Finite Group Databasebeta

← #13093049

Subgroups — #13093049

Order 1472.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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⋊D23C212yesmaximal, complemented
2C2×D4×D23C212yesmaximal, complemented
3C23×#120C211yesmaximal, characteristic, complemented
4#10625111C211yesmaximal, characteristic, complemented
5#10625110C212yesmaximal, complemented
6#10625108C211yesmaximal, characteristic, complemented
7#10625055C212yesmaximal, complemented
8#10625053C212yesmaximal, complemented
9(C22⋊D23)⋊C4C212yesmaximal, complemented
10C24×C23C2211yescharacteristic, complemented
11C23×D23C2212yescomplemented
12C23×D23C2211yescharacteristic, complemented
13C2×D4×C23C2212yescomplemented
14C2×D4×C23C2211yescharacteristic, complemented
15C2×C22⋊D23—28no—
16C2×C22⋊D23C2212yes—
17C2×C22⋊D23C2212yescomplemented
18C2×C22⋊D23C2212yescomplemented
19C2×C22⋊D23C2212yescomplemented
20C2×C22⋊D23C2212yescomplemented
21C22×Dic23C2212yescomplemented
22D4×D23—28no—
23C2×D92C2212yescomplemented
24C2×D92C2211yescharacteristic, complemented
25C2×C4×D23C2212yescomplemented
26C22⋊C4×C23C2212yescomplemented
27C22⋊C4×C23C2211yescharacteristic, complemented
28C22⋊Dic23C2211yescharacteristic, complemented
29(C2×D23)⋊C4C2212yescomplemented
30(C2×D23)⋊C4C2212yescomplemented
31Dic23⋊C4C2212yescomplemented
32C4×Dic23C2211yescharacteristic, complemented
33C23×C23—24no—
34C23×C23—41no—
35C23×C23C2311yescharacteristic
36C23×C23C2311yescharacteristic
37C23×C23C2312yes—
38C22×D23—24no—
39C22×D23—24no—
40C22×D23—24no—
41C22×D23—41no—
42C22×D23—42no—
43C22×D23C2312yes—
44C22×D23C2312yes—
45D4×C23—24no—
46D4×C23—41no—
47C2×C4×C23C2311yescharacteristic
48C2×C4×C23C2312yes—
49C22⋊D23—24no—
50C22⋊D23—24no—
51C22⋊D23—24no—
52C22⋊D23—24no—
53C22⋊D23D418yescomplemented
54C2×Dic23—24no—
55C2×Dic23C2312yes—
56C2×Dic23C2312yes—
57D92—24no—
58D92—41no—
59C4×D23—24no—
60C22×C23—22no—
61C22×C23—22no—
62C22×C23—22no—
63C22×C23—22no—
64C22×C23—21no—
65C22×C23—21no—
66C22×C23—44no—
67C22×C23C2×D414yescomplemented
68C22×C23C2411yescharacteristic
69C2×D23—22no—
70C2×D23—22no—
71C2×D23—22no—
72C2×D23—22no—
73C2×D23—22no—
74C2×D23—44no—
75C2×D23—48no—
76C2×D23C2×D414yescomplemented
77C4×C23—22no—
78C4×C23—21no—
79Dic23—22no—
80Dic23C2×D414yescomplemented
81D42—231nomaximal, Sylow
82C2×C23—24no—
83C2×C23—41no—
84C2×C23—41no—
85C2×C23C22×D412yes—
86C2×C23H(5,𝔽2)11yescharacteristic
87D23—24no—
88D23—42no—
89C22×D4—232no—
90C22×D4—232no—
91#127—231no—
92#121—232no—
93#121—232no—
94#120—231no—
95#120—232no—
96#120D2311yescharacteristic, complemented
97C4×D4—232no—
98C23D4211yescharacteristic, complemented, Sylow
99C24—231no—
100C24—232no—
101C24C2×D2311yescharacteristic, complemented
102C2×D4—232no—
103C2×D4—232no—
104C2×D4—232no—
105C2×D4—232no—
106C2×D4—232no—
107C2×D4—232no—
108C2×D4—231no—
109C2×D4—468no—
110C2×D4—468no—
111C2×D4C2×D2312yescomplemented
112C2×D4C2×D2311yescharacteristic, complemented
113C22×C4—232no—
114C22×C4—232no—
115C4⋊C4—232no—
116C22⋊C4—231no—
117C22⋊C4—232no—
118C22⋊C4—232no—
119C22⋊C4C2×D2312yescomplemented
120C22⋊C4C2×D2311yescharacteristic, complemented
121C42—231no—
122C23—24no—
123C23—41no—
124C23—232no—
125C23—232no—
126C23—464no—
127C23—464no—
128C23—464no—
129C23—922no—
130C23—921no—
131C23C22×D2311yescharacteristic
132C23C22×D2311yescharacteristic
133C23C22×D2312yes—
134D4—24no—
135D4—41no—
136D4—238no—
137D4—464no—
138D4—464no—
139D4—464no—
140D4—464no—
141D4—464no—
142D4—921no—
143C2×C4—232no—
144C2×C4—232no—
145C2×C4—464no—
146C2×C4—464no—
147C2×C4C22×D2311yescharacteristic
148C2×C4C22×D2312yes—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—234no—
157C22—462no—
158C22—462no—
159C22—462no—
160C22—462no—
161C22—462no—
162C22—928no—
163C22—924no—
164C22D4×D2314yescomplemented
165C22C23×D2311yescharacteristic
166C4—22no—
167C4—21no—
168C4—234no—
169C4—462no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—464no—
174C2—922no—
175C2C2×D4×D2312yes—
176C2#1062515511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D23
commutator, socle, cosocleC22×C23C24
FittingC23×#120C2
hypercenter#120D23
hypocenterC23D42
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. #10625111C2semidirect
5. #10625110C2semidirect
6. #10625108C2semidirect
7. #10625055C2semidirect
8. #10625053C2semidirect
9. (C22⋊D23)⋊C4C2semidirect
10. C24×C23C22semidirect
11. C23×D23C22semidirect
12. C23×D23C22semidirect
13. C2×D4×C23C22semidirect
14. C2×D4×C23C22semidirect
17. C2×C22⋊D23C22semidirect
18. C2×C22⋊D23C22semidirect
19. C2×C22⋊D23C22semidirect
20. C2×C22⋊D23C22semidirect
21. C22×Dic23C22semidirect
23. C2×D92C22semidirect
24. C2×D92C22semidirect
25. C2×C4×D23C22semidirect
26. C22⋊C4×C23C22semidirect
27. C22⋊C4×C23C22semidirect
28. C22⋊Dic23C22semidirect
29. (C2×D23)⋊C4C22semidirect
30. (C2×D23)⋊C4C22semidirect
31. Dic23⋊C4C22semidirect
32. C4×Dic23C22semidirect
53. C22⋊D23D4semidirect
67. C22×C23C2×D4semidirect
76. C2×D23C2×D4semidirect
80. Dic23C2×D4semidirect
96. #120D23semidirect
98. C23D42semidirect
101. C24C2×D23semidirect
111. C2×D4C2×D23semidirect
112. C2×D4C2×D23semidirect
119. C22⋊C4C2×D23semidirect
120. C22⋊C4C2×D23semidirect
164. C22D4×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
#106251112
#106251102
#106251082
#106250552
#106250532
(C22⋊D23)⋊C42
D4223