Finite Group Databasebeta

← C3×GL(2,ℤ/4) #64258

Subgroups — C3×GL(2,ℤ/4) #64258

Order 288.

134 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 24 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)C311yesmaximal, characteristic, direct product factor
5C3×#120—31nomaximal
6C2×C3×A4—21no—
7C2×C3×A4C2211yescharacteristic
8C3×S4—21no—
9C3×#67—41nomaximal
10C24×C3D311yescharacteristic, complemented
11C22×A4—21no—
12C22×A4C2×C311yescharacteristic, complemented
13C2×S4C2×C311yescharacteristic, complemented
14C2×C3×D4—31no—
15C2×C3×D4—31no—
16C2×C3×D4—31no—
17SL(2,ℤ/4)C2×C311yescharacteristic, complemented
18C3×C22⋊C4—31no—
19C3×C22⋊C4—31no—
20C3×C22⋊C4—31no—
21C22×C32—41no—
22C2×C3×D3—41no—
23C3×A4D411yescharacteristic, complemented
24C3×Dic3—41no—
25#120—31noSylow
26C23×C3—21no—
27C23×C3—31no—
28C23×C3—31no—
29C23×C3—31no—
30C23×C3—61no—
31C23×C3C2×D311yescharacteristic
32C2×A4—22no—
33C2×A4—21no—
34C2×A4—21no—
35C2×A4C22×C311yescharacteristic
36S4—21no—
37C3×D4—61no—
38C3×D4—62no—
39C3×D4—61no—
40C3×D4—61no—
41C3×D4—61no—
42C2×C3×C4—31no—
43C2×C3×C4—31no—
44C2×C3×C4—31no—
45#67—41no—
46C2×C32—41no—
47C2×C32—81no—
48C3×D3—81no—
49C24C3×D311yescharacteristic, complemented
50C2×D4—31no—
51C2×D4—31no—
52C2×D4—31no—
53C22⋊C4—31no—
54C22⋊C4—31no—
55C22⋊C4—31no—
56C22×C3—31no—
57C22×C3—31no—
58C22×C3—32no—
59C22×C3—31no—
60C22×C3—41no—
61C22×C3—61no—
62C22×C3—61no—
63C22×C3—61no—
64C22×C3—61no—
65C22×C3—61no—
66C22×C3—61no—
67C22×C3—81no—
68C22×C3#6711yescharacteristic, complemented
69C22×C3S411yescharacteristic, complemented
70C2×D3—41no—
71A4—21no—
72A4C3×D411yescharacteristic, complemented
73C3×C4—61no—
74C3×C4—61no—
75C3×C4—61no—
76Dic3—41no—
77C32—41noSylow
78C23—21no—
79C23—31no—
80C23—31no—
81C23—31no—
82C23—61no—
83C23C2×C3×D311yescharacteristic
84D4—61no—
85D4—62no—
86D4—61no—
87D4—61no—
88D4—61no—
89C2×C4—31no—
90C2×C4—31no—
91C2×C4—31no—
92C2×C3—21no—
93C2×C3—31no—
94C2×C3—31no—
95C2×C3—41no—
96C2×C3—61no—
97C2×C3—81no—
98C2×C3—82no—
99C2×C3—81no—
100C2×C3—121no—
101C2×C3C2×S411yescharacteristic
102D3—81no—
103C22—31no—
104C22—31no—
105C22—32no—
106C22—31no—
107C22—61no—
108C22—61no—
109C22—61no—
110C22—61no—
111C22—61no—
112C22—61no—
113C22C3×#6711yescharacteristic, complemented
114C22C3×S411yescharacteristic, complemented
115C4—61no—
116C4—61no—
117C4—61no—
118C3—41no—
119C3—81no—
120C3GL(2,ℤ/4)11yescharacteristic, direct product factor
121C2—21no—
122C2—31no—
123C2—31no—
124C2—61no—
125C2—121no—
126C2C2×C3×S411yescharacteristic
subgroupG/H
centerC2×C3C2×S4
commutator, cosocleC2×A4C22×C3
FrattiniC2C2×C3×S4
FittingC24×C3D3
socleC23×C3C2×D3
hypercenterC22×C3S4
hypocenterA4C3×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C3 ◁ C22×C3

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
4. GL(2,ℤ/4)C3direct (N × H)
10. C24×C3D3semidirect
12. C22×A4C2×C3semidirect
13. C2×S4C2×C3semidirect
17. SL(2,ℤ/4)C2×C3semidirect
23. C3×A4D4semidirect
49. C24C3×D3semidirect
68. C22×C3#67semidirect
69. C22×C3S4semidirect
72. A4C3×D4semidirect
113. C22C3×#67semidirect
114. C22C3×S4semidirect
120. C3GL(2,ℤ/4)direct (N × H)

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
C3×#1203
C3×#674