Finite Group Databasebeta

← #5545

Subgroups — #5545

Order 192.

338 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×D3×D4C211yesmaximal, characteristic
2C3×C4×D4C211yesmaximal, characteristic, complemented
3D4×Dic3C211yesmaximal, characteristic, complemented
4C4×#67C212yesmaximal, complemented
5C2×(C2×D3)⋊C4C212yesmaximal, complemented
6#893C211yesmaximal, characteristic, complemented
7#892C211yesmaximal, characteristic, complemented
8#67⋊C4C212yesmaximal, complemented
9D3×C22⋊C4C212yesmaximal, complemented
10C4×D12C211yesmaximal, characteristic, complemented
11#872C211yesmaximal, characteristic, complemented
12#518—31nomaximal, Sylow
13C23×D3C2212yes—
14C2×C3×D4C2211yescharacteristic
15C22×C3×C4C2212yescomplemented
16C2×#67C2212yes—
17C22×Dic3C2212yescomplemented
18D3×D4C418yescomplemented
19C2×D12C2211yescharacteristic
20C2×C4×D3C2211yescharacteristic
21C2×C4×D3C2212yescomplemented
22C2×C4×D3C2212yescomplemented
23C3×C4⋊C4C2211yescharacteristic, complemented
24C3×C22⋊C4C2212yescomplemented
25C3×C42C2211yescharacteristic, complemented
26#217C2212yescomplemented
27(C2×D3)⋊C4C2211yescharacteristic
28(C2×D3)⋊C4C2211yescharacteristic
29(C2×D3)⋊C4C2212yes—
30(C2×D3)⋊C4C2212yescomplemented
31(C2×D3)⋊C4C2212yescomplemented
32C4⋊Dic3C2211yescharacteristic, complemented
33Dic3⋊C4C2211yescharacteristic, complemented
34Dic3⋊C4C2211yescharacteristic, complemented
35C4×Dic3C2211yescharacteristic, complemented
36C4×Dic3C2212yescomplemented
37C22×D4—31no—
38C4×D4—31no—
39C4×D4—32no—
40C4×D4—31no—
41C4×D4—32no—
42C4×D4—31no—
43C4×D4D311yescharacteristic, complemented
44#117—31no—
45#117—31no—
46C2×C22⋊C4—32no—
47C2×C22⋊C4—32no—
48C23×C3C2312yes—
49C22×D3—42no—
50C22×D3—42no—
51C22×D3C2×C418yescomplemented
52C22×D3C2311yescharacteristic
53C22×D3C2312yes—
54C3×D4C2×C414yescomplemented
55C2×C3×C4—42no—
56C2×C3×C4C2312yes—
57C2×C3×C4C2311yescharacteristic
58C2×C3×C4C2311yescharacteristic
59C2×C3×C4C2311yescharacteristic
60#67C2×C418yescomplemented
61C2×Dic3—42no—
62C2×Dic3C2311yescharacteristic
63C2×Dic3C2311yescharacteristic
64C2×Dic3C2312yes—
65C2×Dic3C2311yescharacteristic
66D12C2×C414yescomplemented
67C4×D3—42no—
68C4×D3—42no—
69C4×D3C2×C414yescomplemented
70C24—32no—
71C2×D4—32no—
72C2×D4—38no—
73C2×D4—31no—
74C2×D4C2×D311yescharacteristic
75C22×C4—31no—
76C22×C4—32no—
77C22×C4—32no—
78C22×C4—32no—
79C22×C4C2×D312yescomplemented
80C4⋊C4—31no—
81C4⋊C4—31no—
82C4⋊C4—31no—
83C4⋊C4C2×D311yescharacteristic, complemented
84C22⋊C4—32no—
85C22⋊C4—32no—
86C22⋊C4—31no—
87C22⋊C4—31no—
88C22⋊C4—32no—
89C22⋊C4—32no—
90C22⋊C4C2×D312yescomplemented
91C42—32no—
92C42—31no—
