Finite Group Databasebeta

← #27136063

Subgroups — #27136063

Order 12352.

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

normal subgroups only

This group has 268 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#20259945C211yesmaximal, characteristic, complemented
2#20259944C211yesmaximal, characteristic, complemented
3#20259927C211yesmaximal, characteristic, complemented
4#20259926C212yesmaximal, complemented
5#20259923C212yesmaximal, complemented
6#20259920C212yesmaximal, complemented
7#20259919C211yesmaximal, characteristic, complemented
8C2×#15690540C211yesmaximal, characteristic, complemented
9D8×D193C212yesmaximal, complemented
10#20259907C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C193C211yesmaximal, characteristic, complemented
12#15690556C2212yescomplemented
13#15690555—21no—
14#15690555C2211yescharacteristic
15#15690555C2211yescharacteristic, complemented
16#15690554—21no—
17#15690554C2212yescomplemented
18C2×D772—21no—
19C2×D772C2211yescharacteristic, complemented
20C2×C22⋊D193—22no—
21D4×D193—21no—
22D4×D193—22no—
23D4×D193C2212yescomplemented
24D4×D193C2212yescomplemented
25#15690545C2211yescharacteristic, complemented
26#15690544C2212yescomplemented
27#15690543C2211yescharacteristic, complemented
28#15690542C2211yescharacteristic, complemented
29D1544C2212yescomplemented
30#15690540C2211yescharacteristic, complemented
31#15690540C2212yescomplemented
32#15690540C2212yescomplemented
33#15690538C2212yescomplemented
34C2×#13095393C2211yescharacteristic, complemented
35C8×D193C2212yescomplemented
36#41×C193C2211yescharacteristic, complemented
37C2×D4×C193C2211yescharacteristic, complemented
38SD16×C193C2212yescomplemented
39D8×C193C2212yescomplemented
40M16×C193C2211yescharacteristic, complemented
41C22×D193—21no—
42C22×D193—22no—
43C22×D193—22no—
44Dic386D411yescharacteristic
45Dic386C2311yescharacteristic
46D772—22no—
47D772—21no—
48D772D411yescharacteristic
49D772C2312yes—
50D772C2311yescharacteristic
51C22⋊D193—21no—
52C22⋊D193—22no—
53C22⋊D193—22no—
54C22⋊D193—22no—
55C22⋊D193D412yes—
56C2×Dic193—22no—
57C4×D193—21no—
58C4×D193—22no—
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—22no—
77C2×D193—82no—
78C2×D193—82no—
79C2×D193—81no—
80C2×D193C2×D412yes—
81C4×C193—21no—
82C4×C193C2×D411yescharacteristic
83C4×C193C2411yescharacteristic
84Dic193—21no—
85Dic193C2×D412yes—
86C2×C193—21no—
87C2×C193—41no—
88C2×C193—42no—
89C2×C193C22×D411yescharacteristic
90D193—22no—
91D193—41no—
92D193—42no—
93C193#57611yescharacteristic, complemented, Sylow
94#576—1931nomaximal, Sylow
95H(5,𝔽2)—1931no—
96H(5,𝔽2)—1931no—
97AGL(1,ℤ/8)—1932no—
98AGL(1,ℤ/8)—1932no—
99AGL(1,ℤ/8)—1931no—
100AGL(1,ℤ/8)D19311yescharacteristic, complemented
101#135—1931no—
102#135—1932no—
103C2×D8—1931no—
104C2×D8—1932no—
105#131—1931no—
106#41—3861no—
107#41—3861no—
108#41—1932no—
109#41—1932no—
110#41—1931no—
111#41—1931no—
112#41C2×D19311yescharacteristic, complemented
113C2×D4—3861no—
114C2×D4—3862no—
115C2×D4—3861no—
116C2×D4—3862no—
117C2×D4—1932no—
118C2×D4—1932no—
119C2×D4—1931no—
120C2×D4C2×D19311yescharacteristic, complemented
121Q16—1931no—
122SD16—1931no—
123SD16—1931no—
124SD16—1932no—
125SD16C2×D19312yescomplemented
126D8—1932no—
127D8—1931no—
128D8—1932no—
129D8—1932no—
130D8C2×D19312yescomplemented
131M16—1932no—
132M16C2×D19311yescharacteristic, complemented
133C2×C8—1931no—
134C2×C8—1932no—
135C23—21no—
136C23—3862no—
137C23—3862no—
138C23—3861no—
139Q8—1931no—
140Q8—1931no—
141Q8C22×D19311yescharacteristic
142D4—21no—
143D4—21no—
144D4—3862no—
145D4—3862no—
146D4—3861no—
147D4—3862no—
148D4—3862no—
149D4—3861no—
150D4—1932no—
151D4—1931no—
152D4—1932no—
153D4—1931no—
154D4C22×D19312yes—
155D4C22×D19311yescharacteristic
156C2×C4—21no—
157C2×C4—3862no—
158C2×C4—3862no—
159C2×C4—3861no—
160C2×C4—1932no—
161C2×C4C22×D19311yescharacteristic
162C8—1931no—
163C8—1931no—
164C8C22×D19312yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—15441no—
169C22—15442no—
170C22—15442no—
171C22—3862no—
172C22—3861no—
173C22—1932no—
174C22D4×D19311yescharacteristic
175C4—21no—
176C4—3861no—
177C4—1932no—
178C4D4×D19311yescharacteristic
179C4C23×D19311yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—7722no—
184C2—7721no—
185C2—3862no—
186C2C2×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#576
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. #20259945C2semidirect
2. #20259944C2semidirect
3. #20259927C2semidirect
4. #20259926C2semidirect
5. #20259923C2semidirect
6. #20259920C2semidirect
7. #20259919C2semidirect
8. C2×#15690540C2semidirect
9. D8×D193C2semidirect
10. #20259907C2semidirect
11. AGL(1,ℤ/8)×C193C2semidirect
12. #15690556C22semidirect
15. #15690555C22semidirect
17. #15690554C22semidirect
19. C2×D772C22semidirect
23. D4×D193C22semidirect
24. D4×D193C22semidirect
25. #15690545C22semidirect
26. #15690544C22semidirect
27. #15690543C22semidirect
28. #15690542C22semidirect
29. D1544C22semidirect
30. #15690540C22semidirect
31. #15690540C22semidirect
32. #15690540C22semidirect
33. #15690538C22semidirect
34. C2×#13095393C22semidirect
35. C8×D193C22semidirect
36. #41×C193C22semidirect
37. C2×D4×C193C22semidirect
38. SD16×C193C22semidirect
39. D8×C193C22semidirect
40. M16×C193C22semidirect
93. C193#576semidirect
100. AGL(1,ℤ/8)D193semidirect
112. #41C2×D193semidirect
120. C2×D4C2×D193semidirect
125. SD16C2×D193semidirect
130. D8C2×D193semidirect
132. M16C2×D193semidirect

Maximal subgroups

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

subgroupindex
#202599452
#202599442
#202599272
#202599262
#202599232
#202599202
#202599192
C2×#156905402
D8×D1932
#202599072
AGL(1,ℤ/8)×C1932
#576193