Finite Group Databasebeta

← C2×C4×C22⋊A4 #87538

Subgroups — C2×C4×C22⋊A4 #87538

Order 384.

1852 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 1852 conjugacy classes of subgroups — too many to draw. Only its 64 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×C22⋊A4C211yesmaximal, characteristic
2C4×C22⋊A4C212yesmaximal, direct product factor
3C25×C4C311yesmaximal, characteristic, complemented, Sylow
4C2×C22⋊A4C412yesdirect product factor
5C2×C22⋊A4C2211yescharacteristic
6C2×C4×A4—45nomaximal
7C26C2×C311yescharacteristic
8C24×C4—35no—
9C24×C4—35no—
10C24×C4—310no—
11C24×C4C2×C312yescomplemented
12C22⋊A4C2×C411yescharacteristic, direct product factor
13C22×A4—45no—
14C4×A4—410no—
15C25—35no—
16C25—310no—
17C25—35no—
18C25C3×C412yescomplemented
19C25C22×C311yescharacteristic
20C23×C4—360no—
21C23×C4—360no—
22C23×C4—330no—
23C23×C4—310no—
24C23×C4—310no—
25C23×C4—35no—
26C23×C4—310no—
27C23×C4—310no—
28C23×C4—310no—
29C23×C4A415yescomplemented
30C2×A4—410no—
31C2×A4—45no—
32C2×C3×C4—161no—
33C24—310no—
34C24—35no—
35C24—330no—
36C24—310no—
37C24—360no—
38C24—360no—
39C24—310no—
40C24—310no—
41C24—310no—
42C24—35no—
43C24—35no—
44C24C2×C3×C411yescharacteristic, complemented
45C24C2×A415yes—
46C22×C4—330no—
47C22×C4—3120no—
48C22×C4—360no—
49C22×C4—360no—
50C22×C4—330no—
51C22×C4—360no—
52C22×C4—320no—
53C22×C4—310no—
54C22×C4—35no—
55C22×C4—35no—
56C22×C4—310no—
57C22×C4C2×A4110yescomplemented
58C22×C3—161no—
59A4—45no—
60C3×C4—162no—
61C23—35no—
62C23—310no—
63C23—330no—
64C23—310no—
65C23—360no—
66C23—360no—
67C23—330no—
68C23—330no—
69C23—3120no—
70C23—360no—
71C23—320no—
72C23—310no—
73C23—35no—
74C23—35no—
75C23—35no—
76C23C4×A4110yescomplemented
77C23C22×A415yes—
78C2×C4—360no—
79C2×C4—310no—
80C2×C4—360no—
81C2×C4—35no—
82C2×C4—310no—
83C2×C4—310no—
84C2×C4—310no—
85C2×C4C22⋊A411yescharacteristic, direct product factor
86C2×C3—162no—
87C2×C3—161no—
88C22—310no—
89C22—35no—
90C22—310no—
91C22—360no—
92C22—360no—
93C22—330no—
94C22—310no—
95C22—310no—
96C22—310no—
97C22—35no—
98C22—35no—
99C22C2×C4×A415yescomplemented
100C22C2×C22⋊A411yescharacteristic
101C4—310no—
102C4C2×C22⋊A412yesdirect product factor
103C3—161noSylow
104C2—310no—
105C2—35no—
106C2—35no—
107C2C4×C22⋊A412yesdirect product factor
108C2C22×C22⋊A411yescharacteristic
subgroupG/H
center, hypercenterC2×C4C22⋊A4
commutator, hypocenterC24C2×C3×C4
FrattiniC2C22×C22⋊A4
FittingC25×C4C3
socleC26C2×C3
prime radicalC22×C22⋊A4C2
cosocleC25C22×C3
LCSG ▷ C24
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊A4C2direct (N × H)
3. C25×C4C3semidirect
4. C2×C22⋊A4C4direct (N × H)
11. C24×C4C2×C3semidirect
12. C22⋊A4C2×C4direct (N × H)
18. C25C3×C4semidirect
29. C23×C4A4semidirect
44. C24C2×C3×C4semidirect
57. C22×C4C2×A4semidirect
76. C23C4×A4semidirect
85. C2×C4C22⋊A4direct (N × H)
99. C22C2×C4×A4semidirect
102. C4C2×C22⋊A4direct (N × H)
107. C2C4×C22⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊A42
C4×C22⋊A42
C25×C43
C2×C4×A44