Finite Group Databasebeta

← #71248

Subgroups — #71248

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
2C3×#490C211yesmaximal, characteristic, complemented
3#5074C212yesmaximal, complemented
4#4543C211yesmaximal, characteristic, complemented
5#4464C212yesmaximal, complemented
6#1656—31nomaximal, Sylow
7C3×#127C2211yescharacteristic
8C3×#125C2211yescharacteristic
9C2×#214—22no—
10C2×#213—22no—
11#896—22no—
12#895—22no—
13#894—22no—
14#875C2212yes—
15C3×#102—22no—
16C3×#97C2211yescharacteristic
17#803C2212yes—
18#562—31no—
19#496—32no—
20#490D311yescharacteristic, complemented
21#330—31no—
22#329—32no—
23C2×C3×D4—21no—
24C2×C3×D4—22no—
25C2×D12—22no—
26C2×D12D412yes—
27C2×C4×D3—22no—
28C2×Dic6D412yes—
29C2×C3×C8—21no—
30C3×C4⋊C4—22no—
31C3×C4⋊C4D412yes—
32C3×C42C2311yescharacteristic
33#214—44no—
34#213—44no—
35(C2×D3)⋊C4—22no—
36(C2×D3)⋊C4—22no—
37(C2×D3)⋊C4—22no—
38Dic3⋊C4—22no—
39C2×C3⋊C8—22no—
40C2×SD16—62no—
41C2×D8—62no—
42#127C2×D311yescharacteristic
43#125C2×D311yescharacteristic
44#124—32no—
45#123—62no—
46C22⋊Q8—62no—
47#121—62no—
48#102—22no—
49#97—32no—
50#97C2×D311yescharacteristic
51C23×C3—41no—
52C22×D3—22no—
53C3×D4—42no—
54C3×D4—42no—
55C3×D4—42no—
56C2×C3×C4—22no—
57C2×C3×C4C2×D412yes—
58C2×C3×C4C2×D411yescharacteristic
59C2×Dic3—22no—
60D12—42no—
61D12—82no—
62C4×D3—82no—
63Dic6—42no—
64C3×C8—41no—
65C3⋊C8—24no—
66C2×Q8—32no—
67C2×D4—22no—
68C2×D4—21no—
69C2×D4—32no—
70C2×D4—62no—
71C22×C4—62no—
72SD16—124no—
73D8—124no—
74C2×C8—21no—
75C2×C8—62no—
76C4⋊C4—22no—
77C4⋊C4—62no—
78C4⋊C4#6712yes—
79C22⋊C4—62no—
80C22⋊C4—62no—
81C22⋊C4—62no—
82C42C22×D311yescharacteristic
83C22×C3—82no—
84C22×C3—81no—
85C22×C3#12011yescharacteristic
86C2×D3—42no—
87C2×D3—42no—
88C2×D3—42no—
89C3×C4—22no—
90C3×C4—21no—
91C3×C4—42no—
92Dic3—42no—
93C23—41no—
94C23—62no—
95Q8—122no—
96D4—42no—
97D4—42no—
98D4—42no—
99D4—122no—
100D4—242no—
101C2×C4—22no—
102C2×C4—62no—
103C2×C4—242no—
104C2×C4D3×D412yes—
105C2×C4C2×#6711yescharacteristic
106C8—41no—
107C8—64no—
108C2×C3—161no—
109C2×C3#45312yes—
110C2×C3#45411yescharacteristic
111D3—82no—
112C22—82no—
113C22—81no—
114C22—122no—
115C22—122no—
116C22—122no—
117C22#93711yescharacteristic
118C4—22no—
119C4—21no—
120C4—42no—
121C4—122no—
122C3#165611yescharacteristic, complemented, Sylow
123C2—161no—
124C2—242no—
125C2#506211yescharacteristic
126C2#519812yes—
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#490C2
socleC22×C3#120
hypercenter#490D3
hypocenterC3#1656
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #5700C2semidirect
2. C3×#490C2semidirect
3. #5074C2semidirect
4. #4543C2semidirect
5. #4464C2semidirect
20. #490D3semidirect
122. C3#1656semidirect

Maximal subgroups

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

subgroupindex
#57002
C3×#4902
#50742
#45432
#44642
#16563