Finite Group Databasebeta

← A4×#125 #85588

Subgroups — A4×#125 #85588

Order 384.

292 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 66 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×C4⋊C4C214yesmaximal
2A4×C4⋊C4C212yesmaximal
3C42×A4C211yesmaximal, characteristic
4C22×#125C311yesmaximal, characteristic, complemented, Sylow
5C2×C4×A4C2212yes—
6C2×C4×A4C2214yes—
7C2×C4×A4C2211yescharacteristic
8C3×#125—41nomaximal
9C2×#125—31no—
10C2×#125—32no—
11C2×#125—34no—
12C2×#125—31no—
13C22×C4⋊C4C2×C314yes—
14C22×C4⋊C4C2×C312yes—
15C22×C42C2×C311yescharacteristic
16C22×A4C2311yescharacteristic
17C4×A4—22no—
18C4×A4—24no—
19C4×A4Q812yes—
20C3×C4⋊C4—42no—
21C3×C4⋊C4—44no—
22C3×C42—41no—
23C23×C4C22×C314yes—
24C23×C4C22×C312yes—
25C23×C4C22×C311yescharacteristic
26#125—38no—
27#125—34no—
28#125—32no—
29#125—34no—
30#125—32no—
31#125—31no—
32#125A411yescharacteristic, direct product factor
33C2×C4⋊C4—32no—
34C2×C4⋊C4—34no—
35C2×C4⋊C4—32no—
36C2×C4⋊C4—34no—
37C2×C4⋊C4—34no—
38C2×C4⋊C4—38no—
39C2×C42—31no—
40C2×C42—31no—
41C2×C42—32no—
42C2×A4C2×Q811yescharacteristic
43C2×A4#4112yes—
44C2×C3×C4—41no—
45C2×C3×C4—42no—
46C2×C3×C4—44no—
47C24C23×C311yescharacteristic
48C22×C4—24no—
49C22×C4—22no—
50C22×C4—31no—
51C22×C4—31no—
52C22×C4—32no—
53C22×C4—32no—
54C22×C4—34no—
55C22×C4—32no—
56C22×C4—34no—
57C22×C4—64no—
58C22×C4—62no—
59C22×C4C3×Q812yes—
60C4⋊C4—34no—
61C4⋊C4—38no—
62C4⋊C4—34no—
63C4⋊C4—38no—
64C4⋊C4—34no—
65C4⋊C4—32no—
66C4⋊C4C2×A414yes—
67C4⋊C4C2×A412yes—
68C42—32no—
69C42—32no—
70C42—31no—
71C42C2×A411yescharacteristic
72C22×C3—41no—
73A4#12511yescharacteristic, direct product factor
74C3×C4—42no—
75C3×C4—82no—
76C3×C4—84no—
77C23—31no—
78C23—32no—
79C23—31no—
80C23C2×C3×Q811yescharacteristic
81C23C3×#4112yes—
82C2×C4—32no—
83C2×C4—31no—
84C2×C4—32no—
85C2×C4—32no—
86C2×C4—32no—
87C2×C4—32no—
88C2×C4—34no—
89C2×C4—62no—
90C2×C4—64no—
91C2×C4—62no—
92C2×C4—64no—
93C2×C4—62no—
94C2×C4—64no—
95C2×C4—62no—
96C2×C4—64no—
97C2×C4C22×A412yes—
98C2×C4C22×A414yes—
99C2×C4C22×A411yescharacteristic
100C2×C3—41no—
101C2×C3—42no—
102C22—31no—
103C22—32no—
104C22—32no—
105C22—31no—
106C22—31no—
107C22—32no—
108C22—32no—
109C22C3×#12511yescharacteristic, complemented
110C22C23×A411yescharacteristic
111C4—22no—
112C4—24no—
113C4—32no—
114C4—62no—
115C4—64no—
116C4Q8×A412yes—
117C3—41noSylow
118C2—31no—
119C2—32no—
120C2—31no—
121C2C2×Q8×A411yescharacteristic
122C2A4×#4112yes—
subgroupG/H
center, FrattiniC22C23×A4
commutator, socle, cosocleC24C23×C3
FittingC22×#125C3
prime radicalC22×A4C23
hypercenter#125A4
hypocenterC22C3×#125
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
4. C22×#125C3semidirect
32. #125A4direct (N × H)
73. A4#125direct (N × H)
109. C22C3×#125semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊C42
C42×A42
C22×#1253
C3×#1254