Finite Group Databasebeta

← #64269

Subgroups — #64269

Order 288.

156 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 27 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
1C22×C3×A4C211yesmaximal, characteristic, complemented
2C2×C3⋊S4C211yesmaximal, characteristic, complemented
3C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4GL(2,ℤ/4)—33nomaximal
5#953—31nomaximal
6C2×C3×A4—21no—
7C2×C3×A4C2211yescharacteristic
8C3⋊S4—21no—
9#638—41nomaximal
10C24×C3D311yescharacteristic, complemented
11C22×A4D313yescomplemented
12C2×S4—33no—
13C2×#67—31no—
14C2×#67—31no—
15C2×#67—31no—
16SL(2,ℤ/4)—33no—
17#217—31no—
18#217—31no—
19#217—31no—
20C22×C32—41no—
21C2×C3⋊D3—41no—
22C3×A4D411yescharacteristic, complemented
23C3⋊Dic3—41no—
24#120—91noSylow
25C23×C3—21no—
26C23×C3—31no—
27C23×C3—31no—
28C23×C3—61no—
29C23×C3C2×D311yescharacteristic
30C22×D3—31no—
31C2×A4—23no—
32C2×A4C2×D313yes—
33S4—63no—
34#67—61no—
35#67—62no—
36#67—61no—
37#67—61no—
38#67—61no—
39#67—123no—
40C2×Dic3—31no—
41C2×Dic3—31no—
42C2×Dic3—31no—
43C2×C32—41no—
44C2×C32—81no—
45C3⋊D3—81no—
46C24C3⋊D311yescharacteristic, complemented
47C2×D4—91no—
48C2×D4—91no—
49C2×D4—91no—
50C22⋊C4—91no—
51C22⋊C4—91no—
52C22⋊C4—91no—
53C22×C3—31no—
54C22×C3—31no—
55C22×C3—32no—
56C22×C3—31no—
57C22×C3—43no—
58C22×C3—61no—
59C22×C3—61no—
60C22×C3—61no—
61C22×C3#6711yescharacteristic, complemented
62C22×C3S411yescharacteristic, complemented
63C2×D3—61no—
64C2×D3—61no—
65C2×D3—61no—
66C2×D3—123no—
67A4#6713yescomplemented
68Dic3—61no—
69Dic3—61no—
70Dic3—61no—
71Dic3—123no—
72C32—41noSylow
73C23—21no—
74C23—31no—
75C23—31no—
76C23—61no—
77C23—91no—
78C23C2×C3⋊D311yescharacteristic
79D4—181no—
80D4—182no—
81D4—181no—
82D4—181no—
83D4—181no—
84C2×C4—91no—
85C2×C4—91no—
86C2×C4—91no—
87C2×C3—21no—
88C2×C3—31no—
89C2×C3—31no—
90C2×C3—43no—
91C2×C3—61no—
92C2×C3—83no—
93C2×C3C2×S411yescharacteristic
94D3—121no—
95D3—243no—
96C22—31no—
97C22—31no—
98C22—32no—
99C22—31no—
100C22—61no—
101C22—61no—
102C22—61no—
103C22—181no—
104C22—181no—
105C22—181no—
106C22#63811yescharacteristic, complemented
107C22C3⋊S411yescharacteristic, complemented
108C4—181no—
109C4—181no—
110C4—181no—
111C3—43no—
112C3GL(2,ℤ/4)11yescharacteristic, complemented
113C2—21no—
114C2—31no—
115C2—31no—
116C2—61no—
117C2—361no—
118C2C2×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊S4
commutator, cosocleC2×C3×A4C22
FittingC24×C3D3
socleC23×C3C2×D3
hypercenterC22C3⋊S4
hypocenterC3×A4D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×A4C2semidirect
2. C2×C3⋊S4C2semidirect
3. C3⋊SL(2,ℤ/4)C2semidirect
10. C24×C3D3semidirect
11. C22×A4D3semidirect
22. C3×A4D4semidirect
46. C24C3⋊D3semidirect
61. C22×C3#67semidirect
62. C22×C3S4semidirect
67. A4#67semidirect
106. C22#638semidirect
107. C22C3⋊S4semidirect
112. C3GL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C22×C3×A42
C2×C3⋊S42
C3⋊SL(2,ℤ/4)2
GL(2,ℤ/4)3
#9533
#6384