Finite Group Databasebeta

← #13092977

Subgroups — #13092977

Order 1472.

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×C22⋊D23C211yesmaximal, characteristic, complemented
2C2×D4×D23C211yesmaximal, characteristic, complemented
3C23×#120C211yesmaximal, characteristic, complemented
4#10625124C211yesmaximal, characteristic, complemented
5#10625109C212yesmaximal, complemented
6#10625108C212yesmaximal, complemented
7#10625106C211yesmaximal, characteristic, complemented
8#10625057C211yesmaximal, characteristic, complemented
9#10625055C212yesmaximal, complemented
10#10625054C212yesmaximal, complemented
11C22⋊C4×D23C211yesmaximal, characteristic, complemented
12C24×C23C2211yescharacteristic, complemented
13C23×D23C2211yescharacteristic
14C23×D23C2211yescharacteristic, complemented
15C2×D4×C23C2211yescharacteristic
16C2×D4×C23C2212yescomplemented
17C2×C22⋊D23—24no—
18C2×C22⋊D23C2212yes—
19C2×C22⋊D23C2211yescharacteristic
20C2×C22⋊D23C2212yescomplemented
21C2×C22⋊D23C2211yescharacteristic, complemented
22C2×C22⋊D23C2212yescomplemented
23C22×Dic23C2211yescharacteristic, complemented
24D4×D23—24no—
25C2×D92C2211yescharacteristic, complemented
26C2×C4×D23C2211yescharacteristic, complemented
27C2×C4×D23C2212yescomplemented
28C22⋊C4×C23C2211yescharacteristic, complemented
29C22⋊C4×C23C2212yescomplemented
30C22⋊Dic23C2211yescharacteristic, complemented
31C22⋊Dic23C2212yescomplemented
32C22⋊Dic23C2212yescomplemented
33(C2×D23)⋊C4C2212yescomplemented
34(C2×D23)⋊C4C2212yescomplemented
35C4⋊Dic23C2212yescomplemented
36Dic23⋊C4C2212yescomplemented
37C23×C23—22no—
38C23×C23—42no—
39C23×C23C2311yescharacteristic
40C23×C23C2312yes—
41C23×C23C2311yescharacteristic
42C22×D23—22no—
43C22×D23—42no—
44C22×D23—42no—
45C22×D23D414yescomplemented
46C22×D23C2311yescharacteristic
47C22×D23C2311yescharacteristic
48C22×D23C2311yescharacteristic
49D4×C23—22no—
50D4×C23—42no—
51C2×C4×C23C2311yescharacteristic
52C2×C4×C23C2312yes—
53C22⋊D23—22no—
54C22⋊D23—22no—
55C22⋊D23—24no—
56C22⋊D23—24no—
57C22⋊D23—42no—
58C2×Dic23—22no—
59C2×Dic23C2312yes—
60C2×Dic23C2312yes—
61C2×Dic23C2311yescharacteristic
62D92—22no—
63C4×D23—22no—
64C4×D23—42no—
65C22×C23—22no—
66C22×C23—22no—
67C22×C23—21no—
68C22×C23—22no—
69C22×C23—22no—
70C22×C23—22no—
71C22×C23—44no—
72C22×C23C2×D412yes—
73C22×C23C2411yescharacteristic
74C2×D23—22no—
75C2×D23—22no—
76C2×D23—21no—
77C2×D23—22no—
78C2×D23—44no—
79C2×D23—44no—
80C2×D23C2×D412yes—
81C2×D23C2×D412yes—
82C4×C23—21no—
83C4×C23—22no—
84Dic23—21no—
85Dic23—22no—
86Dic23—22no—
87#534—231nomaximal, Sylow
88C2×C23—22no—
89C2×C23—41no—
90C2×C23—42no—
91C2×C23C22×D411yescharacteristic
92C2×C23H(5,𝔽2)12yes—
93D23—22no—
94D23—22no—
95D23—41no—
96C22×D4—231no—
97C22×D4—231no—
98#123—231no—
99#123—232no—
100#123—231no—
101#121—232no—
102#121—232no—
103#120—231no—
104#120—232no—
