Finite Group Databasebeta

← #68367

Subgroups — #68367

Order 384.

196 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D3×#125C211yesmaximal, characteristic
2#4705C212yesmaximal
3C3×#330C211yesmaximal, characteristic, complemented
4#4543C211yesmaximal, characteristic, complemented
5#4465C212yesmaximal, complemented
6#1534—31nomaximal, Sylow
7C3×#125C2211yescharacteristic
8D3×C4⋊C4—22no—
9D3×C4⋊C4C412yes—
10#890C412yes—
11#889—22no—
12C42×D3C2211yescharacteristic
13#870C2211yescharacteristic
14C3×#97C2212yes—
15(C2×D3)⋊C8—22no—
16Dic3⋊C8—22no—
17#803C2212yes—
18C2×#125—31no—
19#432—32no—
20#330—31no—
21#330—32no—
22#330D311yescharacteristic, complemented
23C2×C4×D3—22no—
24C2×C4×D3C2×C411yescharacteristic
25C2×C4×D3D412yes—
26C2×C3×C8—22no—
27C3×C4⋊C4—22no—
28C3×C4⋊C4C2×C412yes—
29C3×C42C2311yescharacteristic
30C4⋊Dic3—22no—
31C4⋊Dic3C2×C412yes—
32Dic3⋊C4—22no—
33Dic3⋊C4—22no—
34Dic3⋊C4—22no—
35C4×Dic3C2×C411yescharacteristic
36C4×Dic3D412yes—
37C2×C3⋊C8—22no—
38#125—32no—
39#125—31no—
40#125—62no—
41#125C2×D311yescharacteristic
42C2×C4⋊C4—32no—
43C2×C4⋊C4—62no—
44C2×C42—31no—
45C4⋊C8—62no—
46C22⋊C8—62no—
47#97—32no—
48#97C2×D312yes—
49C22×D3C22⋊C411yescharacteristic
50C2×C3×C4—22no—
51C2×C3×C4C22×C411yescharacteristic
52C2×C3×C4C2×D412yes—
53C2×Dic3—22no—
54C2×Dic3C22⋊C412yes—
55C2×Dic3C22⋊C411yescharacteristic
56C4×D3—22no—
57C4×D3—42no—
58C4×D3—44no—
59C3×C8—42no—
60C3⋊C8—42no—
61C22×C4—31no—
62C22×C4—32no—
63C22×C4—62no—
64C2×C8—22no—
65C2×C8—62no—
66C4⋊C4—22no—
67C4⋊C4—32no—
68C4⋊C4—62no—
69C4⋊C4—62no—
70C4⋊C4—62no—
71C4⋊C4—62no—
72C4⋊C4C4×D312yes—
73C42—31no—
74C42—32no—
75C42C22×D311yescharacteristic
76C22×C3C2×C22⋊C411yescharacteristic
77C2×D3—22no—
78C2×D3#10112yes—
79C3×C4—21no—
80C3×C4—22no—
81C3×C4—42no—
82Dic3—21no—
83Dic3—22no—
84Dic3—42no—
85C23—31no—
86C2×C4—22no—
87C2×C4—31no—
88C2×C4—32no—
89C2×C4—62no—
90C2×C4—62no—
91C2×C4—122no—
92C2×C4—124no—
93C2×C4C2×C4×D311yescharacteristic
94C2×C4D3×D412yes—
95C8—42no—
96C8—122no—
97C2×C3C2×#10111yescharacteristic
98C2×C3#42112yes—
99D3—22no—
100C22—32no—
101C22—62no—
102C22D3×C22⋊C411yescharacteristic
103C4—21no—
104C4—22no—
105C4—42no—
106C4—61no—
107C4—62no—
108C4—122no—
109C3#153411yescharacteristic, complemented, Sylow
110C2—62no—
111C2D3×#10111yescharacteristic
112C2#482112yes—
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC42C22×D3
FittingC3×#330C2
socleC22×C3C2×C22⋊C4
prime radicalC22×D3C22⋊C4
cosocleC3×C42C23
hypercenter#330D3
hypocenterC3#1534
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
3. C3×#330C2semidirect
4. #4543C2semidirect
5. #4465C2semidirect
22. #330D3semidirect
109. C3#1534semidirect

Maximal subgroups

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

subgroupindex
D3×#1252
#47052
C3×#3302
#45432
#44652
#15343