Finite Group Databasebeta

← #5751

Subgroups — #5751

Order 192.

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

normal subgroups only

This group has 258 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
1#1010C211yesmaximal, characteristic, complemented
2#1009C211yesmaximal, characteristic, complemented
3C3×#131C211yesmaximal, characteristic, complemented
4#909C213yesmaximal, complemented
5#908C213yesmaximal, complemented
6#904C213yesmaximal, complemented
7C2×#204C213yesmaximal, complemented
8#577—31nomaximal, Sylow
9C3×#41C2211yescharacteristic
10#239—21no—
11D3×Q8—21no—
12#237—23no—
13D3×D4—23no—
14#235C2213yes—
15#235C2213yescomplemented
16C2×D12C2213yes—
17C2×Dic6C2213yescomplemented
18C3×M16C2213yescomplemented
19C2×C3×C8C2213yescomplemented
20Dic12C2213yescomplemented
21D24C2213yescomplemented
22#204C2213yescomplemented
23#204C2216yescomplemented
24#204C2211yescharacteristic, complemented
252−1+4—31no—
26H(5,𝔽2)—31no—
27#137—33no—
28AGL(1,ℤ/8)—33no—
29#135—33no—
30C2×SD16—33no—
31#131D311yescharacteristic, complemented
32C22×D3—23no—
33C3×Q8D411yescharacteristic
34C3×D4D413yes—
35C2×C3×C4C2313yes—
36#67—23no—
37#67—23no—
38C2×Dic3—23no—
39D12—23no—
40D12C2313yes—
41D12C2311yescharacteristic
42C4×D3—23no—
43C4×D3—23no—
44Dic6—23no—
45Dic6C2313yes—
46Dic6C2311yescharacteristic
47C3×C8C2311yescharacteristic
48C3×C8C2313yes—
49#41—33no—
50#41—33no—
51#41—61no—
52#41—63no—
53#41C2×D311yescharacteristic
54C2×Q8—33no—
55C2×Q8—61no—
56C2×D4—33no—
57C2×D4—63no—
58Q16—33no—
59SD16—33no—
60SD16—36no—
61SD16—31no—
62D8—33no—
63M16C2×D313yescomplemented
64C2×C8C2×D313yescomplemented
65C22×C3C2×D413yes—
66C2×D3—21no—
67C2×D3—23no—
68C2×D3—83no—
69C3×C4C2×D413yes—
70C3×C4C2411yescharacteristic
71Dic3—23no—
72Dic3—21no—
73C23—63no—
74Q8—33no—
75Q8—31no—
76Q8—63no—
77Q8D1211yescharacteristic
78D4—31no—
79D4—33no—
80D4—63no—
81D4—63no—
82D4—63no—
83D4D1213yes—
84C2×C4—63no—
85C2×C4—63no—
86C2×C4—63no—
87C2×C4C22×D313yes—
88C8C22×D311yescharacteristic
89C8C22×D313yes—
90C2×C3—23no—
91C2×C3C22×D411yescharacteristic
92D3—41no—
93D3—43no—
94C22—61no—
95C22—63no—
96C22—243no—
97C22C2×D1213yes—
98C4—61no—
99C4—63no—
100C4C2×D1213yes—
101C4C23×D311yescharacteristic
102C3#57711yescharacteristic, complemented, Sylow
103C2—23no—
104C2—123no—
105C2—121no—
106C2C22×D1211yescharacteristic
subgroupG/H
centerC2C22×D12
commutator, cosocleC3×C4C24
FrattiniC4C23×D3
FittingC3×#131C2
socleC2×C3C22×D4
hypercenter#131D3
hypocenterC3#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ #41 ◁ #131

Complements

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

NH ≅ G/Ntype
1. #1010C2semidirect
2. #1009C2semidirect
3. C3×#131C2semidirect
4. #909C2semidirect
5. #908C2semidirect
6. #904C2semidirect
7. C2×#204C2semidirect
15. #235C22semidirect
17. C2×Dic6C22semidirect
18. C3×M16C22semidirect
19. C2×C3×C8C22semidirect
20. Dic12C22semidirect
21. D24C22semidirect
22. #204C22semidirect
23. #204C22semidirect
24. #204C22semidirect
31. #131D3semidirect
63. M16C2×D3semidirect
64. C2×C8C2×D3semidirect
102. C3#577semidirect

Maximal subgroups

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

subgroupindex
#10102
#10092
C3×#1312
#9092
#9082
#9042
C2×#2042
#5773