Finite Group Databasebeta

← C3⋊D3×#126 #10596174

Subgroups — C3⋊D3×#126 #10596174

Order 576.

738 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C32×#126C211yesmaximal, characteristic, complemented
2#64107C213yesmaximal, complemented
3C4⋊C4×C3⋊D3C213yesmaximal, complemented
4C22⋊C4×C3⋊D3C213yesmaximal
5#64088C213yesmaximal, complemented
6#64085C211yesmaximal, characteristic, complemented
7C42×C3⋊D3C211yesmaximal, characteristic, complemented
8D3×#126—34nomaximal
9C23×C3⋊D3C2211yescharacteristic
10C2×C4×C3⋊D3C2213yes—
11C2×C4×C3⋊D3C2213yes—
12C32×C4⋊C4C2213yescomplemented
13C32×C22⋊C4C2213yes—
14C32×C42C2211yescharacteristic, complemented
15#3757C2213yes—
16#3752C2213yes—
17#3752C2213yes—
18C4⋊(C3⋊Dic3)C2213yescomplemented
19#3750C2213yescomplemented
20#3750C2213yescomplemented
21C4×C3⋊Dic3C2213yescomplemented
22C3×#126D314yescomplemented
23#898—312no—
24D3×C4⋊C4—312no—
25D3×C22⋊C4—312no—
26#879—312no—
27#876—34no—
28C42×D3—34no—
29C23×C32C2311yescharacteristic
30C22×C3⋊D3—26no—
31C22×C3⋊D3C2311yescharacteristic
32C22×C3⋊D3C2311yescharacteristic
33C2×C32×C4C2313yes—
34C2×C32×C4C2313yes—
35C2×C3⋊Dic3C2313yes—
36C2×C3⋊Dic3C2313yes—
37C4×C3⋊D3—26no—
38C4×C3⋊D3—26no—
39C2×#126—91noSylow
40C23×D3—34no—
41C2×C4×D3—312no—
42C2×C4×D3—312no—
43C3×C4⋊C4C2×D3112yescomplemented
44C3×C22⋊C4C2×D3112yes—
45C3×C42C2×D314yescomplemented
46#217—312no—
47(C2×D3)⋊C4—312no—
48(C2×D3)⋊C4—312no—
49C4⋊Dic3—312no—
50Dic3⋊C4—312no—
51Dic3⋊C4—312no—
52C4×Dic3—312no—
53C22×C32—23no—
54C22×C32C2411yescharacteristic
55C2×C3⋊D3—23no—
56C2×C3⋊D3—44no—
57C2×C3⋊D3#4116yes—
58C32×C4—23no—
59C32×C4—23no—
60C3⋊Dic3—23no—
61C3⋊Dic3—23no—
62#126—93no—
63#126—93no—
64#126—91no—
65#126C3⋊D311yescharacteristic, direct product factor
66C2×C4⋊C4—93no—
67C2×C22⋊C4—93no—
68C2×C42—91no—
69C23×C3C22×D314yes—
70C22×D3—34no—
71C22×D3—34no—
72C22×D3—624no—
73C2×C3×C4C22×D3112yes—
74C2×C3×C4C22×D3112yes—
75C2×Dic3—312no—
76C2×Dic3—312no—
77C4×D3—624no—
78C4×D3—624no—
79C2×C32—41no—
80C2×C32C2×#4113yes—
81C3⋊D3—41no—
82C3⋊D3#12614yesdirect product factor
83C24—91no—
84C22×C4—93no—
85C22×C4—93no—
86C4⋊C4—93no—
87C4⋊C4—93no—
88C4⋊C4—93no—
89C4⋊C4C2×C3⋊D313yescomplemented
90C22⋊C4—93no—
91C22⋊C4—93no—
92C22⋊C4—93no—
93C22⋊C4C2×C3⋊D313yes—
94C42—93no—
95C42C2×C3⋊D311yescharacteristic, complemented
96C22×C3—212no—
97C22×C3C23×D314yes—
98C2×D3—324no—
99C2×D3—612no—
100C2×D3—1216no—
101C3×C4—212no—
102C3×C4—212no—
103Dic3—612no—
104Dic3—612no—
105C32C2×#12611yescharacteristic, complemented, Sylow
106C23—91no—
107C23—91no—
108C23—186no—
109C23C22×C3⋊D311yescharacteristic
110C2×C4—93no—
111C2×C4—93no—
112C2×C4—186no—
113C2×C4—186no—
114C2×C4C22×C3⋊D313yes—
115C2×C4C22×C3⋊D313yes—
116C2×C3—44no—
117C2×C3D3×#41112yes—
118D3—316no—
119D3—124no—
120C22—23no—
121C22—96no—
122C22—183no—
123C22—364no—
124C22C23×C3⋊D311yescharacteristic
125C4—23no—
126C4—23no—
127C4—183no—
128C4—183no—
129C3D3×#12614yescomplemented
130C2—41no—
131C2—94no—
132C2—361no—
133C2#41×C3⋊D313yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#126C2
prime radicalC23×C3⋊D3C22
hypercenter#126C3⋊D3
hypocenterC32C2×#126
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C32×#126C2semidirect
2. #64107C2semidirect
3. C4⋊C4×C3⋊D3C2semidirect
5. #64088C2semidirect
6. #64085C2semidirect
7. C42×C3⋊D3C2semidirect
12. C32×C4⋊C4C22semidirect
14. C32×C42C22semidirect
18. C4⋊(C3⋊Dic3)C22semidirect
19. #3750C22semidirect
20. #3750C22semidirect
21. C4×C3⋊Dic3C22semidirect
22. C3×#126D3semidirect
43. C3×C4⋊C4C2×D3semidirect
45. C3×C42C2×D3semidirect
65. #126C3⋊D3direct (N × H)
82. C3⋊D3#126direct (N × H)
89. C4⋊C4C2×C3⋊D3semidirect
95. C42C2×C3⋊D3semidirect
105. C32C2×#126semidirect
129. C3D3×#126semidirect

Maximal subgroups

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

subgroupindex
C32×#1262
#641072
C4⋊C4×C3⋊D32
C22⋊C4×C3⋊D32
#640882
#640852
C42×C3⋊D32
D3×#1263