Finite Group Databasebeta

← #68370

Subgroups — #68370

Order 384.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#5700C211yesmaximal, characteristic, complemented
2#5067C211yesmaximal, characteristic, complemented
3#4712C212yesmaximal, complemented
4C3×#330C211yesmaximal, characteristic, complemented
5#4464C212yesmaximal, complemented
6#1656—31nomaximal, Sylow
7C3×#125C2211yescharacteristic
8C2×D24—22no—
9C2×#204—22no—
10#896—22no—
11#895—22no—
12#894—22no—
13#875C2212yes—
14#874C2211yescharacteristic
15C3×#97C2212yes—
16#809—22no—
17#803C2211yescharacteristic
18#562—31no—
19#496—32no—
20#490—31no—
21#330D311yescharacteristic, complemented
22#329—32no—
23C2×D12—21no—
24C2×D12—22no—
25C2×D12—22no—
26C2×D12D412yes—
27C2×C4×D3—22no—
28C2×Dic6D412yes—
29C2×C3×C8—22no—
30C3×C4⋊C4—22no—
31C3×C4⋊C4D412yes—
32C3×C42C2311yescharacteristic
33(C2×D3)⋊C4—22no—
34(C2×D3)⋊C4—22no—
35(C2×D3)⋊C4—22no—
36Dic3⋊C4—22no—
37C2×C3⋊C8—21no—
38D24—44no—
39#204—44no—
40C2×SD16—62no—
41C2×D8—62no—
42#127—31no—
43#125C2×D311yescharacteristic
44#124—32no—
45#123—62no—
46C22⋊Q8—62no—
47#121—62no—
48#102—62no—
49#97—31no—
50#97C2×D312yes—
51C22×D3—22no—
52C22×D3—41no—
53C2×C3×C4—22no—
54C2×C3×C4C2×D411yescharacteristic
55C2×C3×C4C2×D412yes—
56C2×Dic3—22no—
57D12—42no—
58D12—42no—
59D12—42no—
60D12—42no—
61D12—82no—
62C4×D3—82no—
63Dic6—42no—
64C3×C8—24no—
65C3⋊C8—41no—
66C2×Q8—32no—
67C2×D4—32no—
68C2×D4—62no—
69C2×D4—61no—
70C2×D4—62no—
71C22×C4—62no—
72SD16—124no—
73D8—124no—
74C2×C8—22no—
75C2×C8—61no—
76C4⋊C4—22no—
77C4⋊C4—62no—
78C4⋊C4D1212yes—
79C22⋊C4—62no—
80C22⋊C4—62no—
81C22⋊C4—62no—
82C42C22×D311yescharacteristic
83C22×C3#12011yescharacteristic
84C2×D3—42no—
85C2×D3—42no—
86C2×D3—42no—
87C2×D3—82no—
88C2×D3—81no—
89C3×C4—22no—
90C3×C4—21no—
91C3×C4—42no—
92Dic3—42no—
93C23—62no—
94C23—121no—
95Q8—122no—
96D4—122no—
97D4—122no—
98D4—122no—
99D4—122no—
100D4—242no—
101C2×C4—22no—
102C2×C4—62no—
103C2×C4—242no—
104C2×C4C2×D1211yescharacteristic
105C2×C4D3×D412yes—
106C8—24no—
107C8—121no—
108C2×C3#45312yes—
109C2×C3#45411yescharacteristic
110D3—82no—
111D3—161no—
112C22—122no—
113C22—122no—
114C22—122no—
115C22—241no—
116C22—242no—
117C22#88211yescharacteristic
118C4—21no—
119C4—22no—
120C4—42no—
121C4—122no—
122C3#165611yescharacteristic, complemented, Sylow
123C2—242no—
124C2—481no—
125C2#475311yescharacteristic
126C2#482212yes—
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#330C2
socleC22×C3#120
hypercenter#330D3
hypocenterC3#1656
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. #5700C2semidirect
2. #5067C2semidirect
3. #4712C2semidirect
4. C3×#330C2semidirect
5. #4464C2semidirect
21. #330D3semidirect
122. C3#1656semidirect

Maximal subgroups

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

subgroupindex
#57002
#50672
#47122
C3×#3302
#44642
#16563