Finite Group Databasebeta

← #68491

Subgroups — #68491

Order 384.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#5734C211yesmaximal, characteristic, complemented
2#5085C211yesmaximal, characteristic, complemented
3#4843C212yesmaximal, complemented
4C3×#333C211yesmaximal, characteristic, complemented
5#4483C212yesmaximal, complemented
6#1624—31nomaximal, Sylow
7C3×C4⋊Q8C2211yescharacteristic
8#947—22no—
9C2×#215—22no—
10C2×D24—22no—
11#895—22no—
12#892—22no—
13#874C2211yescharacteristic
14C4×D12C2212yes—
15C3×C4⋊C8C2212yes—
16#821—22no—
17C4×C3⋊C8C2211yescharacteristic
18#555—31no—
19#492—31no—
20#459—32no—
21#333D311yescharacteristic, complemented
22D4⋊C8—32no—
23C2×C3×Q8D412yes—
24C2×D12—22no—
25C2×D12—21no—
26C2×D12—22no—
27C2×D12D412yes—
28C2×C4×D3—22no—
29C2×C3×C8—22no—
30C3×C4⋊C4—22no—
31C3×C42C2311yescharacteristic
32#215—44no—
33(C2×D3)⋊C4—24no—
34(C2×D3)⋊C4—22no—
35C4⋊Dic3D412yes—
36C4×Dic3—22no—
37C2×C3⋊C8—21no—
38D24—44no—
39C2×SD16—62no—
40C2×D8—62no—
41C4⋊Q8C2×D311yescharacteristic
42#127—31no—
43#124—62no—
44#121—62no—
45C4×D4—32no—
46#117—62no—
47C4⋊C8C2×D312yes—
48#102—62no—
49C4×C8—31no—
50C22×D3—22no—
51C22×D3—41no—
52C3×Q8—42no—
53C2×C3×C4—22no—
54C2×C3×C4C2×D411yescharacteristic
55C2×C3×C4C2×D412yes—
56C2×Dic3—22no—
57D12—22no—
58D12—22no—
59D12—42no—
60D12—42no—
61D12—42no—
62D12—82no—
63C4×D3—42no—
64C4×D3—42no—
65C3×C8—42no—
66C3⋊C8—22no—
67C2×Q8D1212yes—
68C2×D4—32no—
69C2×D4—62no—
70C2×D4—61no—
71C2×D4—62no—
72C22×C4—62no—
73SD16—124no—
74D8—124no—
75C2×C8—22no—
76C2×C8—61no—
77C4⋊C4—22no—
78C4⋊C4—32no—
79C22⋊C4—64no—
80C22⋊C4—62no—
81C42—62no—
82C42C22×D311yescharacteristic
83C22×C3#12011yescharacteristic
84C2×D3—42no—
85C2×D3—42no—
86C2×D3—42no—
87C2×D3—82no—
88C2×D3—81no—
89C3×C4—21no—
90C3×C4—42no—
91C3×C4#13512yes—
92C3×C4AGL(1,ℤ/8)12yes—
93Dic3—42no—
94C23—62no—
95C23—121no—
96Q8—42no—
97D4—62no—
98D4—62no—
99D4—122no—
100D4—122no—
101D4—122no—
102D4—242no—
103C2×C4—22no—
104C2×C4—62no—
105C2×C4—122no—
106C2×C4—122no—
107C2×C4C2×D1211yescharacteristic
108C2×C4D3×D412yes—
109C8—42no—
110C8—62no—
111C2×C3#44912yes—
112C2×C3#45311yescharacteristic
113D3—82no—
114D3—161no—
115C22—122no—
116C22—122no—
117C22—122no—
118C22—241no—
119C22—242no—
120C22#88211yescharacteristic
121C4—21no—
122C4—42no—
123C4—122no—
124C4#91412yes—
125C4#91912yes—
126C3#162411yescharacteristic, complemented, Sylow
127C2—242no—
128C2—481no—
129C2#474711yescharacteristic
130C2#481012yes—
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#333C2
socleC22×C3#120
hypercenter#333D3
hypocenterC3#1624
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #5734C2semidirect
2. #5085C2semidirect
3. #4843C2semidirect
4. C3×#333C2semidirect
5. #4483C2semidirect
21. #333D3semidirect
126. C3#1624semidirect

Maximal subgroups

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

subgroupindex
#57342
#50852
#48432
C3×#3332
#44832
#16243