Finite Group Databasebeta

← #85547

Subgroups — #85547

Order 384.

160 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 10 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#1896—31nomaximal, Sylow
3#1020—41nomaximal
4#857—42nomaximal
5#525—31no—
6C22×C42D311yescharacteristic, complemented
7C2×#104—32no—
8#407—31no—
9#339—32no—
10C22⋊A4—41no—
11C42⋊C3—42no—
12C42⋊C3—81no—
13C2×#41—31no—
14C23×C4—31no—
15C2×M16—32no—
16#123—61no—
17C22⋊Q8—32no—
18#121—32no—
19#120—61no—
20C4×D4—62no—
21#117—31no—
22C2×C42—32no—
23C2×C42—61no—
24#104—62no—
25#104—62no—
26C22⋊C8—62no—
27S4—161no—
28S4—162no—
29C24S411yescharacteristic
30#41—62no—
31#41—62no—
32C2×Q8—31no—
33C2×D4—31no—
34C2×D4—62no—
35C2×D4—61no—
36C22×C4—31no—
37C22×C4—32no—
38C22×C4—31no—
39C22×C4—32no—
40C22×C4—62no—
41C22×C4—62no—
42C22×C4—61no—
43M16—62no—
44M16—62no—
45C2×C8—62no—
46C4⋊C4—32no—
47C4⋊C4—62no—
48C22⋊C4—62no—
49C22⋊C4—61no—
50C22⋊C4—62no—
51C42—21no—
52C42—32no—
53C42—61no—
54C42—61no—
55C42S412yescomplemented
56A4—162no—
57A4—161no—
58A4—321no—
59C23—32no—
60C23—31no—
61C23—61no—
62C23—61no—
63Q8—62no—
64D4—62no—
65D4—122no—
66D4—121no—
67D4—122no—
68D4—121no—
69C2×C4—31no—
70C2×C4—31no—
71C2×C4—32no—
72C2×C4—32no—
73C2×C4—32no—
74C2×C4—64no—
75C2×C4—64no—
76C2×C4—61no—
77C2×C4—61no—
78C2×C4—62no—
79C2×C4—61no—
80C2×C4—122no—
81C8—122no—
82D3—641no—
83C22—21no—
84C22—31no—
85C22—31no—
86C22—32no—
87C22—62no—
88C22—61no—
89C22—121no—
90C22—122no—
91C22#85712yescomplemented
92C22#102011yescharacteristic
93C4—32no—
94C4—32no—
95C4—62no—
96C4—121no—
97C4—122no—
98C3—641noSylow
99C2—31no—
100C2—32no—
101C2—61no—
102C2—241no—
subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22⋊(C42⋊C3)C2
FrattiniC22#1020
FittingC22×C42D3
socleC24S4
LCSG ▷ C22⋊(C42⋊C3)
UCSC1

Complements

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

NH ≅ G/Ntype
1. C22⋊(C42⋊C3)C2semidirect
6. C22×C42D3semidirect
55. C42S4semidirect
91. C22#857semidirect

Maximal subgroups

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

subgroupindex
C22⋊(C42⋊C3)2
#18963
#10204
#8574