93C42C2×D311yescharacteristic, complemented
94C22×C3—22no—
95C22×C3—22no—
96C22×C3C22×C414yes—
97C22×C3C2411yescharacteristic
98C2×D3—22no—
99C2×D3—22no—
100C2×D3—21no—
101C2×D3—21no—
102C2×D3—48no—
103C2×D3C22×C412yes—
104C2×D3C22×C414yes—
105C3×C4—21no—
106C3×C4—22no—
107C3×C4—21no—
108C3×C4C22×C412yes—
109Dic3—21no—
110Dic3—22no—
111Dic3—21no—
112Dic3C22×C412yes—
113C23—31no—
114C23—32no—
115C23—38no—
116C23—122no—
117C23—122no—
118C23C22×D312yes—
119D4—38no—
120D4—34no—
121D4C4×D314yescomplemented
122C2×C4—31no—
123C2×C4—34no—
124C2×C4—31no—
125C2×C4—32no—
126C2×C4—31no—
127C2×C4—42no—
128C2×C4—122no—
129C2×C4—122no—
130C2×C4—122no—
131C2×C4C22×D312yes—
132C2×C4C22×D311yescharacteristic
133C2×C4C22×D311yescharacteristic
134C2×C4C22×D311yescharacteristic
135C2×C3—24no—
136C2×C3C23×C411yescharacteristic
137C2×C3H(5,𝔽2)11yescharacteristic
138C2×C3H(5,𝔽2)11yescharacteristic
139D3—22no—
140D3—24no—
141C22—22no—
142C22—22no—
143C22—32no—
144C22—34no—
145C22—62no—
146C22—61no—
147C22—62no—
148C22—61no—
149C22—128no—
150C22C2×C4×D314yes—
151C22C23×D311yescharacteristic
152C4—21no—
153C4—22no—
154C4—21no—
155C4—32no—
156C4—61no—
157C4—62no—
158C4—61no—
159C4C2×C4×D312yes—
160C3#51811yescharacteristic, complemented, Sylow
161C2—24no—
162C2—62no—
163C2—64no—
164C2C22×C4×D311yescharacteristic
165C2#100411yescharacteristic
166C2#100911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3
commutatorC2×C3C23×C4
FittingC3×C4×D4C2
socle, cosocleC22×C3C24
prime radicalC2×D3×D4C2
hypercenterC4×D4D3
hypocenterC3#518
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C3×C4×D4C2semidirect
3. D4×Dic3C2semidirect
4. C4×#67C2semidirect
5. C2×(C2×D3)⋊C4C2semidirect
6. #893C2semidirect
7. #892C2semidirect
8. #67⋊C4C2semidirect
9. D3×C22⋊C4C2semidirect
10. C4×D12C2semidirect
11. #872C2semidirect
15. C22×C3×C4C22semidirect
17. C22×Dic3C22semidirect
18. D3×D4C4semidirect
21. C2×C4×D3C22semidirect
22. C2×C4×D3C22semidirect
23. C3×C4⋊C4C22semidirect
24. C3×C22⋊C4C22semidirect
25. C3×C42C22semidirect
26. #217C22semidirect
30. (C2×D3)⋊C4C22semidirect
31. (C2×D3)⋊C4C22semidirect
32. C4⋊Dic3C22semidirect
33. Dic3⋊C4C22semidirect
34. Dic3⋊C4C22semidirect
35. C4×Dic3C22semidirect
36. C4×Dic3C22semidirect
43. C4×D4D3semidirect
51. C22×D3C2×C4semidirect
54. C3×D4C2×C4semidirect
60. #67C2×C4semidirect
66. D12C2×C4semidirect
69. C4×D3C2×C4semidirect
79. C22×C4C2×D3semidirect
83. C4⋊C4C2×D3semidirect
90. C22⋊C4C2×D3semidirect
93. C42C2×D3semidirect
121. D4C4×D3semidirect
160. C3#518semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D42
C3×C4×D42
D4×Dic32
C4×#672
C2×(C2×D3)⋊C42
#8932
#8922
#67⋊C42
D3×C22⋊C42
C4×D122
#8722
#5183