Finite Group Databasebeta

← #71251

Subgroups — #71251

Order 384.

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

normal subgroups only

This group has 196 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
2C3×#491C211yesmaximal, characteristic, complemented
3#5089C212yesmaximal, complemented
4#4543C211yesmaximal, characteristic, complemented
5#4464C212yesmaximal, complemented
6#1662—31nomaximal, Sylow
7C3×C4⋊Q8C2211yescharacteristic
8C3×#125C2211yescharacteristic
9#896—22no—
10#895—22no—
11#894—22no—
12#875C2212yes—
13C3×#103—22no—
14C3×#97C2211yescharacteristic
15#810—22no—
16#809—22no—
17#803C2212yes—
18#562—31no—
19#491D311yescharacteristic, complemented
20#489—32no—
21#330—31no—
22#329—32no—
23C2×C3×Q8—21no—
24C2×D12—22no—
25C2×D12D412yes—
26C2×C4×D3—22no—
27C2×Dic6D412yes—
28C2×C3×C8—21no—
29C3×C4⋊C4—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—22no—
38C4⋊Q8C2×D311yescharacteristic
39#125C2×D311yescharacteristic
40#124—32no—
41#123—62no—
42C22⋊Q8—62no—
43#121—62no—
44#103—22no—
45#103—62no—
46#102—62no—
47#97—32no—
48#97C2×D311yescharacteristic
49C22×D3—22no—
50C3×Q8—42no—
51C2×C3×C4—22no—
52C2×C3×C4—41no—
53C2×C3×C4C2×D412yes—
54C2×C3×C4C2×D411yescharacteristic
55C2×Dic3—22no—
56D12—42no—
57D12—82no—
58C4×D3—82no—
59Dic6—42no—
60C3×C8—41no—
61C3⋊C8—42no—
62C2×Q8—21no—
63C2×Q8—32no—
64C2×D4—32no—
65C2×D4—62no—
66C22×C4—62no—
67C2×C8—21no—
68C2×C8—62no—
69C4⋊C4—22no—
70C4⋊C4—22no—
71C4⋊C4—62no—
72C4⋊C4#6712yes—
73C22⋊C4—62no—
74C22⋊C4—62no—
75C22⋊C4—62no—
76C42C22×D311yescharacteristic
77C22×C3#12011yescharacteristic
78C2×D3—42no—
79C2×D3—42no—
80C2×D3—42no—
81C3×C4—21no—
82C3×C4—22no—
83C3×C4—42no—
84C3×C4—81no—
85Dic3—42no—
86C23—62no—
87Q8—42no—
88Q8—122no—
89D4—122no—
90D4—242no—
91C2×C4—22no—
92C2×C4—41no—
93C2×C4—62no—
94C2×C4—242no—
95C2×C4D3×D412yes—
96C2×C4C2×#6711yescharacteristic
97C8—41no—
98C8—122no—
99C2×C3#45411yescharacteristic
100C2×C3#45512yes—
101D3—82no—
102C22—122no—
103C22—122no—
104C22—122no—
105C22#93711yescharacteristic
106C4—22no—
107C4—21no—
108C4—42no—
109C4—81no—
110C4—122no—
111C3#166211yescharacteristic, complemented, Sylow
112C2—242no—
113C2#506211yescharacteristic
114C2#520212yes—
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#491C2
socleC22×C3#120
prime radical#5700C2
hypercenter#491D3
hypocenterC3#1662
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. C3×#491C2semidirect
3. #5089C2semidirect
4. #4543C2semidirect
5. #4464C2semidirect
19. #491D3semidirect
111. C3#1662semidirect

Maximal subgroups

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

subgroupindex
#57002
C3×#4912
#50892
#45432
#44642
#16623