Finite Group Databasebeta

← #5609

Subgroups — #5609

Order 192.

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
1C2×D3×D4C211yesmaximal, characteristic, complemented
2C2×D3×D4C212yesmaximal, complemented
3C22×D12C211yesmaximal, characteristic, complemented
4C3×#121C211yesmaximal, characteristic, complemented
5#940C211yesmaximal, characteristic, complemented
6#937C212yesmaximal, complemented
7C4×#67C211yesmaximal, characteristic, complemented
8#895C211yesmaximal, characteristic, complemented
9#893C211yesmaximal, characteristic, complemented
10#884C212yesmaximal, complemented
11#882C212yesmaximal, complemented
12D42—31nomaximal, Sylow
13C23×D3C2212yescomplemented
14C23×D3C2212yescomplemented
15C2×C3×D4C2212yescomplemented
16C2×C3×D4C2211yescharacteristic, complemented
17C22×C3×C4C2211yescharacteristic, complemented
18C2×#67C2211yescharacteristic
19C2×#67C2212yescomplemented
20C2×#67C2212yescomplemented
21C2×#67C2212yescomplemented
22D3×D4—28no—
23D3×D4—24no—
24C2×D12—24no—
25C2×D12C2211yescharacteristic
26C2×D12C2212yescomplemented
27C2×D12C2212yescomplemented
28C2×D12C2211yescharacteristic, complemented
29C2×C4×D3C2212yescomplemented
30C2×C4×D3C2211yescharacteristic, complemented
31C3×C4⋊C4C2211yescharacteristic, complemented
32C3×C22⋊C4C2212yescomplemented
33#217C2211yescharacteristic, complemented
34(C2×D3)⋊C4C2211yescharacteristic, complemented
35(C2×D3)⋊C4C2212yescomplemented
36(C2×D3)⋊C4C2212yescomplemented
37Dic3⋊C4C2211yescharacteristic, complemented
38C4×Dic3C2211yescharacteristic, complemented
39C22×D4—32no—
40C22×D4—31no—
41C22×D4—31no—
42#127—31no—
43#121—31no—
44#121—32no—
45#121D311yescharacteristic, complemented
46#120—32no—
47#120—32no—
48C4×D4—31no—
49C4×D4—31no—
50C23×C3C2312yes—
51C23×C3C2311yescharacteristic
52C22×D3—24no—
53C22×D3—24no—
54C22×D3—24no—
55C22×D3—24no—
56C22×D3—42no—
57C22×D3—42no—
58C22×D3C2312yes—
59C22×D3C2312yes—
60C22×D3C2311yescharacteristic
61C3×D4—24no—
62C3×D4—22no—
63C2×C3×C4—22no—
64C2×C3×C4C2311yescharacteristic
65C2×C3×C4C2312yes—
66C2×C3×C4C2311yescharacteristic
67#67—24no—
68#67—24no—
69#67—42no—
70#67D414yescomplemented
71C2×Dic3C2311yescharacteristic
72C2×Dic3C2311yescharacteristic
73C2×Dic3C2311yescharacteristic
74D12—24no—
75D12—24no—
76D12—22no—
77D12D414yescomplemented
78C4×D3—22no—
79C4×D3—24no—
80C24—32no—
81C24—32no—
82C2×D4—31no—
83C2×D4—32no—
84C2×D4—32no—
85C2×D4—32no—
86C2×D4—31no—
87C2×D4—32no—
88C2×D4—32no—
89C2×D4—31no—
90C2×D4—64no—
91C2×D4—68no—
92C2×D4—64no—
93C2×D4C2×D312yescomplemented
94C2×D4C2×D311yescharacteristic, complemented
95C22×C4—32no—
96C22×C4—31no—
97C22×C4C2×D311yescharacteristic, complemented
98C4⋊C4—31no—
99C4⋊C4C2×D311yescharacteristic, complemented
100C22⋊C4—31no—
101C22⋊C4—32no—
102C22⋊C4—31no—
103C22⋊C4—32no—
104C22⋊C4C2×D312yescomplemented
105C42—31no—
106C22×C3—21no—
107C22×C3—21no—
108C22×C3—22no—
109C22×C3—22no—
110C22×C3—22no—
111C22×C3C2×D412yescomplemented
112C22×C3C2411yescharacteristic
113C2×D3—21no—
