Finite Group Databasebeta

← #71265

Subgroups — #71265

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
2C3×#492C211yesmaximal, characteristic, complemented
3#5072C212yesmaximal, complemented
4#4546C211yesmaximal, characteristic, complemented
5#4459C212yesmaximal, complemented
6#1624—31nomaximal, Sylow
7C2×C3×SD16—22no—
8C3×C4⋊Q8C2211yescharacteristic
9C3×#127C2211yescharacteristic
10#947—22no—
11C2×#213—22no—
12#895—22no—
13#892—22no—
14C4×D12C2212yes—
15C3×C4×C8C2211yescharacteristic
16#832—22no—
17C4⋊(C3⋊C8)C2212yes—
18#555—31no—
19#492D311yescharacteristic, complemented
20#459—32no—
21#333—31no—
22D4⋊C8—32no—
23C2×C3×Q8D412yes—
24C2×C3×D4—22no—
25C2×C3×D4—21no—
26C2×D12—22no—
27C2×D12D412yes—
28C2×C4×D3—22no—
29C3×SD16—44no—
30C2×C3×C8—21no—
31C3×C4⋊C4—22no—
32C3×C42C2311yescharacteristic
33#213—44no—
34(C2×D3)⋊C4—24no—
35(C2×D3)⋊C4—22no—
36C4⋊Dic3D412yes—
37C4×Dic3—22no—
38C2×C3⋊C8—22no—
39C2×SD16—22no—
40C2×D8—62no—
41C4⋊Q8C2×D311yescharacteristic
42#127C2×D311yescharacteristic
43#124—62no—
44#121—62no—
45C4×D4—32no—
46#117—62no—
47C4⋊C8—32no—
48#102—62no—
49C4×C8C2×D311yescharacteristic
50C23×C3—41no—
51C22×D3—22no—
52C3×Q8—42no—
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—22no—
61D12—22no—
62D12—82no—
63C4×D3—42no—
64C4×D3—42no—
65C3×C8—22no—
66C3⋊C8—42no—
67C2×Q8#6712yes—
68C2×D4—22no—
69C2×D4—21no—
70C2×D4—32no—
71C2×D4—62no—
72C22×C4—62no—
73SD16—44no—
74D8—124no—
75C2×C8—21no—
76C2×C8—62no—
77C4⋊C4—22no—
78C4⋊C4—32no—
79C22⋊C4—64no—
80C22⋊C4—62no—
81C42—62no—
82C42C22×D311yescharacteristic
83C22×C3—82no—
84C22×C3—81no—
85C22×C3#12011yescharacteristic
86C2×D3—42no—
87C2×D3—42no—
88C2×D3—42no—
89C3×C4—21no—
90C3×C4—42no—
91C3×C4#13512yes—
92C3×C4AGL(1,ℤ/8)12yes—
93Dic3—42no—
94C23—41no—
95C23—62no—
96Q8—42no—
97D4—42no—
98D4—42no—
99D4—42no—
100D4—62no—
101D4—62no—
102D4—242no—
103C2×C4—22no—
104C2×C4—62no—
105C2×C4—122no—
106C2×C4—122no—
107C2×C4D3×D412yes—
108C2×C4C2×#6711yescharacteristic
109C8—22no—
110C8—122no—
111C2×C3—161no—
112C2×C3#44912yes—
113C2×C3#45311yescharacteristic
114D3—82no—
115C22—82no—
116C22—81no—
117C22—122no—
118C22—122no—
119C22—122no—
120C22#93711yescharacteristic
121C4—21no—
122C4—42no—
123C4—122no—
124C4#91412yes—
125C4#91612yes—
126C3#162411yescharacteristic, complemented, Sylow
127C2—161no—
128C2—242no—
129C2#507711yescharacteristic
130C2#517212yes—
subgroupG/H
centerC22#937
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#492C2
socleC22×C3#120
hypercenter#492D3
hypocenterC3#1624
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #5734C2semidirect
2. C3×#492C2semidirect
3. #5072C2semidirect
4. #4546C2semidirect
5. #4459C2semidirect
19. #492D3semidirect
126. C3#1624semidirect

Maximal subgroups

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

subgroupindex
#57342
C3×#4922
#50722
#45462
#44592
#16243