×C4semidirect
34. D776C2×C4semidirect
86. C97#2956semidirect
93. C4×M16D97semidirect
108. #97C2×D97semidirect
109. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615352
#202615012
#202608582
#202608532
#202607842
C2×#156907382
C4×M16×C972
#295697