Finite Group Databasebeta

← #27136055

Subgroups — #27136055

Order 12352.

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

normal subgroups only

This group has 262 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
1C2×#15690554C211yesmaximal, characteristic, complemented
2#41×D193C211yesmaximal, characteristic, complemented
3#20259932C211yesmaximal, characteristic, complemented
4#20259929C211yesmaximal, characteristic, complemented
5#20259928C212yesmaximal, complemented
6#20259925C211yesmaximal, characteristic, complemented
7#20259923C212yesmaximal, complemented
8#20259920C212yesmaximal, complemented
9#20259918C212yesmaximal, complemented
10M16×D193C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C193C211yesmaximal, characteristic, complemented
12#15690556C2211yescharacteristic, complemented
13#15690555—21no—
14#15690555C2211yescharacteristic
15#15690554—21no—
16#15690554—21no—
17#15690554—21no—
18#15690554C2212yes—
19#15690554C2211yescharacteristic, complemented
20#15690554C2212yescomplemented
21C2×Dic386C2211yescharacteristic, complemented
22Q8×D193C2211yescharacteristic, complemented
23C2×C22⋊D193—21no—
24D4×D193—21no—
25D4×D193C2211yescharacteristic, complemented
26C22×Dic193—21no—
27C2×C4×D193—21no—
28C2×C4×D193C2211yescharacteristic
29Dic772C2212yescomplemented
30#15690545C2212yescomplemented
31#15690544C2212yescomplemented
32#15690543C2212yescomplemented
33#15690543C2212yescomplemented
34#15690540C2212yescomplemented
35#15690539C2211yescharacteristic, complemented
36#15690538C2212yescomplemented
37C8×D193C2212yescomplemented
38#41×C193C2211yescharacteristic
39C2×D4×C193C2211yescharacteristic, complemented
40SD16×C193C2212yescomplemented
41D8×C193C2212yescomplemented
42M16×C193C2211yescharacteristic, complemented
43C22×D193—21no—
44C22×D193D411yescharacteristic
45Dic386—21no—
46Dic386—21no—
47Dic386C2311yescharacteristic
48Dic386C2312yes—
49D772—21no—
50D772C2311yescharacteristic
51C22⋊D193—21no—
52C22⋊D193—21no—
53C22⋊D193—22no—
54C22⋊D193—22no—
55C22⋊D193—21no—
56C2×Dic193—21no—
57C2×Dic193—21no—
58C2×Dic193—22no—
59C2×Dic193—22no—
60C2×Dic193D411yescharacteristic
61C4×D193—21no—
62C4×D193—21no—
63C4×D193—21no—
64C4×D193D411yescharacteristic
65C4×D193D411yescharacteristic
66C4×D193C2311yescharacteristic
67C4×D193C2311yescharacteristic
68#13095393C2312yes—
69C23×C193—21no—
70Q8×C193C2311yescharacteristic
71D4×C193—21no—
72D4×C193—21no—
73D4×C193C2311yescharacteristic
74D4×C193C2312yes—
75C2×C4×C193—21no—
76C2×C4×C193C2311yescharacteristic
77C8×C193C2312yes—
78C22×C193—21no—
79C22×C193—22no—
80C22×C193—42no—
81C22×C193C2×D411yescharacteristic
82C2×D193—21no—
83C2×D193—41no—
84C2×D193—42no—
85C2×D193C2×D411yescharacteristic
86C2×D193C2×D411yescharacteristic
87C4×C193—21no—
88C4×C193C2×D411yescharacteristic
89C4×C193C2411yescharacteristic
90Dic193—21no—
91Dic193—22no—
92Dic193C2×D411yescharacteristic
93Dic193C2×D411yescharacteristic
94C2×C193—21no—
95C2×C193—41no—
96C2×C193—42no—
97C2×C193C22×D411yescharacteristic
98D193—21no—
99D193—21no—
100D193—41no—
101C193#57511yescharacteristic, complemented, Sylow