105#120D2311yescharacteristic, complemented
106C2×C22⋊C4—231no—
107C23#53411yescharacteristic, complemented, Sylow
108C24—231no—
109C24—231no—
110C24C2×D2311yescharacteristic, complemented
111C2×D4—232no—
112C2×D4—231no—
113C2×D4—231no—
114C2×D4—232no—
115C2×D4—232no—
116C2×D4—231no—
117C2×D4—464no—
118C2×D4—464no—
119C2×D4C2×D2311yescharacteristic
120C2×D4C2×D2312yescomplemented
121C22×C4—231no—
122C22×C4—231no—
123C22×C4—232no—
124C4⋊C4—232no—
125C4⋊C4—232no—
126C22⋊C4—231no—
127C22⋊C4—232no—
128C22⋊C4—232no—
129C22⋊C4—232no—
130C22⋊C4—232no—
131C22⋊C4C2×D2311yescharacteristic, complemented
132C22⋊C4C2×D2312yescomplemented
133C23—22no—
134C23—42no—
135C23—231no—
136C23—231no—
137C23—234no—
138C23—231no—
139C23—462no—
140C23—922no—
141C23—922no—
142C23C22×D2311yescharacteristic
143C23C22×D2312yes—
144C23C22×D2311yescharacteristic
145D4—22no—
146D4—42no—
147D4—462no—
148D4—462no—
149D4—464no—
150D4—464no—
151D4—462no—
152D4—922no—
153C2×C4—232no—
154C2×C4—231no—
155C2×C4—232no—
156C2×C4—462no—
157C2×C4—462no—
158C2×C4—922no—
159C2×C4C22×D2312yes—
160C2×C4C22×D2311yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—232no—
169C22—232no—
170C22—462no—
171C22—462no—
172C22—461no—
173C22—462no—
174C22—924no—
175C22—924no—
176C22D4×D2312yes—
177C22C23×D2311yescharacteristic
178C4—21no—
179C4—22no—
180C4—461no—
181C4—462no—
182C4—462no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—462no—
187C2—462no—
188C2—921no—
189C2C2×D4×D2311yescharacteristic
190C2#1062515512yes—
subgroupG/H
center, FrattiniC22C23×D23
commutator, socle, cosocleC22×C23C24
FittingC23×#120C2
hypercenter#120D23
hypocenterC23#534
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D23C2semidirect
2. C2×D4×D23C2semidirect
3. C23×#120C2semidirect
4. #10625124C2semidirect
5. #10625109C2semidirect
6. #10625108C2semidirect
7. #10625106C2semidirect
8. #10625057C2semidirect
9. #10625055C2semidirect
10. #10625054C2semidirect
11. C22⋊C4×D23C2semidirect
12. C24×C23C22semidirect
14. C23×D23C22semidirect
16. C2×D4×C23C22semidirect
20. C2×C22⋊D23C22semidirect
21. C2×C22⋊D23C22semidirect
22. C2×C22⋊D23C22semidirect
23. C22×Dic23C22semidirect
25. C2×D92C22semidirect
26. C2×C4×D23C22semidirect
27. C2×C4×D23C22semidirect
28. C22⋊C4×C23C22semidirect
29. C22⋊C4×C23C22semidirect
30. C22⋊Dic23C22semidirect
31. C22⋊Dic23C22semidirect
32. C22⋊Dic23C22semidirect
33. (C2×D23)⋊C4C22semidirect
34. (C2×D23)⋊C4C22semidirect
35. C4⋊Dic23C22semidirect
36. Dic23⋊C4C22semidirect
45. C22×D23D4semidirect
105. #120D23semidirect
107. C23#534semidirect
110. C24C2×D23semidirect
120. C2×D4C2×D23semidirect
131. C22⋊C4C2×D23semidirect
132. C22⋊C4C2×D23semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D232
C2×D4×D232
C23×#1202
#106251242
#106251092
#106251082
#106251062
#106250572
#106250552
#106250542
C22⋊C4×D232
#53423