Finite Group Databasebeta

← #27136069

Subgroups — #27136069

Order 12352.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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#20259946C211yesmaximal, characteristic, complemented
2#20259944C211yesmaximal, characteristic, complemented
3#20259932C211yesmaximal, characteristic, complemented
4#20259928C212yesmaximal, complemented
5#20259923C212yesmaximal, complemented
6#20259919C211yesmaximal, characteristic, complemented
7#20259918C212yesmaximal, complemented
8C2×#15690543C211yesmaximal, characteristic, complemented
9SD16×D193C212yesmaximal, complemented
10#20259907C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C193C211yesmaximal, characteristic, complemented
12#15690555—21no—
13#15690555C2211yescharacteristic
14#15690555C2211yescharacteristic
15#15690554—21no—
16#15690554—22no—
17#15690554C2212yes—
18#15690554C2212yescomplemented
19C2×Dic386—21no—
20C2×Dic386C2211yescharacteristic, complemented
21Q8×D193C2212yescomplemented
22C2×C22⋊D193—22no—
23D4×D193—21no—
24D4×D193C2212yes—
25Dic772C2212yescomplemented
26#15690545C2211yescharacteristic, complemented
27#15690544C2212yescomplemented
28#15690543C2212yescomplemented
29#15690543C2212yescomplemented
30#15690543C2211yescharacteristic, complemented
31#15690542C2211yescharacteristic, complemented
32#15690540C2211yescharacteristic, complemented
33#15690538C2212yescomplemented
34C2×#13095393C2211yescharacteristic, complemented
35C8×D193C2212yescomplemented
36#41×C193C2211yescharacteristic, complemented
37C2×D4×C193C2211yescharacteristic
38SD16×C193C2212yescomplemented
39D8×C193C2212yescomplemented
40M16×C193C2211yescharacteristic, complemented
41C22×D193—22no—
42Dic386—21no—
43Dic386—22no—
44Dic386D411yescharacteristic
45Dic386C2312yes—
46Dic386C2311yescharacteristic
47D772D411yescharacteristic
48D772C2311yescharacteristic
49C22⋊D193—21no—
50C22⋊D193—22no—
51C22⋊D193—22no—
52C22⋊D193—22no—
53C22⋊D193D412yes—
54C2×Dic193—21no—
55C2×Dic193—22no—
56C2×Dic193—22no—
57C4×D193—22no—
58C4×D193—21no—
59C4×D193C2312yes—
60#13095393C2311yescharacteristic
61#13095393C2311yescharacteristic
62C23×C193—21no—
63Q8×C193C2311yescharacteristic
64D4×C193—21no—
65D4×C193—21no—
66D4×C193C2311yescharacteristic
67D4×C193C2312yes—
68C2×C4×C193—21no—
69C2×C4×C193C2311yescharacteristic
70C8×C193C2312yes—
71C22×C193—21no—
72C22×C193—22no—
73C22×C193—81no—
74C22×C193C2×D411yescharacteristic
75C2×D193—21no—
76C2×D193—82no—
77C2×D193C2×D412yes—
78C4×C193—21no—
79C4×C193C2×D411yescharacteristic
80C4×C193C2411yescharacteristic
81Dic193—21no—
82Dic193—22no—
83Dic193C2×D412yes—
84C2×C193—21no—
85C2×C193—42no—
86C2×C193—41no—
87C2×C193C22×D411yescharacteristic
88D193—22no—
89D193—41no—
90C193#57711yescharacteristic, complemented, Sylow
91#577—1931nomaximal, Sylow
922−1+4—1931no—
93H(5,𝔽2)—1931no—
94#137—1932no—
95#137—1931no—
96AGL(1,ℤ/8)—1932no—
97AGL(1,ℤ/8)D19311yescharacteristic, complemented
