Finite Group Databasebeta

← #5676

Subgroups — #5676

Order 192.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#239C211yesmaximal, characteristic, complemented
2C2×D3×D4C211yesmaximal, characteristic, complemented
3C3×#124C211yesmaximal, characteristic, complemented
4#946C211yesmaximal, characteristic, complemented
5#938C211yesmaximal, characteristic, complemented
6#884C212yesmaximal, complemented
7#883C212yesmaximal, complemented
8#882C212yesmaximal, complemented
9D3×C22⋊C4C212yesmaximal, complemented
10C4×D12C211yesmaximal, characteristic, complemented
11C4×D12C211yesmaximal, characteristic, complemented
12#546—31nomaximal, Sylow
13C23×D3C2212yes—
14C2×C3×Q8C2211yescharacteristic, complemented
15C2×C3×D4C2211yescharacteristic
16C2×#67C2212yes—
17C2×#67C2212yescomplemented
18#239—24no—
19D3×D4—24no—
20C2×D12C2211yescharacteristic
21C2×D12C2211yescharacteristic
22C2×D12C2212yescomplemented
23C2×C4×D3C2211yescharacteristic
24C2×C4×D3C2211yescharacteristic, complemented
25C2×C4×D3C2212yescomplemented
26C2×C4×D3C2212yescomplemented
27C3×C22⋊C4C2212yescomplemented
28C3×C22⋊C4C2212yescomplemented
29C3×C42C2211yescharacteristic, complemented
30#217C2212yescomplemented
31(C2×D3)⋊C4C2211yescharacteristic
32(C2×D3)⋊C4C2212yes—
33(C2×D3)⋊C4C2212yescomplemented
34(C2×D3)⋊C4C2211yescharacteristic, complemented
35C4⋊Dic3C2211yescharacteristic, complemented
36C4⋊Dic3C2211yescharacteristic, complemented
37Dic3⋊C4C2212yescomplemented
38C2×#41—31no—
39C22×D4—31no—
40#124D311yescharacteristic, complemented
41#123—32no—
42C22⋊Q8—31no—
43#121—31no—
44#121—32no—
45#120—32no—
46C4×D4—31no—
47C4×D4—31no—
48C2×C22⋊C4—32no—
49C23×C3C2312yes—
50C22×D3—24no—
51C22×D3—24no—
52C22×D3—42no—
53C22×D3C2311yescharacteristic
54C22×D3C2312yes—
55C22×D3C2311yescharacteristic
56C3×Q8—22no—
57C3×D4—22no—
58C2×C3×C4C2312yes—
59C2×C3×C4C2311yescharacteristic
60C2×C3×C4C2311yescharacteristic
61C2×C3×C4C2311yescharacteristic
62#67—24no—
63#67—42no—
64C2×Dic3C2311yescharacteristic
65C2×Dic3C2312yes—
66C2×Dic3C2311yescharacteristic
67D12—24no—
68D12—22no—
69D12D414yescomplemented
70C4×D3—24no—
71C4×D3—22no—
72C4×D3—22no—
73C4×D3—42no—
74C24—32no—
75#41—64no—
76C2×Q8C2×D311yescharacteristic, complemented
77C2×D4—32no—
78C2×D4—32no—
79C2×D4—31no—
80C2×D4—31no—
81C2×D4—32no—
82C2×D4—64no—
83C2×D4C2×D311yescharacteristic
84C22×C4—32no—
85C22×C4—31no—
86C22×C4—31no—
87C22×C4—32no—
88C4⋊C4—31no—
89C4⋊C4—31no—
90C4⋊C4—32no—
91C22⋊C4—32no—
92C22⋊C4—31no—
93C22⋊C4—32no—
94C22⋊C4—32no—
95C22⋊C4—31no—
96C22⋊C4C2×D312yescomplemented
97C22⋊C4C2×D312yescomplemented
