Finite Group Databasebeta

← #5053

Subgroups — #5053

Order 192.

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

normal subgroups only

This group has 160 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
2C3×#123C211yesmaximal, characteristic, complemented
3#929C212yesmaximal, complemented
4#834C211yesmaximal, characteristic, complemented
5#806C212yesmaximal, complemented
6#458—31nomaximal, Sylow
7C2×C3×D4C2211yescharacteristic
8C22×C3×C4C2211yescharacteristic, complemented
9C2×#67—21no—
10C2×#67C2212yes—
11#237—22no—
12D3×D4—22no—
13#235—21no—
14C3×C4⋊C4—21no—
15C3×C22⋊C4—21no—
16C3×C22⋊C4C2211yescharacteristic, complemented
17#217C2212yescomplemented
18(C2×D3)⋊C4—22no—
19Dic3⋊C4—22no—
20H(5,𝔽2)—31no—
21#123—32no—
22#123D311yescharacteristic, complemented
23#99—31no—
24#99—32no—
25C23×C3D411yescharacteristic
26C23×C3C2311yescharacteristic
27C22×D3—21no—
28C22×D3D412yes—
29C3×D4—22no—
30C2×C3×C4—22no—
31C2×C3×C4—21no—
32C2×C3×C4—21no—
33C2×C3×C4D411yescharacteristic
34#67—22no—
35#67—22no—
36#67—22no—
37C2×Dic3—21no—
38C2×Dic3—22no—
39C2×Dic3D412yes—
40D12—21no—
41C4×D3—22no—
42Dic6—21no—
43#41—62no—
44#41—61no—
45C2×D4—32no—
46C2×D4—62no—
47C2×D4—61no—
48C2×D4C2×D311yescharacteristic
49C22×C4C2×D311yescharacteristic, complemented
50C4⋊C4—21no—
51C4⋊C4—62no—
52C22⋊C4—21no—
53C22⋊C4—32no—
54C22⋊C4—62no—
55C22⋊C4C2×D311yescharacteristic, complemented
56C22×C3—21no—
57C22×C3—41no—
58C22×C3—41no—
59C22×C3C2×D412yes—
60C22×C3C2×D411yescharacteristic
61C2×D3—22no—
62C2×D3—42no—
63C2×D3—81no—
64C3×C4—21no—
65C3×C4—41no—
66C3×C4—41no—
67Dic3—22no—
68Dic3—42no—
69C23—32no—
70C23—61no—
71C23#6711yescharacteristic
72C23C22×D311yescharacteristic
73Q8—61no—
74D4—22no—
75D4—61no—
76D4—62no—
77D4—62no—
78D4—62no—
79C2×C4—22no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—32no—
83C2×C4—62no—
84C2×C4—61no—
85C2×C4—62no—
86C2×C4#6711yescharacteristic
87C2×C3—22no—
88C2×C3—21no—
89C2×C3—41no—
90C2×C3#12011yescharacteristic
91D3—42no—
92C22—21no—
93C22—41no—
94C22—41no—
95C22—62no—
96C22—122no—
97C22—241no—
98C22D3×D412yes—
99C22C2×#6711yescharacteristic
100C4—21no—
101C4—41no—
102C4—41no—
103C4—62no—
104C4—122no—
105C3#45811yescharacteristic, complemented, Sylow
106C2—21no—
107C2—22no—
108C2—41no—
109C2—122no—
110C2#93711yescharacteristic
subgroupG/H
centerC2#937
commutator, cosocleC23×C3C23
FrattiniC23C22×D3
FittingC3×#123C2
socleC2×C3#120
prime radical#1004C2
hypercenter#123D3
hypocenterC3#458
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C3×#123C2semidirect
3. #929C2semidirect
4. #834C2semidirect
5. #806C2semidirect
8. C22×C3×C4C22semidirect
16. C3×C22⋊C4C22semidirect
17. #217C22semidirect
22. #123D3semidirect
49. C22×C4C2×D3semidirect
55. C22⋊C4C2×D3semidirect
105. C3#458semidirect

Maximal subgroups

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

subgroupindex
#10042
C3×#1232
#9292
#8342
#8062
#4583