Finite Group Databasebeta

← C2×#384⋊C3 #73170

Subgroups — C2×#384⋊C3 #73170

Order 384.

188 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 29 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⋊C3C211yesmaximal, characteristic
2#384⋊C3C212yesmaximal, direct product factor
3C2×#384C311yesmaximal, characteristic, complemented, Sylow
4C2×C4×A4—41nomaximal
5C2×C42⋊C3C412yescomplemented
6C2×C42⋊C3C2211yescharacteristic
7C22×C4⋊C4—31no—
8C22×C4⋊C4—31no—
9C22×C42C2×C311yescharacteristic
10#384—32no—
11#384C2×C312yescomplemented
12C22×A4—41no—
13C4×A4—42no—
14C42⋊C3C2×C411yescharacteristic, complemented
15C23×C4—31no—
16C23×C4—31no—
17C23×C4A411yescharacteristic
18C2×C4⋊C4—32no—
19C2×C4⋊C4—32no—
20C2×C4⋊C4—32no—
21C2×C4⋊C4—32no—
22C2×C4⋊C4—62no—
23C2×C4⋊C4—62no—
24C2×C4⋊C4—62no—
25C2×C4⋊C4—62no—
26C2×C42—31no—
27C2×C42—32no—
28C2×C42C3×C412yescomplemented
29C2×C42C22×C311yescharacteristic
30C2×A4—42no—
31C2×A4—41no—
32C2×C3×C4—161no—
33C24C2×A411yescharacteristic
34C22×C4—32no—
35C22×C4—32no—
36C22×C4—34no—
37C22×C4—32no—
38C22×C4—32no—
39C22×C4—31no—
40C22×C4—31no—
41C22×C4—32no—
42C22×C4—61no—
43C22×C4—62no—
44C22×C4—61no—
45C22×C4—61no—
46C22×C4—62no—
47C22×C4—61no—
48C22×C4C2×A412yes—
49C4⋊C4—64no—
50C4⋊C4—64no—
51C4⋊C4—64no—
52C4⋊C4—64no—
53C42—32no—
54C42—31no—
55C42—32no—
56C42C2×C3×C411yescharacteristic, complemented
57C22×C3—161no—
58A4—41no—
59C3×C4—162no—
60C23—31no—
61C23—32no—
62C23—31no—
63C23C4×A412yes—
64C23C22×A411yescharacteristic
65C2×C4—32no—
66C2×C4—32no—
67C2×C4—34no—
68C2×C4—34no—
69C2×C4—32no—
70C2×C4—32no—
71C2×C4—34no—
72C2×C4—34no—
73C2×C4—31no—
74C2×C4—32no—
75C2×C4—31no—
76C2×C4—41no—
77C2×C4—62no—
78C2×C4—62no—
79C2×C4—64no—
80C2×C4—121no—
81C2×C4—121no—
82C2×C4—122no—
83C2×C4—121no—
84C2×C3—162no—
85C2×C3—161no—
86C22—32no—
87C22—31no—
88C22—32no—
89C22—32no—
90C22—32no—
91C22—31no—
92C22—31no—
93C22#127⋊C311yescharacteristic
94C22C2×C4×A411yescharacteristic
95C4—32no—
96C4—32no—
97C4—34no—
98C4—42no—
99C4—122no—
100C3—161noSylow
101C2—31no—
102C2—32no—
103C2—31no—
104C2#384⋊C312yesdirect product factor
105C2C2×#127⋊C311yescharacteristic
subgroupG/H
center, hypercenterC22#127⋊C3
commutator, hypocenterC42C2×C3×C4
FrattiniC23C22×A4
FittingC2×#384C3
socleC24C2×A4
prime radicalC22×C42⋊C3C2
cosocleC2×C42C22×C3
LCSG ▷ C42
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. #384⋊C3C2direct (N × H)
3. C2×#384C3semidirect
5. C2×C42⋊C3C4semidirect
11. #384C2×C3semidirect
14. C42⋊C3C2×C4semidirect
28. C2×C42C3×C4semidirect
56. C42C2×C3×C4semidirect
104. C2#384⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C42⋊C32
#384⋊C32
C2×#3843
C2×C4×A44