Finite Group Databasebeta

← A4×#3708 #13165843

Subgroups — A4×#3708 #13165843

Order 1728.

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

normal subgroups only

This group has 142 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
1A4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C22×#3708C311yesmaximal, characteristic, complemented
3C32×C4×A4C411yescharacteristic
4C3×#3708—41nomaximal
5C22×C3⋊(C3⋊C8)C2×C311yescharacteristic
6C2×#3708—31no—
7C2×#3708—31no—
8A4×C3⋊C8—62no—
9C2×C32×A4C811yescharacteristic
10C3×C3⋊(C3⋊C8)—41no—
11A4×C16—91nomaximal
12C22×C32×C4C3×C411yescharacteristic
13C3×C4×A4—22no—
14C3×C4×A4—42no—
15C2×C3⋊(C3⋊C8)—31no—
16C2×C3⋊(C3⋊C8)—31no—
17#3708—31no—
18#3708A411yescharacteristic, direct product factor
19C32×A4C1611yescharacteristic, complemented
20C33×C4—41no—
21C22×C3⋊C8—62no—
22C8×A4—91no—
23C23×C32C3×C811yescharacteristic
24C2×C3×A4—22no—
25C2×C3×A4—42no—
26C2×C32×C4—31no—
27C2×C32×C4—31no—
28C3⋊(C3⋊C8)—31no—
29C3⋊(C3⋊C8)C2×A411yescharacteristic
30C3×C3⋊C8—242no—
31C22×C16—91noSylow
32C2×C33—41no—
33C22×C3×C4—22no—
34C4×A4—42no—
35C4×A4#15711yescharacteristic
36C2×C3⋊C8—182no—
37C2×C3⋊C8—182no—
38C3×C16—361no—
39C22×C32—31no—
40C22×C32—31no—
41C22×C32C3×C1611yescharacteristic, complemented
42C3×A4—22no—
43C3×A4—42no—
44C32×C4—31no—
45C32×C4—82no—
46C32×C4—162no—
47C32×C4C4×A411yescharacteristic
48C22×C8—91no—
49C2×C16—271no—
50C2×C16—271no—
51C33—41noSylow
52C23×C3—22no—
53C2×A4—42no—
54C2×A4#62211yescharacteristic
55C2×C3×C4—62no—
56C2×C3×C4—62no—
57C3×C8—361no—
58C3⋊C8—62no—
59C3⋊C8—182no—
60C2×C32—31no—
61C2×C32—31no—
62C2×C32—82no—
63C2×C32—162no—
64C2×C32C8×A411yescharacteristic
65C22×C4C3×#15711yescharacteristic
66C2×C8—271no—
67C2×C8—271no—
68C16—91no—
69C16—271no—
70C22×C3—22no—
71C22×C3—62no—
72C22×C3—62no—
73A4—42no—
74A4#370811yescharacteristic, direct product factor
75C3×C4—22no—
76C3×C4—41no—
77C3×C4—62no—
78C3×C4—162no—
79C32—82no—
80C32—162no—
81C32A4×C1611yescharacteristic, complemented
82C23C3×#62211yescharacteristic
83C2×C4—31no—
84C2×C4—31no—
85C8—91no—
86C8—271no—
87C2×C3—22no—
88C2×C3—41no—
89C2×C3—62no—
90C2×C3—62no—
91C2×C3—162no—
92C22—31no—
93C22—31no—
94C22C3×#370811yescharacteristic, complemented
95C4—31no—
96C4A4×#15711yescharacteristic
97C3—22no—
98C3—41no—
99C3—162no—
100C2—31no—
101C2—31no—
102C2A4×#62211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4A4×#157
commutator, hypocenterC22×C32C3×C16
FittingC22×C32×C4C3×C4
socleC23×C32C3×C8
prime radicalC2×C32×A4C8
cosocleC22×C3⋊(C3⋊C8)C2×C3
LCSG ▷ C22×C32
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C22×#3708C3semidirect
18. #3708A4direct (N × H)
19. C32×A4C16semidirect
41. C22×C32C3×C16semidirect
74. A4#3708direct (N × H)
81. C32A4×C16semidirect
94. C22C3×#3708semidirect

Maximal subgroups

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

subgroupindex
A4×C3⋊(C3⋊C8)2
C22×#37083
C3×#37084
A4×C169