Finite Group Databasebeta

← #27135731

Subgroups — #27135731

Order 12352.

338 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×D4×D193C211yesmaximal, characteristic
2C4×D772C211yesmaximal, characteristic, complemented
3D4×Dic193C211yesmaximal, characteristic, complemented
4#20259861C211yesmaximal, characteristic, complemented
5C4×C22⋊D193C212yesmaximal, complemented
6#20259859C212yesmaximal, complemented
7#20259857C211yesmaximal, characteristic, complemented
8#20259856C211yesmaximal, characteristic, complemented
9C2×#15690532C212yesmaximal, complemented
10C22⋊C4×D193C212yesmaximal, complemented
11C4×D4×C193C211yesmaximal, characteristic, complemented
12C23×D193C2212yes—
13C2×D772C2211yescharacteristic
14C2×C22⋊D193C2212yes—
15D4×D193C418yescomplemented
16C22×Dic193C2212yescomplemented
17C2×C4×D193C2211yescharacteristic
18C2×C4×D193C2212yescomplemented
19C2×C4×D193C2212yescomplemented
20C4⋊Dic193C2211yescharacteristic, complemented
21#15690534C2211yescharacteristic, complemented
22#15690534C2211yescharacteristic, complemented
23C22⋊Dic193C2212yescomplemented
24#15690532C2211yescharacteristic
25#15690532C2212yes—
26#15690532C2211yescharacteristic
27#15690532C2212yescomplemented
28#15690532C2212yescomplemented
29C4×Dic193C2211yescharacteristic, complemented
30C4×Dic193C2212yescomplemented
31C2×D4×C193C2211yescharacteristic
32C22×C4×C193C2212yescomplemented
33C4⋊C4×C193C2211yescharacteristic, complemented
34C22⋊C4×C193C2212yescomplemented
35C42×C193C2211yescharacteristic, complemented
36C22×D193—42no—
37C22×D193—42no—
38C22×D193C2×C418yescomplemented
39C22×D193C2311yescharacteristic
40C22×D193C2312yes—
41D772C2×C414yescomplemented
42C22⋊D193C2×C418yescomplemented
43C2×Dic193—42no—
44C2×Dic193C2311yescharacteristic
45C2×Dic193C2312yes—
46C2×Dic193C2311yescharacteristic
47C2×Dic193C2311yescharacteristic
48C4×D193—42no—
49C4×D193—42no—
50C4×D193C2×C414yescomplemented
51C23×C193C2312yes—
52D4×C193C2×C414yescomplemented
53C2×C4×C193—42no—
54C2×C4×C193C2311yescharacteristic
55C2×C4×C193C2311yescharacteristic
56C2×C4×C193C2312yes—
57C2×C4×C193C2311yescharacteristic
58C22×C193—22no—
59C22×C193—22no—
60C22×C193C22×C414yes—
61C22×C193C2411yescharacteristic
62C2×D193—22no—
63C2×D193—22no—
64C2×D193—21no—
65C2×D193—21no—
66C2×D193—48no—
67C2×D193C22×C412yes—
68C2×D193C22×C414yes—
69C4×C193—21no—
70C4×C193—22no—
71C4×C193—21no—
72C4×C193C22×C412yes—
73Dic193—21no—
74Dic193—22no—
75Dic193—21no—
76Dic193C22×C412yes—
77C2×C193—24no—
78C2×C193C23×C411yescharacteristic
79C2×C193H(5,𝔽2)11yescharacteristic
80C2×C193H(5,𝔽2)11yescharacteristic
81D193—22no—
82D193—24no—
83C193#51811yescharacteristic, complemented, Sylow
84#518—1931nomaximal, Sylow
85C22×D4—1931no—
86C4×D4—1931no—
87C4×D4—1932no—
88C4×D4—1931no—
89C4×D4—1932no—
90C4×D4—1931no—
91C4×D4D19311yescharacteristic, complemented
92#117—1931no—
93#117—1931no—
94C2×C22⋊C4—1932no—
