Finite Group Databasebeta

← #13092982

Subgroups — #13092982

Order 1472.

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

normal subgroups only

This group has 334 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×#67290C211yesmaximal, characteristic, complemented
3C23×#120C211yesmaximal, characteristic, complemented
4#10625124C211yesmaximal, characteristic, complemented
5#10625111C212yesmaximal, complemented
6#10625110C212yesmaximal, complemented
7#10625108C211yesmaximal, characteristic, complemented
8#10625056C212yesmaximal, complemented
9#10625055C212yesmaximal, complemented
10#10625053C211yesmaximal, characteristic, complemented
11#10625048C211yesmaximal, characteristic, complemented
12C24×C23C2211yescharacteristic
13C23×D23C2211yescharacteristic
14C2×D4×C23C2211yescharacteristic
15C2×D4×C23C2212yescomplemented
16C2×C22⋊D23—24no—
17C2×C22⋊D23C2211yescharacteristic
18C2×C22⋊D23C2212yes—
19C2×C22⋊D23C2212yes—
20C2×C22⋊D23C2211yescharacteristic, complemented
21C2×C22⋊D23C2212yescomplemented
22C2×C22⋊D23C2212yescomplemented
23C22×Dic23C2211yescharacteristic
24C22×Dic23C2211yescharacteristic, complemented
25#67290—24no—
26C2×D92C2212yescomplemented
27C2×C4×D23C2211yescharacteristic, complemented
28C2×Dic46C2211yescharacteristic, complemented
29C22⋊C4×C23C2211yescharacteristic
30C22⋊C4×C23C2212yescomplemented
31C22⋊Dic23C2211yescharacteristic
32C22⋊Dic23C2212yescomplemented
33(C2×D23)⋊C4C2212yescomplemented
34(C2×D23)⋊C4C2212yescomplemented
35Dic23⋊C4C2212yescomplemented
36C4×Dic23C2212yescomplemented
37C23×C23—22no—
38C23×C23—42no—
39C23×C23C2312yes—
40C23×C23C2311yescharacteristic
41C23×C23C2311yescharacteristic
42C22×D23—22no—
43C22×D23—42no—
44C22×D23C2312yes—
45C22×D23C2311yescharacteristic
46D4×C23—22no—
47D4×C23—42no—
48C2×C4×C23C2312yes—
49C2×C4×C23C2311yescharacteristic
50C22⋊D23—22no—
51C22⋊D23—22no—
52C22⋊D23—24no—
53C22⋊D23—24no—
54C22⋊D23—42no—
55C22⋊D23—42no—
56C2×Dic23—22no—
57C2×Dic23D414yescomplemented
58C2×Dic23C2312yes—
59C2×Dic23C2311yescharacteristic
60C2×Dic23C2311yescharacteristic
61C2×Dic23C2311yescharacteristic
62D92—42no—
63C4×D23—22no—
64Dic46—22no—
65C22×C23—22no—
66C22×C23—21no—
67C22×C23—22no—
68C22×C23—22no—
69C22×C23—22no—
70C22×C23—22no—
71C22×C23—44no—
72C22×C23C2×D412yes—
73C22×C23C2411yescharacteristic
74C2×D23—21no—
75C2×D23—22no—
76C2×D23—22no—
77C2×D23—22no—
78C2×D23—22no—
79C2×D23—44no—
80C4×C23—21no—
81C4×C23—22no—
82Dic23—21no—
83Dic23—22no—
84Dic23C2×D412yes—
85Dic23C2×D412yes—
86#535—231nomaximal, Sylow
87C2×C23—22no—
88C2×C23—41no—
89C2×C23—42no—
90C2×C23C22×D411yescharacteristic
91C2×C23H(5,𝔽2)12yes—
92D23—41no—
93D23—42no—
94C2×#41—231no—
95C22×D4—231no—
96#127—232no—
97#124—232no—
98#121—232no—
99#121—232no—
100#120—231no—
101#120—231no—
102#120—231no—
103#120D2311yescharacteristic, complemented
104#117—231no—
105C23#53511yescharacteristic, complemented, Sylow
106C24—231no—
107C24C2×D2311yescharacteristic
108#41—464no—
109C2×Q8—231no—
110C2×D4—231no—
111C2×D4—232no—
112C2×D4—232no—
113C2×D4—231no—
114C2×D4—232no—
115C2×D4—232no—
116C2×D4—232no—
117C2×D4—464no—
118C2×D4C2×D2311yescharacteristic
119C2×D4C2×D2312yescomplemented
120C22×C4—231no—
121C22×C4—231no—
122C22×C4—231no—
123C4⋊C4—232no—
124C22⋊C4—231no—
125C22⋊C4—232no—
126C22⋊C4—232no—
127C22⋊C4—232no—
128C22⋊C4C2×D2311yescharacteristic
129C22⋊C4C2×D2312yescomplemented
130C42—232no—
131C23—22no—
132C23—42no—
133C23—232no—
134C23—231no—
135C23—462no—
136C23—922no—
137C23C22×D2312yes—
138C23C22×D2311yescharacteristic
139C23C22×D2311yescharacteristic
140Q8—462no—
141D4—22no—
142D4—42no—
143D4—464no—
144D4—464no—
145D4—462no—
146D4—462no—
147D4—922no—
148D4—922no—
149D4—922no—
150C2×C4—234no—
151C2×C4—231no—
152C2×C4—231no—
153C2×C4—232no—
154C2×C4—231no—
155C2×C4—462no—
156C2×C4—462no—
157C2×C4C22×D2312yes—
158C2×C4C22×D2311yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—461no—
167C22—462no—
168C22—462no—
169C22—462no—
170C22—462no—
171C22—924no—
172C22D4×D2312yes—
173C22C23×D2311yescharacteristic
174C4—21no—
175C4—22no—
176C4—232no—
177C4—232no—
178C4—461no—
179C4—462no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—922no—
184C2—921no—
185C2C2×D4×D2311yescharacteristic
186C2#1062515512yes—
subgroupG/H
center, FrattiniC22C23×D23
commutator, socle, cosocleC22×C23C24
FittingC23×#120C2
hypercenter#120D23
hypocenterC23#535
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×#67290C2semidirect
3. C23×#120C2semidirect
4. #10625124C2semidirect
5. #10625111C2semidirect
6. #10625110C2semidirect
7. #10625108C2semidirect
8. #10625056C2semidirect
9. #10625055C2semidirect
10. #10625053C2semidirect
11. #10625048C2semidirect
15. C2×D4×C23C22semidirect
20. C2×C22⋊D23C22semidirect
21. C2×C22⋊D23C22semidirect
22. C2×C22⋊D23C22semidirect
24. C22×Dic23C22semidirect
26. C2×D92C22semidirect
27. C2×C4×D23C22semidirect
28. C2×Dic46C22semidirect
30. C22⋊C4×C23C22semidirect
32. C22⋊Dic23C22semidirect
33. (C2×D23)⋊C4C22semidirect
34. (C2×D23)⋊C4C22semidirect
35. Dic23⋊C4C22semidirect
36. C4×Dic23C22semidirect
57. C2×Dic23D4semidirect
103. #120D23semidirect
105. C23#535semidirect
119. C2×D4C2×D23semidirect
129. 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×#672902
C23×#1202
#106251242
#106251112
#106251102
#106251082
#106250562
#106250552
#106250532
#106250482
#53523