Finite Group Databasebeta

← #13093061

Subgroups — #13093061

Order 1472.

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

normal subgroups only

This group has 334 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×D4×C23C211yesmaximal, characteristic, complemented
2C2×#67290C211yesmaximal, characteristic, complemented
3#10625124C212yesmaximal, complemented
4C2×C22⋊Dic23C212yesmaximal, complemented
5#10625110C212yesmaximal, complemented
6#10625109C211yesmaximal, characteristic, complemented
7#10625107C211yesmaximal, characteristic, complemented
8#10625106C212yesmaximal, complemented
9D4×Dic23C211yesmaximal, characteristic, complemented
10C4×C22⋊D23C211yesmaximal, characteristic, complemented
11#10625095C211yesmaximal, characteristic, complemented
12C24×C23C2212yes—
13C2×D4×C23—24no—
14C2×D4×C23C2212yes—
15C2×D4×C23C2211yescharacteristic
16C2×D4×C23C2211yescharacteristic
17C22×C4×C23C2211yescharacteristic
18C2×C22⋊D23C2211yescharacteristic
19C2×C22⋊D23C2212yescomplemented
20C2×C22⋊D23C2212yescomplemented
21C22×Dic23C2212yescomplemented
22C22×Dic23C2212yescomplemented
23#67290—24no—
24C2×C4×D23C2211yescharacteristic, complemented
25C2×Dic46C2211yescharacteristic, complemented
26C22⋊Dic23C2212yes—
27C22⋊Dic23C2211yescharacteristic
28C22⋊Dic23C2212yescomplemented
29C22⋊Dic23C2212yescomplemented
30C22⋊Dic23C2212yescomplemented
31C22⋊Dic23C2212yescomplemented
32(C2×D23)⋊C4C2211yescharacteristic, complemented
33C4⋊Dic23C2211yescharacteristic, complemented
34Dic23⋊C4C2212yescomplemented
35Dic23⋊C4C2211yescharacteristic, complemented
36C4×Dic23C2211yescharacteristic, complemented
37C23×C23—24no—
38C23×C23—24no—
39C23×C23—42no—
40C23×C23C2312yes—
41C23×C23C2312yes—
42C23×C23C2311yescharacteristic
43C22×D23C2311yescharacteristic
44D4×C23—24no—
45D4×C23—22no—
46D4×C23D414yescomplemented
47C2×C4×C23—22no—
48C2×C4×C23C2311yescharacteristic
49C2×C4×C23C2311yescharacteristic
50C22⋊D23—24no—
51C22⋊D23—22no—
52C22⋊D23—42no—
53C2×Dic23—24no—
54C2×Dic23—42no—
55C2×Dic23C2311yescharacteristic
56C2×Dic23C2311yescharacteristic
57C2×Dic23C2312yes—
58C2×Dic23C2312yes—
59C2×Dic23C2311yescharacteristic
60C4×D23—22no—
61Dic46—22no—
62C22×C23—21no—
63C22×C23—22no—
64C22×C23—22no—
65C22×C23—21no—
66C22×C23—22no—
67C22×C23—22no—
68C22×C23—22no—
69C22×C23—48no—
70C22×C23C2×D414yes—
71C22×C23#4112yescomplemented
72C22×C23C2411yescharacteristic
73C2×D23—21no—
74C2×D23—21no—
75C2×D23—21no—
76C4×C23—21no—
77C4×C23C2×D412yes—
78Dic23—22no—
79Dic23—21no—
80Dic23—21no—
81Dic23—21no—
82Dic23—22no—
83#546—231nomaximal, Sylow
84C2×C23—22no—
85C2×C23—24no—
86C2×C23—42no—
87C2×C23C22×D411yescharacteristic
88C2×C23C2×#4111yescharacteristic
89C2×C23H(5,𝔽2)11yescharacteristic
90D23—41no—
91C2×#41—231no—
92C22×D4D2311yescharacteristic, complemented
93#124—231no—