98C42C2×D311yescharacteristic, complemented
99C22×C3—22no—
100C22×C3—22no—
101C22×C3—22no—
102C22×C3C2411yescharacteristic
103C2×D3—21no—
104C2×D3—22no—
105C2×D3—22no—
106C2×D3—21no—
107C2×D3—21no—
108C2×D3—21no—
109C2×D3—21no—
110C2×D3—48no—
111C2×D3C2×D414yes—
112C2×D3#4112yescomplemented
113C3×C4—22no—
114C3×C4—21no—
115C3×C4—21no—
116C3×C4C2×D412yes—
117Dic3—21no—
118Dic3—21no—
119Dic3—22no—
120C23—31no—
121C23—32no—
122C23—31no—
123C23—64no—
124C23—64no—
125C23—122no—
126C23C22×D312yes—
127Q8—22no—
128D4—22no—
129D4—34no—
130D4—64no—
131D4—62no—
132D4—64no—
133D4—122no—
134C2×C4—32no—
135C2×C4—31no—
136C2×C4—31no—
137C2×C4—64no—
138C2×C4—62no—
139C2×C4—62no—
140C2×C4—122no—
141C2×C4C22×D311yescharacteristic
142C2×C4C22×D311yescharacteristic
143C2×C4C22×D312yes—
144C2×C4C22×D311yescharacteristic
145C2×C3—42no—
146C2×C3C22×D411yescharacteristic
147C2×C3C2×#4111yescharacteristic
148C2×C3H(5,𝔽2)11yescharacteristic
149D3—22no—
150D3—24no—
151D3—41no—
152C22—22no—
153C22—22no—
154C22—22no—
155C22—34no—
156C22—32no—
157C22—61no—
158C22—62no—
159C22—62no—
160C22—61no—
161C22—61no—
162C22—61no—
163C22—61no—
164C22—128no—
165C22C23×D311yescharacteristic
166C4—22no—
167C4—21no—
168C4—21no—
169C4—61no—
170C4—61no—
171C4—62no—
172C4D3×D412yes—
173C3#54611yescharacteristic, complemented, Sylow
174C2—42no—
175C2—62no—
176C2—64no—
177C2—121no—
178C2C2×D3×D411yescharacteristic
179C2D3×#4111yescharacteristic
180C2#100911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3
commutator, socle, cosocleC22×C3C24
FittingC3×#124C2
hypercenter#124D3
hypocenterC3#546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#239C2semidirect
2. C2×D3×D4C2semidirect
3. C3×#124C2semidirect
4. #946C2semidirect
5. #938C2semidirect
6. #884C2semidirect
7. #883C2semidirect
8. #882C2semidirect
9. D3×C22⋊C4C2semidirect
10. C4×D12C2semidirect
11. C4×D12C2semidirect
14. C2×C3×Q8C22semidirect
17. C2×#67C22semidirect
22. C2×D12C22semidirect
24. C2×C4×D3C22semidirect
25. C2×C4×D3C22semidirect
26. C2×C4×D3C22semidirect
27. C3×C22⋊C4C22semidirect
28. C3×C22⋊C4C22semidirect
29. C3×C42C22semidirect
30. #217C22semidirect
33. (C2×D3)⋊C4C22semidirect
34. (C2×D3)⋊C4C22semidirect
35. C4⋊Dic3C22semidirect
36. C4⋊Dic3C22semidirect
37. Dic3⋊C4C22semidirect
40. #124D3semidirect
69. D12D4semidirect
76. C2×Q8C2×D3semidirect
96. C22⋊C4C2×D3semidirect
97. C22⋊C4C2×D3semidirect
98. C42C2×D3semidirect
112. C2×D3#41semidirect
173. C3#546semidirect

Maximal subgroups

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

subgroupindex
C2×#2392
C2×D3×D42
C3×#1242
#9462
#9382
#8842
#8832
#8822
D3×C22⋊C42
C4×D122
#5463