Finite Group Databasebeta

← #4174⋊C4 #10619546

Subgroups — #4174⋊C4 #10619546

Order 648.

134 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 22 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
1C2×#4174C211yesmaximal, characteristic
2#66330C211yesmaximal, characteristic, complemented
3#66322C211yesmaximal, characteristic, complemented
4D3×C3⋊Dic3—31nomaximal
5#6199—31nomaximal
6C2×AΣL(1,𝔽27)C2211yescharacteristic
7#4174C412yescomplemented
8C2×C32×D3—31no—
9C3⋊(C3⋊Dic3)D311yescharacteristic, complemented
10C3×C3⋊Dic3—31no—
11C2×#272D311yescharacteristic
123−1+2⋊C4—61no—
13(C3⋊Dic3)⋊C3—31no—
14(C3⋊Dic3)⋊C3—31no—
15AΣL(1,𝔽27)C2×C411yescharacteristic, complemented, Sylow
16C2×C3⋊Dic3—91no—
17D3×Dic3—91no—
18D3×Dic3—91no—
19D3×Dic3—91no—
20D3×Dic3—91no—
21D3×Dic3—91no—
22C2×C33C2×D311yescharacteristic
23C32×D3—32no—
24C2×3−1+2—21no—
25C2×H(3,𝔽3)C2×D311yescharacteristic
26#272Dic312yescomplemented
27C22×C32—91no—
28C2×C3×D3—31no—
29C2×C3×D3—31no—
30C2×C3×D3—31no—
31C2×C3×D3—31no—
32C2×C3×D3—181no—
33C3⋊Dic3—31no—
34C3⋊Dic3—31no—
35C3⋊Dic3—31no—
36C3⋊Dic3—91no—
37C3⋊Dic3—91no—
38C3⋊Dic3—181no—
39C3⋊Dic3—181no—
40C3×Dic3—91no—
41C3×Dic3—91no—
42C3×Dic3—91no—
43C3×Dic3—91no—
44C3×Dic3—91no—
45Dic9—61no—
46C33C4×D311yescharacteristic, complemented
473−1+2—21no—
48H(3,𝔽3)C2×Dic311yescharacteristic, complemented
49C2×Dic3—271no—
50C2×Dic3—271no—
51C2×Dic3—271no—
52C2×Dic3—271no—
53C4×D3—271no—
54C4×D3—271no—
55C2×C32—31no—
56C2×C32—31no—
57C2×C32—61no—
58C2×C32—61no—
59C2×C32—61no—
60C2×C32—92no—
61C2×C32D3211yescharacteristic
62C3×D3—32no—
63C3×D3—32no—
64C3×D3—32no—
65C3×D3—32no—
66C3×D3—182no—
67C2×C9—121no—
68C22×C3—91no—
69C22×C3—91no—
70C22×C3—91no—
71C22×C3—91no—
72C2×D3—31no—
73C2×D3—181no—
74C3×C4—271no—
75C3×C4—271no—
76Dic3—91no—
77Dic3—91no—
78Dic3—91no—
79Dic3—91no—
80Dic3—91no—
81Dic3—271no—
82Dic3—541no—
83Dic3—541no—
84Dic3—541no—
85Dic3—541no—
86C32—31no—
87C32—31no—
88C32—61no—
89C32—61no—
90C32—61no—
91C32D3×Dic311yescharacteristic
92C9—121no—
93C2×C4—811noSylow
94C2×C3—31no—
95C2×C3—61no—
96C2×C3—61no—
97C2×C3—61no—
98C2×C3—92no—
99C2×C3—92no—
100C2×C3—92no—
101C2×C3—92no—
102C2×C3—361no—
103C2×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic
104D3—32no—
105D3—182no—
106C22—91no—
107C4—271no—
108C4—271no—
109C3—31no—
110C3—61no—
111C3—61no—
112C3—61no—
113C3—361no—
114C3((C3⋊D3)⋊C3)⋊C411yescharacteristic
115C2—92no—
116C2#4174⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2#4174⋊C2
commutator, hypocenterAΣL(1,𝔽27)C2×C4
FrattiniC2×C32D32
Fitting, cosocleC2×AΣL(1,𝔽27)C22
socleC2×C3((C3⋊D3)⋊C3)⋊C2
prime radicalC2×#4174C2
LCSG ▷ AΣL(1,𝔽27)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #66330C2semidirect
3. #66322C2semidirect
7. #4174C4semidirect
9. C3⋊(C3⋊Dic3)D3semidirect
15. AΣL(1,𝔽27)C2×C4semidirect
26. #272Dic3semidirect
46. C33C4×D3semidirect
48. H(3,𝔽3)C2×Dic3semidirect

Maximal subgroups

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

subgroupindex
C2×#41742
#663302
#663222
D3×C3⋊Dic33
#61993