102#575—1931nomaximal, Sylow
103C2×#41—1931no—
104C2×#41—1931no—
105#137—1931no—
106#137—1932no—
107#137—1931no—
108AGL(1,ℤ/8)—1931no—
109AGL(1,ℤ/8)—1932no—
110AGL(1,ℤ/8)D19311yescharacteristic, complemented
111#135—1932no—
112#135—1932no—
113C2×M16—1931no—
114#41—3861no—
115#41—3861no—
116#41—3861no—
117#41—3861no—
118#41—1931no—
119#41—1932no—
120#41—1932no—
121#41—1931no—
122#41—1931no—
123#41C2×D19311yescharacteristic
124C2×Q8—1931no—
125C2×Q8—1931no—
126C2×D4—3861no—
127C2×D4—3861no—
128C2×D4—1931no—
129C2×D4C2×D19311yescharacteristic, complemented
130C22×C4—3861no—
131C22×C4—3861no—
132C22×C4—1931no—
133Q16—1932no—
134Q16—1932no—
135SD16—1932no—
136SD16—1932no—
137SD16—1932no—
138SD16C2×D19312yescomplemented
139D8—1932no—
140D8C2×D19312yescomplemented
141M16—1931no—
142M16—1932no—
143M16C2×D19311yescharacteristic, complemented
144C2×C8—1932no—
145C23—21no—
146C23—3861no—
147C23—1931no—
148Q8—3861no—
149Q8—3861no—
150Q8—1931no—
151Q8—1932no—
152Q8C22×D19311yescharacteristic
153D4—21no—
154D4—21no—
155D4—3861no—
156D4—3861no—
157D4—3862no—
158D4—3862no—
159D4—3861no—
160D4—3861no—
161D4—1931no—
162D4C22×D19312yes—
163D4C22×D19311yescharacteristic
164C2×C4—21no—
165C2×C4—3861no—
166C2×C4—3861no—
167C2×C4—3861no—
168C2×C4—3861no—
169C2×C4—3862no—
170C2×C4—3862no—
171C2×C4—3861no—
172C2×C4—1931no—
173C2×C4—1931no—
174C2×C4—1931no—
175C2×C4—1931no—
176C2×C4—1931no—
177C2×C4C22×D19311yescharacteristic
178C8—1932no—
179C8C22×D19312yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—7721no—
184C22—7722no—
185C22—3861no—
186C22—1931no—
187C22—1931no—
188C22D4×D19311yescharacteristic
189C4—21no—
190C4—3861no—
191C4—3862no—
192C4—1931no—
193C4—1931no—
194C4D4×D19311yescharacteristic
195C4C23×D19311yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—7721no—
200C2—3861no—
201C2—3861no—
202C2C2×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#575
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. C2×#15690554C2semidirect
2. #41×D193C2semidirect
3. #20259932C2semidirect
4. #20259929C2semidirect
5. #20259928C2semidirect
6. #20259925C2semidirect
7. #20259923C2semidirect
8. #20259920C2semidirect
9. #20259918C2semidirect
10. M16×D193C2semidirect
11. AGL(1,ℤ/8)×C193C2semidirect
12. #15690556C22semidirect
19. #15690554C22semidirect
20. #15690554C22semidirect
21. C2×Dic386C22semidirect
22. Q8×D193C22semidirect
25. D4×D193C22semidirect
29. Dic772C22semidirect
30. #15690545C22semidirect
31. #15690544C22semidirect
32. #15690543C22semidirect
33. #15690543C22semidirect
34. #15690540C22semidirect
35. #15690539C22semidirect
36. #15690538C22semidirect
37. C8×D193C22semidirect
39. C2×D4×C193C22semidirect
40. SD16×C193C22semidirect
41. D8×C193C22semidirect
42. M16×C193C22semidirect
101. C193#575semidirect
110. AGL(1,ℤ/8)D193semidirect
129. C2×D4C2×D193semidirect
138. SD16C2×D193semidirect
140. D8C2×D193semidirect
143. M16C2×D193semidirect

Maximal subgroups

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

subgroupindex
C2×#156905542
#41×D1932
#202599322
#202599292
#202599282
#202599252
#202599232
#202599202
#202599182
M16×D1932
AGL(1,ℤ/8)×C1932
#575193