Finite Group Databasebeta

← C22×C8×A4 #85624

Subgroups — C22×C8×A4 #85624

Order 384.

456 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 114 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×C4×A4C211yesmaximal, characteristic
2C2×C8×A4C216yesmaximal, direct product factor
3C24×C8C311yesmaximal, characteristic, complemented, Sylow
4C23×A4C411yescharacteristic
5C2×C4×A4C413yes—
6C2×C4×A4C2213yes—
7C22×C3×C8—41nomaximal
8C8×A4C2214yesdirect product factor
9C24×C4C2×C311yescharacteristic
10C23×C8—31no—
11C23×C8—31no—
12C23×C8—36no—
13C23×C8C2×C316yescomplemented
14C22×A4C814yesdirect product factor
15C22×A4C2×C413yes—
16C22×C3×C4—41no—
17C4×A4C2×C413yes—
18C4×A4C2311yescharacteristic
19C2×C3×C8—46no—
20C25C3×C411yescharacteristic
21C23×C4—31no—
22C23×C4—33no—
23C23×C4—31no—
24C23×C4—33no—
25C23×C4C3×C413yes—
26C23×C4C22×C313yes—
27C22×C8—36no—
28C22×C8—38no—
29C22×C8—31no—
30C22×C8—36no—
31C22×C8—36no—
32C22×C8—36no—
33C22×C8—312no—
34C22×C8A411yescharacteristic, direct product factor
35C22×C8C22×C314yescomplemented
36C23×C3—41no—
37C2×A4C2×C816yesdirect product factor
38C2×A4C22×C411yescharacteristic
39C2×C3×C4—43no—
40C2×C3×C4—43no—
41C3×C8—44no—
42C24—33no—
43C24—34no—
44C24—31no—
45C24C3×C814yescomplemented
46C24C2×C3×C413yes—
47C22×C4—32no—
48C22×C4—36no—
49C22×C4—36no—
50C22×C4—31no—
51C22×C4—33no—
52C22×C4—33no—
53C22×C4—33no—
54C22×C4—33no—
55C22×C4—33no—
56C22×C4—33no—
57C22×C4—36no—
58C22×C4—33no—
59C22×C4—33no—
60C22×C4C2×C3×C413yes—
61C22×C4C2×A411yescharacteristic
62C22×C4C23×C311yescharacteristic
63C2×C8—312no—
64C2×C8—36no—
65C2×C8—312no—
66C2×C8—34no—
67C2×C8—34no—
68C2×C8C2×A416yesdirect product factor
69C22×C3—44no—
70C22×C3—43no—
71A4C22×C811yescharacteristic, direct product factor
72C3×C4—43no—
73C3×C4—41no—
74C23—33no—
75C23—312no—
76C23—36no—
77C23—32no—
78C23—312no—
79C23—34no—
80C23—34no—
81C23—33no—
82C23—33no—
83C23C2×C3×C816yescomplemented
84C23C4×A411yescharacteristic
85C23C22×C3×C411yescharacteristic
86C2×C4—36no—
87C2×C4—36no—
88C2×C4—33no—
89C2×C4—33no—
90C2×C4—33no—
91C2×C4—36no—
92C2×C4—33no—
93C2×C4—33no—
94C2×C4—31no—
95C2×C4—33no—
96C2×C4—31no—
97C2×C4C4×A413yes—
98C2×C4C22×A413yes—
99C8—34no—
100C8C22×A414yesdirect product factor
101C2×C3—41no—
102C2×C3—46no—
103C22—36no—
104C22—31no—
105C22—36no—
106C22—38no—
107C22—33no—
108C22—31no—
109C22—312no—
110C22—36no—
111C22—36no—
112C22C8×A414yesdirect product factor
113C22C22×C3×C811yescharacteristic, complemented
114C22C2×C4×A413yes—
115C4—31no—
116C4—33no—
117C4C2×C4×A413yes—
118C4C23×A411yescharacteristic
119C3—41noSylow
120C2—36no—
121C2—31no—
122C2—31no—
123C2C2×C8×A416yesdirect product factor
124C2C22×C4×A411yescharacteristic
subgroupG/H
center, hypercenterC22×C8A4
commutator, hypocenterC22C22×C3×C8
FrattiniC4C23×A4
FittingC24×C8C3
socleC25C3×C4
prime radicalC23×A4C4
cosocleC22×C4C23×C3
LCSG ▷ C22
UCSC1 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C2×C8×A4C2direct (N × H)
3. C24×C8C3semidirect
8. C8×A4C22direct (N × H)
13. C23×C8C2×C3semidirect
14. C22×A4C8direct (N × H)
34. C22×C8A4direct (N × H)
35. C22×C8C22×C3semidirect
37. C2×A4C2×C8direct (N × H)
45. C24C3×C8semidirect
68. C2×C8C2×A4direct (N × H)
71. A4C22×C8direct (N × H)
83. C23C2×C3×C8semidirect
100. C8C22×A4direct (N × H)
112. C22C8×A4direct (N × H)
113. C22C22×C3×C8semidirect
123. C2C2×C8×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×A42
C2×C8×A42
C24×C83
C22×C3×C84