Finite Group Databasebeta

← C4×#126⋊C3 #73172

Subgroups — C4×#126⋊C3 #73172

Order 384.

126 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 27 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#383⋊C3C211yesmaximal, characteristic, complemented
3C4×C42⋊C3C211yesmaximal, characteristic, complemented
4C4×#126C311yesmaximal, characteristic, complemented, Sylow
5C2×C4×A4—41nomaximal
6#126⋊C3C412yesdirect product factor
7C2×C42⋊C3C2211yescharacteristic
8C2×#126C2×C311yescharacteristic
9#388—31no—
10#387—31no—
11#383C2×C311yescharacteristic, complemented
12C4×C4⋊C4—31no—
13C4×C22⋊C4—31no—
14C43C2×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×C22⋊C4—31no—
25C2×C22⋊C4—31no—
26C2×C42—31no—
27C2×C42—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—61no—
43C22×C4—61no—
44C22×C4C2×A411yescharacteristic
45C22×C4C2×A411yescharacteristic
46C4⋊C4—32no—
47C4⋊C4—32no—
48C4⋊C4—62no—
49C22⋊C4—32no—
50C22⋊C4—32no—
51C22⋊C4—62no—
52C42—31no—
53C42—61no—
54C42—61no—
55C42—61no—
56C42—61no—
57C42—62no—
58C42C2×C3×C411yescharacteristic, complemented
59C22×C3—161no—
60A4—41no—
61C3×C4—161no—
62C3×C4—161no—
63C23—61no—
64C23—61no—
65C23C4×A412yes—
66C23C22×A411yescharacteristic
67C2×C4—31no—
68C2×C4—31no—
69C2×C4—32no—
70C2×C4—31no—
71C2×C4—31no—
72C2×C4—32no—
73C2×C4—41no—
74C2×C4—61no—
75C2×C4—61no—
76C2×C4—61no—
77C2×C4—61no—
78C2×C4—61no—
79C2×C4—61no—
80C2×C4—61no—
81C2×C4—61no—
82C2×C4—61no—
83C2×C4—61no—
84C2×C4—121no—
85C2×C3—162no—
86C2×C3—161no—
87C22—31no—
88C22—31no—
89C22—41no—
90C22—62no—
91C22—121no—
92C22C2×C4×A411yescharacteristic
93C4—31no—
94C4—41no—
95C4—61no—
96C4—61no—
97C4—62no—
98C4—62no—
99C4—121no—
100C4#126⋊C311yescharacteristic, direct product factor
101C3—161noSylow
102C2—31no—
103C2—31no—
104C2—42no—
105C2C2×#126⋊C311yescharacteristic
subgroupG/H
center, hypercenterC4#126⋊C3
commutator, hypocenterC42C2×C3×C4
Frattini, socleC23C22×A4
FittingC4×#126C3
prime radicalC2×#126⋊C3C2
cosocleC2×C42C22×C3
LCSG ▷ C42
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #383⋊C3C2semidirect
3. C4×C42⋊C3C2semidirect
4. C4×#126C3semidirect
6. #126⋊C3C4direct (N × H)
11. #383C2×C3semidirect
14. C43C2×C3semidirect
18. C42⋊C3C2×C4semidirect
21. #126C3×C4semidirect
58. C42C2×C3×C4semidirect
100. C4#126⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#126⋊C32
#383⋊C32
C4×C42⋊C32
C4×#1263
C2×C4×A44