Finite Group Databasebeta

← #67627

Subgroups — #67627

Order 384.

150 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 41 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
1C3×#381C211yesmaximal, characteristic
2#4525C212yesmaximal
3#1317—31nomaximal, Sylow
4C23×C3×C4C411yescharacteristic
5C2×C3×C22⋊C4C411yescharacteristic
6C2×C3×C22⋊C4C2211yescharacteristic
7C3×#95—22no—
8#830—22no—
9#381D311yescharacteristic
10#323—32no—
11C24×C3C2×C411yescharacteristic
12C22×C3×C4—21no—
13C22×C3×C4—21no—
14C22×C3×C4—21no—
15C22×C3×C4—41no—
16C22×C3×C4—41no—
17C22×C3×C4C812yescomplemented
18C22×C3×C4D412yes—
19C3×C22⋊C4—41no—
20C3×C22⋊C4—41no—
21C2×C3⋊C8—42no—
22C23×C4Dic311yescharacteristic
23C2×C22⋊C4Dic311yescharacteristic
24C2×C22⋊C4C2×D311yescharacteristic
25C22⋊C8—62no—
26#95—22no—
27C23×C3—21no—
28C23×C3—21no—
29C23×C3—42no—
30C23×C3C22⋊C411yescharacteristic
31C23×C3C2×C811yescharacteristic
32C23×C3M1611yescharacteristic
33C2×C3×C4—22no—
34C2×C3×C4—22no—
35C2×C3×C4—42no—
36C2×C3×C4—41no—
37C2×C3×C4—41no—
38C2×C3×C4—42no—
39C2×C3×C4—42no—
40C2×C3×C4—42no—
41C2×C3×C4—41no—
42C3⋊C8—82no—
43C24C2×Dic311yescharacteristic
44C22×C4—21no—
45C22×C4—21no—
46C22×C4—21no—
47C22×C4—41no—
48C22×C4—41no—
49C22×C4C3⋊C812yescomplemented
50C22×C4#6712yes—
51C2×C8—122no—
52C22⋊C4—41no—
53C22⋊C4—41no—
54C22×C3—22no—
55C22×C3—21no—
56C22×C3—21no—
57C22×C3—41no—
58C22×C3—41no—
59C22×C3—42no—
60C22×C3—42no—
61C22×C3C22⋊C811yescharacteristic
62C22×C3#9911yescharacteristic
63C22×C3#10011yescharacteristic
64C3×C4—42no—
65C3×C4—42no—
66C3×C4—81no—
67C23—21no—
68C23—21no—
69C23—42no—
70C23C2×C3⋊C811yescharacteristic
71C23#20811yescharacteristic
72C23#21711yescharacteristic
73C2×C4—22no—
74C2×C4—22no—
75C2×C4—42no—
76C2×C4—41no—
77C2×C4—41no—
78C2×C4—42no—
79C2×C4—42no—
80C2×C4—42no—
81C2×C4—41no—
82C8—242no—
83C2×C3—21no—
84C2×C3—21no—
85C2×C3—41no—
86C2×C3—41no—
87C2×C3#32311yescharacteristic
88C2×C3#35212yes—
89C22—22no—
90C22—21no—
91C22—21no—
92C22—41no—
93C22—41no—
94C22—42no—
95C22—42no—
96C22#83011yescharacteristic
97C22#83311yescharacteristic
98C22#83411yescharacteristic
99C4—42no—
100C4—42no—
101C4—81no—
102C3#131711yescharacteristic, complemented, Sylow
103C2—21no—
104C2—21no—
105C2—41no—
106C2—41no—
107C2#452511yescharacteristic
108C2#453912yes—
subgroupG/H
centerC22#834
commutatorC23×C3C2×C8
FrattiniC2×C22⋊C4C2×D3
FittingC3×#381C2
socleC22×C3#99
prime radicalC24×C3C2×C4
cosocleC2×C3×C22⋊C4C22
hypercenter#381D3
hypocenterC3#1317
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #381

Complements

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

NH ≅ G/Ntype
17. C22×C3×C4C8semidirect
49. C22×C4C3⋊C8semidirect
102. C3#1317semidirect

Maximal subgroups

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

subgroupindex
C3×#3812
#45252
#13173