Finite Group Databasebeta

← #5032

Subgroups — #5032

Order 192.

198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 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#1004C211yesmaximal, characteristic, complemented
2C3×#120C211yesmaximal, characteristic, complemented
3#953C212yesmaximal, complemented
4#834C211yesmaximal, characteristic, complemented
5#806C212yesmaximal, complemented
6B(4,𝔽2)—31nomaximal, Sylow
7C24×C3C2211yescharacteristic, complemented
8C2×C3×D4—21no—
9C2×C3×D4C2211yescharacteristic
10C2×#67—21no—
11C2×#67—22no—
12C2×#67C2212yes—
13#237—22no—
14D3×D4—22no—
15#235—21no—
16C3×C22⋊C4—21no—
17C3×C22⋊C4C2211yescharacteristic, complemented
18#217—22no—
19#217C2212yescomplemented
20H(5,𝔽2)—31no—
21#120—32no—
22#120D311yescharacteristic, complemented
23#99—31no—
24#99—32no—
25C23×C3—22no—
26C23×C3—21no—
27C23×C3—42no—
28C23×C3D411yescharacteristic, complemented
29C23×C3C2311yescharacteristic
30C22×D3—21no—
31C22×D3D412yescomplemented
32C3×D4—22no—
33C3×D4—42no—
34C2×C3×C4—21no—
35C2×C3×C4D411yescharacteristic, complemented
36#67—22no—
37#67—22no—
38#67—22no—
39#67—44no—
40C2×Dic3—21no—
41C2×Dic3—22no—
42C2×Dic3D412yescomplemented
43D12—21no—
44C4×D3—22no—
45Dic6—21no—
46C24C2×D311yescharacteristic, complemented
47#41—62no—
48#41—61no—
49C2×D4—21no—
50C2×D4—32no—
51C2×D4—62no—
52C2×D4—61no—
53C2×D4—62no—
54C2×D4C2×D311yescharacteristic
55C22⋊C4—21no—
56C22⋊C4—32no—
57C22⋊C4—62no—
58C22⋊C4C2×D311yescharacteristic, complemented
59C22×C3—24no—
60C22×C3—22no—
61C22×C3—21no—
62C22×C3—41no—
63C22×C3—41no—
64C22×C3—42no—
65C22×C3—41no—
66C22×C3—41no—
67C22×C3C2×D412yes—
68C22×C3C2×D411yescharacteristic
69C2×D3—22no—
70C2×D3—42no—
71C2×D3—81no—
72C3×C4—21no—
73C3×C4—41no—
74Dic3—22no—
75Dic3—42no—
76C23—22no—
77C23—21no—
78C23—32no—
79C23—42no—
80C23—61no—
81C23#6711yescharacteristic, complemented
82C23C22×D311yescharacteristic
83Q8—61no—
84D4—22no—
85D4—42no—
86D4—61no—
87D4—62no—
88D4—62no—
89D4—62no—
90D4—124no—
91C2×C4—21no—
92C2×C4—32no—
93C2×C4—62no—
94C2×C4—61no—
95C2×C4—62no—
96C2×C4#6711yescharacteristic, complemented
97C2×C3—22no—
98C2×C3—21no—
99C2×C3—41no—
100C2×C3—42no—
101C2×C3#12011yescharacteristic
102D3—42no—
103C22—22no—
104C22—24no—
105C22—21no—
106C22—41no—
107C22—41no—
108C22—41no—
109C22—42no—
110C22—41no—
111C22—62no—
112C22—122no—
113C22—241no—
114C22D3×D412yes—
115C22C2×#6711yescharacteristic
116C4—21no—
117C4—41no—
118C4—62no—
119C4—122no—
120C3B(4,𝔽2)11yescharacteristic, complemented, Sylow
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—122no—
126C2#93711yescharacteristic
subgroupG/H
centerC2#937
commutator, cosocleC23×C3C23
FrattiniC23C22×D3
FittingC3×#120C2
socleC2×C3#120
hypercenter#120D3
hypocenterC3B(4,𝔽2)
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #1004C2semidirect
2. C3×#120C2semidirect
3. #953C2semidirect
4. #834C2semidirect
5. #806C2semidirect
7. C24×C3C22semidirect
17. C3×C22⋊C4C22semidirect
19. #217C22semidirect
22. #120D3semidirect
28. C23×C3D4semidirect
31. C22×D3D4semidirect
35. C2×C3×C4D4semidirect
42. C2×Dic3D4semidirect
46. C24C2×D3semidirect
58. C22⋊C4C2×D3semidirect
81. C23#67semidirect
96. C2×C4#67semidirect
120. C3B(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
#10042
C3×#1202
#9532
#8342
#8062
B(4,𝔽2)3