Finite Group Databasebeta

← #5683

Subgroups — #5683

Order 192.

260 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C3×#124C211yesmaximal, characteristic, complemented
2#947C212yesmaximal, complemented
3#937C212yesmaximal, complemented
4#885C214yesmaximal, complemented
5#882C214yesmaximal, complemented
6#875C212yesmaximal, complemented
7H(3,𝔽4)—31nomaximal, Sylow
8C23×D3C2212yescomplemented
9C2×C3×Q8C2211yescharacteristic, complemented
10C2×C3×D4C2211yescharacteristic
11C2×#67C2214yes—
12C2×D12C2212yes—
13C2×D12C2212yes—
14C2×Dic6C2212yescomplemented
15C3×C22⋊C4C2214yescomplemented
16C3×C42C2211yescharacteristic, complemented
17#217C2212yescomplemented
18(C2×D3)⋊C4C2214yescomplemented
19(C2×D3)⋊C4C2214yescomplemented
20(C2×D3)⋊C4C2214yescomplemented
21C4×Dic3C2212yescomplemented
22#124—32no—
23#124—34no—
24#124—32no—
25#124D311yescharacteristic, complemented
26#120—32no—
27#120—34no—
28C23×C3C2312yes—
29C22×D3—42no—
30C22×D3—44no—
31C22×D3C2314yes—
32C3×Q8—41no—
33C3×D4—41no—
34C2×C3×C4C2312yes—
35C2×C3×C4C2311yescharacteristic
36C2×C3×C4C2312yes—
37#67—44no—
38C2×Dic3C2314yes—
39D12—42no—
40D12—42no—
41Dic6—42no—
42C24—32no—
43C2×Q8—32no—
44C2×Q8C2×D311yescharacteristic, complemented
45C2×D4—34no—
46C2×D4—32no—
47C2×D4—32no—
48C2×D4C2×D311yescharacteristic
49C22⋊C4—32no—
50C22⋊C4—34no—
51C22⋊C4—34no—
52C22⋊C4—34no—
53C22⋊C4C2×D314yescomplemented
54C42—32no—
55C42C2×D311yescharacteristic, complemented
56C22×C3—24no—
57C22×C3—22no—
58C22×C3C2411yescharacteristic
59C2×D3—24no—
60C2×D3—24no—
61C2×D3—24no—
62C2×D3—48no—
63C3×C4—22no—
64C3×C4—22no—
65C3×C4—21no—
66Dic3—24no—
67C23—34no—
68C23—122no—
69C23—124no—
70C23C22×D312yes—
71Q8—41no—
72Q8—122no—
73D4—41no—
74D4—122no—
75D4—122no—
76D4—124no—
77C2×C4—34no—
78C2×C4C22×D312yes—
79C2×C4C22×D311yescharacteristic
80C2×C4C22×D312yes—
81C2×C3—42no—
82C2×C3H(5,𝔽2)11yescharacteristic
83C2×C3H(5,𝔽2)12yes—
84D3—44no—
85C22—22no—
86C22—24no—
87C22—64no—
88C22—64no—
89C22—64no—
90C22—128no—
91C22C23×D311yescharacteristic
92C4—21no—
93C4—22no—
94C4—22no—
95C4—64no—
96C3H(3,𝔽4)11yescharacteristic, complemented, Sylow
97C2—42no—
98C2—124no—
99C2#100411yescharacteristic
100C2#100912yes—
subgroupG/H
center, FrattiniC22C23×D3
commutator, socle, cosocleC22×C3C24
FittingC3×#124C2
hypercenter#124D3
hypocenterC3H(3,𝔽4)
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C3×#124C2semidirect
2. #947C2semidirect
3. #937C2semidirect
4. #885C2semidirect
5. #882C2semidirect
6. #875C2semidirect
8. C23×D3C22semidirect
9. C2×C3×Q8C22semidirect
14. C2×Dic6C22semidirect
15. C3×C22⋊C4C22semidirect
16. C3×C42C22semidirect
17. #217C22semidirect
18. (C2×D3)⋊C4C22semidirect
19. (C2×D3)⋊C4C22semidirect
20. (C2×D3)⋊C4C22semidirect
21. C4×Dic3C22semidirect
25. #124D3semidirect
44. C2×Q8C2×D3semidirect
53. C22⋊C4C2×D3semidirect
55. C42C2×D3semidirect
96. C3H(3,𝔽4)semidirect

Maximal subgroups

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

subgroupindex
C3×#1242
#9472
#9372
#8852
#8822
#8752
H(3,𝔽4)3