95C2×C22⋊C4—1932no—
96C24—1932no—
97C2×D4—1932no—
98C2×D4—1938no—
99C2×D4—1931no—
100C2×D4C2×D19311yescharacteristic
101C22×C4—1932no—
102C22×C4—1932no—
103C22×C4—1932no—
104C22×C4—1931no—
105C22×C4C2×D19312yescomplemented
106C4⋊C4—1931no—
107C4⋊C4—1931no—
108C4⋊C4—1931no—
109C4⋊C4C2×D19311yescharacteristic, complemented
110C22⋊C4—1932no—
111C22⋊C4—1932no—
112C22⋊C4—1931no—
113C22⋊C4—1931no—
114C22⋊C4—1932no—
115C22⋊C4—1932no—
116C22⋊C4C2×D19312yescomplemented
117C42—1931no—
118C42—1932no—
119C42C2×D19311yescharacteristic, complemented
120C23—7722no—
121C23—7722no—
122C23—1938no—
123C23—1931no—
124C23—1932no—
125C23C22×D19312yes—
126D4—1938no—
127D4—1934no—
128D4C4×D19314yescomplemented
129C2×C4—42no—
130C2×C4—7722no—
131C2×C4—7722no—
132C2×C4—7722no—
133C2×C4—1931no—
134C2×C4—1932no—
135C2×C4—1931no—
136C2×C4—1931no—
137C2×C4—1934no—
138C2×C4C22×D19312yes—
139C2×C4C22×D19311yescharacteristic
140C2×C4C22×D19311yescharacteristic
141C2×C4C22×D19311yescharacteristic
142C22—22no—
143C22—22no—
144C22—7728no—
145C22—3862no—
146C22—3861no—
147C22—3862no—
148C22—3861no—
149C22—1932no—
150C22—1934no—
151C22C2×C4×D19314yes—
152C22C23×D19311yescharacteristic
153C4—21no—
154C4—22no—
155C4—21no—
156C4—3861no—
157C4—3862no—
158C4—3861no—
159C4—1932no—
160C4C2×C4×D19312yes—
161C2—24no—
162C2—3862no—
163C2—3864no—
164C2C22×C4×D19311yescharacteristic
165C2#2025994411yescharacteristic
166C2#2025994511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D193
commutatorC2×C193C23×C4
FittingC4×D4×C193C2
socle, cosocleC22×C193C24
prime radicalC2×D4×D193C2
hypercenterC4×D4D193
hypocenterC193#518
LCSG ▷ C2×C193 ▷ C193
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×D772C2semidirect
3. D4×Dic193C2semidirect
4. #20259861C2semidirect
5. C4×C22⋊D193C2semidirect
6. #20259859C2semidirect
7. #20259857C2semidirect
8. #20259856C2semidirect
9. C2×#15690532C2semidirect
10. C22⋊C4×D193C2semidirect
11. C4×D4×C193C2semidirect
15. D4×D193C4semidirect
16. C22×Dic193C22semidirect
18. C2×C4×D193C22semidirect
19. C2×C4×D193C22semidirect
20. C4⋊Dic193C22semidirect
21. #15690534C22semidirect
22. #15690534C22semidirect
23. C22⋊Dic193C22semidirect
27. #15690532C22semidirect
28. #15690532C22semidirect
29. C4×Dic193C22semidirect
30. C4×Dic193C22semidirect
32. C22×C4×C193C22semidirect
33. C4⋊C4×C193C22semidirect
34. C22⋊C4×C193C22semidirect
35. C42×C193C22semidirect
38. C22×D193C2×C4semidirect
41. D772C2×C4semidirect
42. C22⋊D193C2×C4semidirect
50. C4×D193C2×C4semidirect
52. D4×C193C2×C4semidirect
83. C193#518semidirect
91. C4×D4D193semidirect
105. C22×C4C2×D193semidirect
109. C4⋊C4C2×D193semidirect
116. C22⋊C4C2×D193semidirect
119. C42C2×D193semidirect
128. D4C4×D193semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D1932
C4×D7722
D4×Dic1932
#202598612
C4×C22⋊D1932
#202598592
#202598572
#202598562
C2×#156905322
C22⋊C4×D1932
C4×D4×C1932
#518193