Finite Group Databasebeta

← C22×A4⋊Q8 #87471

Subgroups — C22×A4⋊Q8 #87471

Order 384.

867 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 867 conjugacy classes of subgroups — too many to draw. Only its 132 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
2C22×SL(2,ℤ/4)C212yesmaximal
3C2×A4⋊Q8C2112yesmaximal, direct product factor
4C22×C22⋊Q8—31nomaximal, Sylow
5C23×A4C2211yescharacteristic
6C22×Dic6—41nomaximal
7C2×C4×A4C2216yes—
8C2×SL(2,ℤ/4)C22112yes—
9A4⋊Q8C22116yesdirect product factor
10C23×Q8—31no—
11C24×C4D311yescharacteristic
12C2×C22⋊Q8—312no—
13C2×C22⋊Q8—312no—
14C22×C4⋊C4—31no—
15C22×C4⋊C4—32no—
16C22×C22⋊C4—32no—
17C22×A4Q814yescomplemented
18C22×A4C2313yes—
19C22×C3×C4—41no—
20C22×Dic3—42no—
21C2×Dic6—412no—
22C4×A4C2314yes—
23SL(2,ℤ/4)C2318yes—
24C25C2×D311yescharacteristic
25C22×Q8—312no—
26C22×Q8—61no—
27C22×Q8—66no—
28C22×Q8—61no—
29C23×C4—31no—
30C23×C4—31no—
31C23×C4—36no—
32C23×C4—32no—
33C23×C4—32no—
34C23×C4C2×D316yes—
35C22⋊Q8—348no—
36C22⋊Q8—316no—
37C2×C4⋊C4—312no—
38C2×C4⋊C4—324no—
39C2×C22⋊C4—312no—
40C2×C22⋊C4—312no—
41C23×C3—41no—
42C2×A4C2×Q816yescomplemented
43C2×A4C2411yescharacteristic
44C2×C3×C4—46no—
45C2×Dic3—412no—
46Dic6—416no—
47C24—31no—
48C24—33no—
49C24—34no—
50C24Dic614yescomplemented
51C24C22×D313yes—
52C2×Q8—316no—
53C2×Q8—612no—
54C2×Q8—624no—
55C2×Q8—612no—
56C22×C4—31no—
57C22×C4—36no—
58C22×C4—36no—
59C22×C4—36no—
60C22×C4—312no—
61C22×C4—312no—
62C22×C4—312no—
63C22×C4—64no—
64C22×C4—63no—
65C22×C4—68no—
66C22×C4—66no—
67C22×C4—62no—
68C22×C4S411yescharacteristic
69C22×C4C22×D314yes—
70C4⋊C4—316no—
71C4⋊C4—332no—
72C22⋊C4—324no—
73C22⋊C4—38no—
74C22×C3—44no—
75C22×C3—43no—
76A4C22×Q811yescharacteristic, complemented
77C3×C4—44no—
78Dic3—48no—
79C23—34no—
80C23—34no—
81C23—33no—
82C23—33no—
83C23—33no—
84C23—312no—
85C23—36no—
86C23—61no—
87C23—66no—
88C23C2×Dic616yescomplemented
89C23C2×S411yescharacteristic
90C23C23×D311yescharacteristic
91Q8—616no—
92Q8—616no—
93C2×C4—36no—
94C2×C4—34no—
95C2×C4—312no—
96C2×C4—34no—
97C2×C4—38no—
98C2×C4—38no—
99C2×C4—66no—
100C2×C4—624no—
101C2×C4—612no—
102C2×C4—612no—
103C2×C4C2×S416yes—
104C2×C3—46no—
105C2×C3—41no—
106C22—36no—
107C22—31no—
108C22—36no—
109C22—36no—
110C22—312no—
111C22—31no—
112C22—33no—
113C22—63no—
114C22—64no—
115C22A4⋊Q814yesdirect product factor
116C22C22×Dic611yescharacteristic, complemented
117C22C22×S413yes—
118C4—34no—
119C4—68no—
120C4—68no—
121C4C22×S414yes—
122C3—41noSylow
123C2—31no—
124C2—31no—
125C2—36no—
126C2C2×A4⋊Q816yesdirect product factor
127C2C23×S411yescharacteristic
subgroupG/H
centerC23C2×S4
commutator, cosocleC2×A4C24
FrattiniC2C23×S4
FittingC24×C4D3
socleC25C2×D3
prime radicalC23×A4C22
hypercenterC22×C4S4
hypocenterA4C22×Q8
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×A4⋊Q8C2direct (N × H)
9. A4⋊Q8C22direct (N × H)
17. C22×A4Q8semidirect
42. C2×A4C2×Q8semidirect
50. C24Dic6semidirect
76. A4C22×Q8semidirect
88. C23C2×Dic6semidirect
115. C22A4⋊Q8direct (N × H)
116. C22C22×Dic6semidirect
126. C2C2×A4⋊Q8direct (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
C22×SL(2,ℤ/4)2
C2×A4⋊Q82
C22×C22⋊Q83
C22×Dic64