94#123—232no—
95C22⋊Q8—231no—
96#121—231no—
97#121—232no—
98#120—232no—
99C4×D4—231no—
100C4×D4—231no—
101C2×C22⋊C4—232no—
102C23#54611yescharacteristic, complemented, Sylow
103C24C2×D2312yes—
104#41—464no—
105C2×Q8—231no—
106C2×D4—24no—
107C2×D4—231no—
108C2×D4—232no—
109C2×D4—232no—
110C2×D4C2×D2312yes—
111C2×D4C2×D2311yescharacteristic
112C2×D4C2×D2311yescharacteristic
113C22×C4—232no—
114C22×C4—232no—
115C22×C4—231no—
116C22×C4C2×D2311yescharacteristic
117C4⋊C4—231no—
118C4⋊C4—231no—
119C4⋊C4—232no—
120C22⋊C4—231no—
121C22⋊C4—232no—
122C22⋊C4—232no—
123C22⋊C4—232no—
124C22⋊C4—232no—
125C22⋊C4—232no—
126C22⋊C4—231no—
127C42—231no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—231no—
132C23C22×D2312yes—
133C23C22×D2311yescharacteristic
134C23C22×D2312yes—
135Q8—462no—
136D4—24no—
137D4—22no—
138D4—462no—
139D4—464no—
140D4—922no—
141D4C22⋊D2314yescomplemented
142C2×C4—22no—
143C2×C4—231no—
144C2×C4—231no—
145C2×C4—232no—
146C2×C4—231no—
147C2×C4—232no—
148C2×C4—464no—
149C2×C4—462no—
150C2×C4—922no—
151C2×C4C22×D2311yescharacteristic
152C2×C4C22×D2311yescharacteristic
153C22—22no—
154C22—22no—
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—461no—
162C22—461no—
163C22—461no—
164C22#6729012yescomplemented
165C22C2×C22⋊D2314yes—
166C22C23×D2311yescharacteristic
167C4—21no—
168C4—461no—
169C4—461no—
170C4—462no—
171C4—461no—
172C4—462no—
173C4C2×C22⋊D2312yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—921no—
178C2C2×#6729011yescharacteristic
179C2#1062515511yescharacteristic
180C2C22×C22⋊D2311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D23
commutator, socle, cosocleC22×C23C24
FittingC22×D4×C23C2
hypercenterC22×D4D23
hypocenterC23#546
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×D4×C23C2semidirect
2. C2×#67290C2semidirect
3. #10625124C2semidirect
4. C2×C22⋊Dic23C2semidirect
5. #10625110C2semidirect
6. #10625109C2semidirect
7. #10625107C2semidirect
8. #10625106C2semidirect
9. D4×Dic23C2semidirect
10. C4×C22⋊D23C2semidirect
11. #10625095C2semidirect
19. C2×C22⋊D23C22semidirect
20. C2×C22⋊D23C22semidirect
21. C22×Dic23C22semidirect
22. C22×Dic23C22semidirect
24. C2×C4×D23C22semidirect
25. C2×Dic46C22semidirect
28. C22⋊Dic23C22semidirect
29. C22⋊Dic23C22semidirect
30. C22⋊Dic23C22semidirect
31. C22⋊Dic23C22semidirect
32. (C2×D23)⋊C4C22semidirect
33. C4⋊Dic23C22semidirect
34. Dic23⋊C4C22semidirect
35. Dic23⋊C4C22semidirect
36. C4×Dic23C22semidirect
46. D4×C23D4semidirect
71. C22×C23#41semidirect
92. C22×D4D23semidirect
102. C23#546semidirect
141. D4C22⋊D23semidirect
164. C22#67290semidirect

Maximal subgroups

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

subgroupindex
C22×D4×C232
C2×#672902
#106251242
C2×C22⋊Dic232
#106251102
#106251092
#106251072
#106251062
D4×Dic232
C4×C22⋊D232
#106250952
#54623