Finite Group Databasebeta

← #14583460

Subgroups — #14583460

Order 2752.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×D172C211yesmaximal, characteristic, complemented
2C2×D4×D43C211yesmaximal, characteristic, complemented
3#13063310C212yesmaximal, complemented
4#13063308C211yesmaximal, characteristic, complemented
5#13063300C212yesmaximal, complemented
6#13063298C212yesmaximal, complemented
7#13063295C212yesmaximal, complemented
8#13063295C211yesmaximal, characteristic, complemented
9#13063293C211yesmaximal, characteristic, complemented
10C22⋊C4×D43C211yesmaximal, characteristic, complemented
11#123×C43C211yesmaximal, characteristic, complemented
12C23×D43C2211yescharacteristic
13C23×D43C2211yescharacteristic, complemented
14C23×D43C2211yescharacteristic, complemented
15C2×D4×C43C2211yescharacteristic
16C22×C4×C43C2211yescharacteristic, complemented
17C2×C22⋊D43C2211yescharacteristic
18C2×C22⋊D43C2211yescharacteristic, complemented
19C2×C22⋊D43C2212yescomplemented
20D4×D43—24no—
21C2×D172—24no—
22C2×D172C2212yes—
23C2×D172C2212yescomplemented
24C2×D172C2211yescharacteristic, complemented
25C2×D172C2212yescomplemented
26C2×C4×D43C2211yescharacteristic, complemented
27C2×C4×D43C2212yescomplemented
28C4⋊C4×C43C2212yescomplemented
29C22⋊C4×C43C2211yescharacteristic, complemented
30C22⋊C4×C43C2212yescomplemented
31C22⋊Dic43C2211yescharacteristic, complemented
32(C2×D43)⋊C4C2212yescomplemented
33(C2×D43)⋊C4C2212yescomplemented
34(C2×D43)⋊C4C2212yescomplemented
35(C2×D43)⋊C4C2212yescomplemented
36Dic43⋊C4C2212yescomplemented
37C23×C43C2311yescharacteristic
38C23×C43C2311yescharacteristic
39C22×D43—22no—
40C22×D43—22no—
41C22×D43—42no—
42C22×D43—42no—
43C22×D43—42no—
44C22×D43D414yescomplemented
45C22×D43C2312yes—
46C22×D43C2311yescharacteristic
47C22×D43C2311yescharacteristic
48C22×D43C2311yescharacteristic
49D4×C43—22no—
50C2×C4×C43—22no—
51C2×C4×C43C2312yes—
52C2×C4×C43C2312yes—
53C2×C4×C43C2311yescharacteristic
54C22⋊D43—22no—
55C22⋊D43—22no—
56C22⋊D43—42no—
57C2×Dic43C2311yescharacteristic
58C2×Dic43C2312yes—
59D172—24no—
60D172—24no—
61D172—22no—
62D172—42no—
63C4×D43—22no—
64C4×D43—42no—
65C22×C43—22no—
66C22×C43—21no—
67C22×C43—22no—
68C22×C43C2×D412yes—
69C22×C43C2411yescharacteristic
70C2×D43—22no—
71C2×D43—22no—
72C2×D43—22no—
73C2×D43—21no—
74C2×D43—22no—
75C2×D43—22no—
76C2×D43—22no—
77C2×D43—44no—
78C2×D43—44no—
79C2×D43—44no—
80C2×D43C2×D412yes—
81C2×D43C2×D412yes—
82C4×C43—21no—
83C4×C43—22no—
84C4×C43—22no—
85Dic43—21no—
86Dic43—22no—
87C2×C43—22no—
88C2×C43—41no—
89C2×C43C22×D411yescharacteristic
90C2×C43H(5,𝔽2)12yes—
91D43—22no—
92D43—22no—
93D43—42no—
94D43—41no—
95#534—431nomaximal, Sylow
96C43#53411yescharacteristic, complemented, Sylow
97C22×D4—431no—
98C22×D4—431no—
99#123—431no—
100#123—432no—
101#123D4311yescharacteristic, complemented
102#121—432no—
103#121—432no—
104#120—431no—