98#135—1931no—
99#135—1932no—
100C2×SD16—1931no—
101C2×SD16—1932no—
102#131—1931no—
103#41—3861no—
104#41—3861no—
105#41—3862no—
106#41—1932no—
107#41—1932no—
108#41—1931no—
109#41—1931no—
110#41C2×D19311yescharacteristic, complemented
111C2×Q8—3861no—
112C2×Q8—1932no—
113C2×Q8—1931no—
114C2×D4—3861no—
115C2×D4—3862no—
116C2×D4—1932no—
117C2×D4C2×D19311yescharacteristic
118Q16—1931no—
119Q16—1932no—
120SD16—1931no—
121SD16—1932no—
122SD16—1932no—
123SD16—1931no—
124SD16—1932no—
125SD16C2×D19312yescomplemented
126D8—1931no—
127D8C2×D19312yescomplemented
128M16—1932no—
129M16C2×D19311yescharacteristic, complemented
130C2×C8—1931no—
131C2×C8—1932no—
132C23—21no—
133C23—3862no—
134Q8—3862no—
135Q8—3861no—
136Q8—1932no—
137Q8—1931no—
138Q8—1931no—
139Q8C22×D19311yescharacteristic
140D4—21no—
141D4—21no—
142D4—3862no—
143D4—3862no—
144D4—3861no—
145D4—3862no—
146D4—1932no—
147D4—1931no—
148D4—1931no—
149D4C22×D19311yescharacteristic
150D4C22×D19312yes—
151C2×C4—21no—
152C2×C4—3862no—
153C2×C4—3861no—
154C2×C4—3862no—
155C2×C4—3862no—
156C2×C4—3861no—
157C2×C4—1932no—
158C2×C4C22×D19311yescharacteristic
159C8—1931no—
160C8—1931no—
161C8C22×D19312yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—15442no—
166C22—3861no—
167C22—1932no—
168C22D4×D19311yescharacteristic
169C4—21no—
170C4—3861no—
171C4—3862no—
172C4—1932no—
173C4D4×D19311yescharacteristic
174C4C23×D19311yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—7721no—
179C2—3862no—
180C2C2×D4×D19311yescharacteristic
subgroupG/H
centerC2C2×D4×D193
commutator, cosocleC4×C193C24
FrattiniC4C23×D193
FittingAGL(1,ℤ/8)×C193C2
socleC2×C193C22×D4
hypercenterAGL(1,ℤ/8)D193
hypocenterC193#577
LCSG ▷ C4×C193 ▷ C2×C193 ▷ C193
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20259946C2semidirect
2. #20259944C2semidirect
3. #20259932C2semidirect
4. #20259928C2semidirect
5. #20259923C2semidirect
6. #20259919C2semidirect
7. #20259918C2semidirect
8. C2×#15690543C2semidirect
9. SD16×D193C2semidirect
10. #20259907C2semidirect
11. AGL(1,ℤ/8)×C193C2semidirect
18. #15690554C22semidirect
20. C2×Dic386C22semidirect
21. Q8×D193C22semidirect
25. Dic772C22semidirect
26. #15690545C22semidirect
27. #15690544C22semidirect
28. #15690543C22semidirect
29. #15690543C22semidirect
30. #15690543C22semidirect
31. #15690542C22semidirect
32. #15690540C22semidirect
33. #15690538C22semidirect
34. C2×#13095393C22semidirect
35. C8×D193C22semidirect
36. #41×C193C22semidirect
38. SD16×C193C22semidirect
39. D8×C193C22semidirect
40. M16×C193C22semidirect
90. C193#577semidirect
97. AGL(1,ℤ/8)D193semidirect
110. #41C2×D193semidirect
125. SD16C2×D193semidirect
127. D8C2×D193semidirect
129. M16C2×D193semidirect

Maximal subgroups

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

subgroupindex
#202599462
#202599442
#202599322
#202599282
#202599232
#202599192
#202599182
C2×#156905432
SD16×D1932
#202599072
AGL(1,ℤ/8)×C1932
#577193