Finite Group Databasebeta

← #1437⋊C3 #68008

Subgroups — #1437⋊C3 #68008

Order 384.

168 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 13 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⋊(C42⋊C3)C211yesmaximal, characteristic, complemented
2#1437C311yesmaximal, characteristic, complemented, Sylow
3C2×C22⋊A4—41nomaximal
4C22×C22⋊C4—31no—
5C22×C42C2×C311yescharacteristic, complemented
6C2×#95—31no—
7C22⋊A4—41no—
8C42⋊C3—82no—
9C25A411yescharacteristic
10C23×C4—31no—
11C23×C4—31no—
12C2×C22⋊C4—32no—
13C2×C22⋊C4—32no—
14C2×C22⋊C4—62no—
15C2×C22⋊C4—62no—
16C2×C42—62no—
17#95—62no—
18C2×A4—161no—
19C2×A4—162no—
20C2×A4—161no—
21C2×A4—161no—
22C24—31no—
23C24—31no—
24C24—61no—
25C24—61no—
26C24—62no—
27C24C2×A411yescharacteristic
28C22×C4—32no—
29C22×C4—32no—
30C22×C4—32no—
31C22×C4—32no—
32C22×C4—62no—
33C22×C4—62no—
34C22×C4—62no—
35C22×C4—61no—
36C22×C4—61no—
37C22×C4—62no—
38C22⋊C4—64no—
39C22⋊C4—64no—
40C42—22no—
41C42—62no—
42A4—161no—
43A4—161no—
44A4—162no—
45A4—161no—
46C23—31no—
47C23—31no—
48C23—32no—
49C23—31no—
50C23—31no—
51C23—41no—
52C23—42no—
53C23—41no—
54C23—61no—
55C23—61no—
56C23—62no—
57C23—62no—
58C23—62no—
59C23—62no—
60C23—61no—
61C23—61no—
62C23—122no—
63C23—121no—
64C23—121no—
65C23C42⋊C311yescharacteristic, complemented
66C2×C4—34no—
67C2×C4—64no—
68C2×C4—64no—
69C2×C4—62no—
70C2×C4—62no—
71C2×C4—62no—
72C2×C4—64no—
73C2×C4—122no—
74C2×C4—121no—
75C2×C4—121no—
76C2×C3—641no—
77C22—31no—
78C22—32no—
79C22—32no—
80C22—32no—
81C22—31no—
82C22—31no—
83C22—31no—
84C22—61no—
85C22—61no—
86C22—62no—
87C22—121no—
88C22—121no—
89C22—122no—
90C22—121no—
91C22—121no—
92C22—122no—
93C22C2×C42⋊C311yescharacteristic
94C22#120⋊C311yescharacteristic
95C22#126⋊C312yes—
96C22#127⋊C311yescharacteristic
97C4—64no—
98C4—122no—
99C3—641noSylow
100C2—31no—
101C2—31no—
102C2—31no—
103C2—32no—
104C2—41no—
105C2—121no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22×C42C2×C3
Frattini, socleC24C2×A4
Fitting#1437C3
LCSG ▷ C22×C42
UCSC1

Complements

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

NH ≅ G/Ntype
1. C22⋊(C42⋊C3)C2semidirect
2. #1437C3semidirect
5. C22×C42C2×C3semidirect
65. C23C42⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22⋊(C42⋊C3)2
#14373
C2×C22⋊A44