114C2×D3—22no—
115C2×D3—22no—
116C2×D3—21no—
117C2×D3—22no—
118C2×D3—22no—
119C2×D3—22no—
120C2×D3—48no—
121C2×D3—48no—
122C2×D3C2×D412yescomplemented
123C2×D3C2×D414yescomplemented
124C3×C4—22no—
125C3×C4—21no—
126C3×C4C2×D412yescomplemented
127Dic3—21no—
128Dic3—21no—
129Dic3C2×D412yescomplemented
130C23—31no—
131C23—32no—
132C23—32no—
133C23—64no—
134C23—64no—
135C23—64no—
136C23—64no—
137C23—122no—
138C23—122no—
139C23C22×D312yes—
140C23C22×D311yescharacteristic
141D4—24no—
142D4—22no—
143D4—34no—
144D4—34no—
145D4—64no—
146D4—64no—
147D4—64no—
148D4—64no—
149D4—62no—
150D4—122no—
151C2×C4—22no—
152C2×C4—31no—
153C2×C4—31no—
154C2×C4—31no—
155C2×C4—64no—
156C2×C4—62no—
157C2×C4C22×D312yes—
158C2×C4C22×D311yescharacteristic
159C2×C4C22×D311yescharacteristic
160C2×C3—22no—
161C2×C3—42no—
162C2×C3C22×D411yescharacteristic
163C2×C3C22×D411yescharacteristic
164C2×C3H(5,𝔽2)11yescharacteristic
165D3—22no—
166D3—24no—
167D3—42no—
168C22—21no—
169C22—21no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—32no—
174C22—34no—
175C22—62no—
176C22—61no—
177C22—62no—
178C22—61no—
179C22—62no—
180C22—62no—
181C22—62no—
182C22—128no—
183C22—128no—
184C22D3×D412yescomplemented
185C22C23×D311yescharacteristic
186C4—21no—
187C4—22no—
188C4—32no—
189C4—61no—
190C4—61no—
191C4D3×D412yescomplemented
192C3D4211yescharacteristic, complemented, Sylow
193C2—22no—
194C2—42no—
195C2—64no—
196C2—62no—
197C2—122no—
198C2C2×D3×D411yescharacteristic
199C2C2×D3×D411yescharacteristic
200C2#100911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3
commutator, socle, cosocleC22×C3C24
FittingC3×#121C2
hypercenter#121D3
hypocenterC3D42
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D3×D4C2semidirect
2. C2×D3×D4C2semidirect
3. C22×D12C2semidirect
4. C3×#121C2semidirect
5. #940C2semidirect
6. #937C2semidirect
7. C4×#67C2semidirect
8. #895C2semidirect
9. #893C2semidirect
10. #884C2semidirect
11. #882C2semidirect
13. C23×D3C22semidirect
14. C23×D3C22semidirect
15. C2×C3×D4C22semidirect
16. C2×C3×D4C22semidirect
17. C22×C3×C4C22semidirect
19. C2×#67C22semidirect
20. C2×#67C22semidirect
21. C2×#67C22semidirect
26. C2×D12C22semidirect
27. C2×D12C22semidirect
28. C2×D12C22semidirect
29. C2×C4×D3C22semidirect
30. C2×C4×D3C22semidirect
31. C3×C4⋊C4C22semidirect
32. C3×C22⋊C4C22semidirect
33. #217C22semidirect
34. (C2×D3)⋊C4C22semidirect
35. (C2×D3)⋊C4C22semidirect
36. (C2×D3)⋊C4C22semidirect
37. Dic3⋊C4C22semidirect
38. C4×Dic3C22semidirect
45. #121D3semidirect
70. #67D4semidirect
77. D12D4semidirect
93. C2×D4C2×D3semidirect
94. C2×D4C2×D3semidirect
97. C22×C4C2×D3semidirect
99. C4⋊C4C2×D3semidirect
104. C22⋊C4C2×D3semidirect
111. C22×C3C2×D4semidirect
122. C2×D3C2×D4semidirect
123. C2×D3C2×D4semidirect
126. C3×C4C2×D4semidirect
129. Dic3C2×D4semidirect
184. C22D3×D4semidirect
191. C4D3×D4semidirect
192. C3D42semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D42
C22×D122
C3×#1212
#9402
#9372
C4×#672
#8952
#8932
#8842
#8822
D423