Finite Group Databasebeta

← #87502

Subgroups — #87502

Order 384.

347 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 21 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
1C4×C22⋊A4C211yesmaximal, characteristic
2#5936C212yesmaximal
3#2848—31nomaximal, Sylow
4C2×C22⋊A4C2211yescharacteristic
5A4⋊Q8—43nomaximal
6C24×C4D311yescharacteristic
7C2×C22⋊Q8—33no—
8#385—36no—
9#380—33no—
10#379—32no—
11C22⋊A4Q811yescharacteristic, complemented
12C4×A4—43no—
13C4×A4—81no—
14SL(2,ℤ/4)—46no—
15C25C2×D311yescharacteristic
16C22×Q8—31no—
17C23×C4—33no—
18C23×C4—33no—
19C23×C4—61no—
20C23×C4—61no—
21C22⋊Q8—63no—
22C22⋊Q8—63no—
23C22⋊Q8—63no—
24C22⋊Q8—63no—
25C2×C4⋊C4—33no—
26C2×C4⋊C4—36no—
27C2×C22⋊C4—36no—
28C2×C22⋊C4—36no—
29#95—36no—
30C2×A4—43no—
31C2×A4—81no—
32Dic6—161no—
33C24—33no—
34C24—33no—
35C24—61no—
36C24—61no—
37C24Dic611yescharacteristic, complemented
38C2×Q8—63no—
39C2×Q8—63no—
40C22×C4—21no—
41C22×C4—31no—
42C22×C4—33no—
43C22×C4—33no—
44C22×C4—33no—
45C22×C4—33no—
46C22×C4—36no—
47C22×C4—32no—
48C22×C4—66no—
49C22×C4—63no—
50C22×C4—63no—
51C22×C4—61no—
52C22×C4—63no—
53C22×C4S413yes—
54C4⋊C4—63no—
55C4⋊C4—63no—
56C4⋊C4—66no—
57C4⋊C4—66no—
58C22⋊C4—66no—
59C22⋊C4—66no—
60C22⋊C4—66no—
61C22⋊C4—66no—
62A4—43no—
63A4—81no—
64C3×C4—161no—
65Dic3—162no—
66C23—21no—
67C23—31no—
68C23—33no—
69C23—33no—
70C23—33no—
71C23—33no—
72C23—33no—
73C23—61no—
74C23—61no—
75C23—63no—
76C23—66no—
77C23—63no—
78C23—63no—
79C23C2×S413yes—
80Q8—123no—
81Q8—121no—
82C2×C4—33no—
83C2×C4—33no—
84C2×C4—33no—
85C2×C4—33no—
86C2×C4—63no—
87C2×C4—66no—
88C2×C4—63no—
89C2×C4—61no—
90C2×C4—61no—
91C2×C4—66no—
92C2×C4—66no—
93C2×C4—612no—
94C2×C3—161no—
95C22—21no—
96C22—33no—
97C22—33no—
98C22—31no—
99C22—33no—
100C22—33no—
101C22—33no—
102C22—63no—
103C22—66no—
104C22—63no—
105C22—63no—
106C22—61no—
107C22—61no—
108C22A4⋊Q813yescomplemented
109C4—33no—
110C4—61no—
111C4—126no—
112C4—122no—
113C4#102011yescharacteristic
114C3—161noSylow
115C2—33no—
116C2—33no—
117C2—61no—
118C2—61no—
119C2C2×#102011yescharacteristic
subgroupG/H
center, FrattiniC2C2×#1020
commutator, prime radical, cosocleC2×C22⋊A4C22
FittingC24×C4D3
socleC25C2×D3
hypercenterC4#1020
hypocenterC22⋊A4Q8
LCSG ▷ C2×C22⋊A4 ▷ C22⋊A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
11. C22⋊A4Q8semidirect
37. C24Dic6semidirect
108. C22A4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊A42
#59362
#28483
A4⋊Q84