105#120—432no—
106#120—431no—
107C2×C22⋊C4—431no—
108C24—431no—
109C24—431no—
110C24—431no—
111C2×D4—431no—
112C2×D4—431no—
113C2×D4—432no—
114C2×D4—432no—
115C2×D4—431no—
116C2×D4—432no—
117C2×D4—432no—
118C2×D4—864no—
119C2×D4—864no—
120C2×D4C2×D4311yescharacteristic
121C22×C4—431no—
122C22×C4—432no—
123C22×C4C2×D4311yescharacteristic, complemented
124C4⋊C4—432no—
125C4⋊C4C2×D4312yescomplemented
126C22⋊C4—431no—
127C22⋊C4—432no—
128C22⋊C4—432no—
129C22⋊C4—432no—
130C22⋊C4—432no—
131C22⋊C4C2×D4311yescharacteristic, complemented
132C22⋊C4C2×D4312yescomplemented
133C23—431no—
134C23—431no—
135C23—434no—
136C23—432no—
137C23—431no—
138C23—862no—
139C23—862no—
140C23—1722no—
141C23—1722no—
142C23—1722no—
143C23C22×D4311yescharacteristic
144C23C22×D4311yescharacteristic
145D4—22no—
146D4—864no—
147D4—864no—
148D4—862no—
149D4—862no—
150D4—862no—
151D4—1722no—
152D4—1722no—
153C2×C4—22no—
154C2×C4—432no—
155C2×C4—431no—
156C2×C4—862no—
157C2×C4—1722no—
158C2×C4C22×D4312yes—
159C2×C4C22×D4311yescharacteristic
160C2×C4C22×D4312yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—432no—
165C22—432no—
166C22—862no—
167C22—862no—
168C22—862no—
169C22—861no—
170C22—862no—
171C22—862no—
172C22—862no—
173C22—1724no—
174C22—1724no—
175C22—1724no—
176C22D4×D4312yes—
177C22C23×D4311yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—861no—
182C4—862no—
183C2—22no—
184C2—41no—
185C2—862no—
186C2—862no—
187C2—1722no—
188C2—1721no—
189C2C2×D4×D4311yescharacteristic
190C2#1306337112yes—
subgroupG/H
center, FrattiniC22C23×D43
commutator, socle, cosocleC22×C43C24
Fitting#123×C43C2
hypercenter#123D43
hypocenterC43#534
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D172C2semidirect
2. C2×D4×D43C2semidirect
3. #13063310C2semidirect
4. #13063308C2semidirect
5. #13063300C2semidirect
6. #13063298C2semidirect
7. #13063295C2semidirect
8. #13063295C2semidirect
9. #13063293C2semidirect
10. C22⋊C4×D43C2semidirect
11. #123×C43C2semidirect
13. C23×D43C22semidirect
14. C23×D43C22semidirect
16. C22×C4×C43C22semidirect
18. C2×C22⋊D43C22semidirect
19. C2×C22⋊D43C22semidirect
23. C2×D172C22semidirect
24. C2×D172C22semidirect
25. C2×D172C22semidirect
26. C2×C4×D43C22semidirect
27. C2×C4×D43C22semidirect
28. C4⋊C4×C43C22semidirect
29. C22⋊C4×C43C22semidirect
30. C22⋊C4×C43C22semidirect
31. C22⋊Dic43C22semidirect
32. (C2×D43)⋊C4C22semidirect
33. (C2×D43)⋊C4C22semidirect
34. (C2×D43)⋊C4C22semidirect
35. (C2×D43)⋊C4C22semidirect
36. Dic43⋊C4C22semidirect
44. C22×D43D4semidirect
96. C43#534semidirect
101. #123D43semidirect
123. C22×C4C2×D43semidirect
125. C4⋊C4C2×D43semidirect
131. C22⋊C4C2×D43semidirect
132. C22⋊C4C2×D43semidirect

Maximal subgroups

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

subgroupindex
C22×D1722
C2×D4×D432
#130633102
#130633082
#130633002
#130632982
#130632952
#130632932
C22⋊C4×D432
#123×C432
#53443