Finite Group Databasebeta

← #5550

Subgroups — #5550

Order 192.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#235C211yesmaximal, characteristic, complemented
2C22×D12C211yesmaximal, characteristic, complemented
3C3×C4×D4C211yesmaximal, characteristic, complemented
4#937C212yesmaximal, complemented
5#930C211yesmaximal, characteristic, complemented
6#924C211yesmaximal, characteristic, complemented
7#894C212yesmaximal, complemented
8#884C212yesmaximal, complemented
9D3×C22⋊C4C212yesmaximal, complemented
10#875C211yesmaximal, characteristic, complemented
11C4×D12C211yesmaximal, characteristic, complemented
12#546—31nomaximal, Sylow
13C23×D3C2212yes—
14C2×C3×D4C2211yescharacteristic
15C22×C3×C4C2211yescharacteristic
16C22×C3×C4C2211yescharacteristic, complemented
17C2×#67C2212yescomplemented
18C2×#67C2212yescomplemented
19#235—24no—
20C2×D12—24no—
21C2×D12C2211yescharacteristic
22C2×D12C2212yes—
23C2×D12C2211yescharacteristic
24C2×C4×D3C2212yescomplemented
25C2×C4×D3C2212yescomplemented
26C2×Dic6C2211yescharacteristic, complemented
27C3×C4⋊C4C2211yescharacteristic, complemented
28C3×C22⋊C4C2211yescharacteristic
29C3×C22⋊C4C2211yescharacteristic, complemented
30C3×C42C2211yescharacteristic, complemented
31#217C2212yes—
32(C2×D3)⋊C4C2212yescomplemented
33(C2×D3)⋊C4C2212yescomplemented
34(C2×D3)⋊C4C2212yescomplemented
35(C2×D3)⋊C4C2212yescomplemented
36C4⋊Dic3C2211yescharacteristic, complemented
37Dic3⋊C4C2212yescomplemented
38C2×#41—31no—
39C22×D4—31no—
40#124—31no—
41#123—32no—
42C22⋊Q8—31no—
43#121—31no—
44#121—32no—
45#120—32no—
46C4×D4—31no—
47C4×D4D311yescharacteristic, complemented
48C2×C22⋊C4—32no—
49C23×C3C2311yescharacteristic
50C23×C3C2311yescharacteristic
51C22×D3—24no—
52C22×D3—24no—
53C22×D3—42no—
54C22×D3C2312yes—
55C22×D3C2312yes—
56C3×D4—22no—
57C2×C3×C4—22no—
58C2×C3×C4—22no—
59C2×C3×C4C2311yescharacteristic
60C2×C3×C4C2311yescharacteristic
61C2×C3×C4C2311yescharacteristic
62C2×C3×C4C2311yescharacteristic
63C2×C3×C4C2311yescharacteristic
64#67—24no—
65#67—42no—
66C2×Dic3C2312yes—
67C2×Dic3C2312yes—
68D12—24no—
69D12—22no—
70D12D414yescomplemented
71C4×D3—24no—
72C4×D3—42no—
73Dic6—22no—
74C24—32no—
75#41—64no—
76C2×Q8—31no—
77C2×D4—32no—
78C2×D4—32no—
79C2×D4—31no—
80C2×D4—32no—
81C2×D4—31no—
82C2×D4—64no—
83C2×D4C2×D311yescharacteristic
84C22×C4—32no—
85C22×C4—32no—
86C22×C4C2×D311yescharacteristic
87C22×C4C2×D311yescharacteristic, complemented
88C4⋊C4—31no—
89C4⋊C4—32no—
90C4⋊C4C2×D311yescharacteristic, complemented
91C22⋊C4—32no—
92C22⋊C4—32no—
93C22⋊C4—32no—
94C22⋊C4—32no—
95C22⋊C4—32no—
96C22⋊C4C2×D311yescharacteristic
97C22⋊C4C2×D311yescharacteristic, complemented
98C42C2×D311yescharacteristic, complemented
99C22×C3—21no—
100C22×C3—21no—
101C22×C3—21no—
102C22×C3—21no—
103C22×C3—21no—
104C22×C3#4112yescomplemented
105C22×C3C2411yescharacteristic
106C2×D3—22no—
107C2×D3—22no—
108C2×D3—22no—
109C2×D3—22no—
110C2×D3—22no—
111C2×D3—48no—
112C2×D3C2×D414yes—
113C3×C4—21no—
114C3×C4—21no—
115C3×C4—21no—
116C3×C4—21no—
117C3×C4C2×D412yes—
118Dic3—22no—
119Dic3—22no—
120C23—32no—
121C23—32no—
122C23—64no—
123C23—64no—
124C23—122no—
125C23C22×D311yescharacteristic
126C23C22×D311yescharacteristic
127Q8—62no—
128D4—22no—
129D4—34no—
130D4—64no—
131D4—64no—
132D4—62no—
133D4—122no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—32no—
137C2×C4—32no—
138C2×C4—64no—
139C2×C4—122no—
140C2×C4C22×D311yescharacteristic
141C2×C4C22×D311yescharacteristic
142C2×C4C22×D311yescharacteristic
143C2×C4C22×D311yescharacteristic
144C2×C4C22×D311yescharacteristic
145C2×C3—22no—
146C2×C3—41no—
147C2×C3C22×D411yescharacteristic
148C2×C3C2×#4111yescharacteristic
149C2×C3H(5,𝔽2)11yescharacteristic
150D3—24no—
151D3—42no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—34no—
158C22—62no—
159C22—62no—
160C22—62no—
161C22—62no—
162C22—62no—
163C22—128no—
164C22#23512yescomplemented
165C22C23×D311yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—62no—
171C4—62no—
172C4D3×D412yes—
173C3#54611yescharacteristic, complemented, Sylow
174C2—22no—
175C2—41no—
176C2—64no—
177C2—122no—
178C2C2×#23511yescharacteristic
179C2C2×D3×D411yescharacteristic
180C2#100911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3
commutator, socle, cosocleC22×C3C24
FittingC3×C4×D4C2
hypercenterC4×D4D3
hypocenterC3#546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#235C2semidirect
2. C22×D12C2semidirect
3. C3×C4×D4C2semidirect
4. #937C2semidirect
5. #930C2semidirect
6. #924C2semidirect
7. #894C2semidirect
8. #884C2semidirect
9. D3×C22⋊C4C2semidirect
10. #875C2semidirect
11. C4×D12C2semidirect
16. C22×C3×C4C22semidirect
17. C2×#67C22semidirect
18. C2×#67C22semidirect
24. C2×C4×D3C22semidirect
25. C2×C4×D3C22semidirect
26. C2×Dic6C22semidirect
27. C3×C4⋊C4C22semidirect
29. C3×C22⋊C4C22semidirect
30. C3×C42C22semidirect
32. (C2×D3)⋊C4C22semidirect
33. (C2×D3)⋊C4C22semidirect
34. (C2×D3)⋊C4C22semidirect
35. (C2×D3)⋊C4C22semidirect
36. C4⋊Dic3C22semidirect
37. Dic3⋊C4C22semidirect
47. C4×D4D3semidirect
70. D12D4semidirect
87. C22×C4C2×D3semidirect
90. C4⋊C4C2×D3semidirect
97. C22⋊C4C2×D3semidirect
98. C42C2×D3semidirect
104. C22×C3#41semidirect
164. C22#235semidirect
173. C3#546semidirect

Maximal subgroups

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

subgroupindex
C2×#2352
C22×D122
C3×C4×D42
#9372
#9302
#9242
#8942
#8842
D3×C22⋊C42
#8752
C4×D122
#5463