Finite Group Databasebeta

← (#126⋊C3)⋊C4 #73180

Subgroups — (#126⋊C3)⋊C4 #73180

Order 384.

116 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 26 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
1C2×#126⋊C3C211yesmaximal, characteristic
2#384⋊C3C211yesmaximal, characteristic, complemented
3#383⋊C3C211yesmaximal, characteristic, complemented
4#2383C311yesmaximal, characteristic, complemented, Sylow
5C2×C4×A4—41nomaximal
6#126⋊C3C412yescomplemented
7C2×C42⋊C3C2211yescharacteristic
8C2×#126C2×C311yescharacteristic
9#389—31no—
10#388—31no—
11#387—31no—
12#385—31no—
13#384C2×C311yescharacteristic, complemented
14#383C2×C311yescharacteristic, complemented
15C22×A4—41no—
16C4×A4—41no—
17C4×A4—41no—
18C42⋊C3C2×C411yescharacteristic, complemented
19C23×C4A411yescharacteristic
20#126—32no—
21#126C3×C412yescomplemented
22C2×C4⋊C4—31no—
23C2×C4⋊C4—31no—
24C2×C4⋊C4—31no—
25C2×C4⋊C4—31no—
26C2×C22⋊C4—31no—
27C2×C22⋊C4—31no—
28C2×C42—31no—
29C2×C42C22×C311yescharacteristic
30#95—31no—
31#95—31no—
32#95—31no—
33#95—31no—
34C2×A4—42no—
35C2×A4—41no—
36C2×C3×C4—161no—
37C24C2×A411yescharacteristic
38C22×C4—31no—
39C22×C4—31no—
40C22×C4—31no—
41C22×C4—31no—
42C22×C4—121no—
43C22×C4C2×A411yescharacteristic
44C22×C4C2×A411yescharacteristic
45C4⋊C4—32no—
46C4⋊C4—32no—
47C4⋊C4—121no—
48C4⋊C4—121no—
49C4⋊C4—121no—
50C22⋊C4—32no—
51C22⋊C4—32no—
52C22⋊C4—121no—
53C42—31no—
54C42—121no—
55C42C2×C3×C411yescharacteristic, complemented
56C22×C3—161no—
57A4—41no—
58C3×C4—161no—
59C3×C4—161no—
60C23—61no—
61C23—61no—
62C23C4×A412yes—
63C23C22×A411yescharacteristic
64C2×C4—31no—
65C2×C4—31no—
66C2×C4—32no—
67C2×C4—41no—
68C2×C4—61no—
69C2×C4—61no—
70C2×C4—61no—
71C2×C4—61no—
72C2×C4—61no—
73C2×C4—61no—
74C2×C4—61no—
75C2×C4—61no—
76C2×C4—61no—
77C2×C4—61no—
78C2×C4—62no—
79C2×C4—121no—
80C2×C3—162no—
81C2×C3—161no—
82C22—31no—
83C22—31no—
84C22—41no—
85C22—62no—
86C22—121no—
87C22C2×C4×A411yescharacteristic
88C4—41no—
89C4—41no—
90C4—61no—
91C4—61no—
92C4—62no—
93C4—121no—
94C4—121no—
95C3—161noSylow
96C2—31no—
97C2—31no—
98C2—42no—
99C2#560⋊C311yescharacteristic
subgroupG/H
center, hypercenterC2#560⋊C3
commutator, hypocenterC42C2×C3×C4
Frattini, socleC23C22×A4
Fitting#2383C3
prime radicalC2×#126⋊C3C2
cosocleC2×C42C22×C3
LCSG ▷ C42
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #384⋊C3C2semidirect
3. #383⋊C3C2semidirect
4. #2383C3semidirect
6. #126⋊C3C4semidirect
13. #384C2×C3semidirect
14. #383C2×C3semidirect
18. C42⋊C3C2×C4semidirect
21. #126C3×C4semidirect
55. C42C2×C3×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#126⋊C32
#384⋊C32
#383⋊C32
#23833
C2×C4×A44