Finite Group Databasebeta

← #72999

Subgroups — #72999

Order 384.

166 conjugacy classes of subgroups; only the 14 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 14 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
1#5931C212yesmaximal, complemented
2#393⋊C3C211yesmaximal, characteristic, complemented
3#1601—31nomaximal, Sylow
4C22×S4—41nomaximal
5C22⋊SL(2,𝔽3)C2211yescharacteristic, complemented
6#534—31no—
7#448—32no—
8#393D311yescharacteristic, complemented
9#327—32no—
10#323—31no—
11C22×A4—41no—
12C2×S4—42no—
13C2×S4—44no—
14GL(2,𝔽3)—82no—
15C22×Q8C2×D311yescharacteristic, complemented
16C22×D4—61no—
17C2×SD16—62no—
18#123—62no—
19#121—32no—
20#121—61no—
21#120—62no—
22C2×C22⋊C4—31no—
23#103—62no—
24C22⋊C8—32no—
25#95—31no—
26C22×D3—161no—
27C2×A4—42no—
28C2×A4—41no—
29S4—44no—
30SL(2,𝔽3)—82no—
31C24—61no—
32C24S411yescharacteristic
33C2×Q8—62no—
34C2×D4—32no—
35C2×D4—62no—
36C2×D4—62no—
37C2×D4—61no—
38C2×D4—122no—
39C2×D4—122no—
40C22×C4—31no—
41C22×C4—31no—
42C22×C4—61no—
43SD16—122no—
44SD16—122no—
45C2×C8—62no—
46C4⋊C4—32no—
47C4⋊C4—61no—
48C22⋊C4—62no—
49C22⋊C4—62no—
50C22⋊C4—62no—
51C22×C3—161no—
52C2×D3—162no—
53C2×D3—164no—
54A4—41no—
55C23—61no—
56C23—61no—
57C23—62no—
58C23—122no—
59C23—241no—
60C23—241no—
61C23GL(2,𝔽3)12yescomplemented
62C23C2×S411yescharacteristic
63Q8—22no—
64Q8—62no—
65D4—122no—
66D4—122no—
67D4—124no—
68D4—122no—
69D4—241no—
70C2×C4—32no—
71C2×C4—61no—
72C2×C4—62no—
73C2×C4—121no—
74C2×C4—122no—
75C2×C4—121no—
76C8—122no—
77C2×C3—162no—
78C2×C3—161no—
79D3—164no—
80C22—31no—
81C22—31no—
82C22—41no—
83C22—62no—
84C22—121no—
85C22—122no—
86C22—122no—
87C22—122no—
88C22—244no—
89C22C2×GL(2,𝔽3)11yescharacteristic
90C4—62no—
91C4—122no—
92C4—121no—
93C3—161noSylow
94C2—31no—
95C2—31no—
96C2—42no—
97C2—242no—
98C2#1020⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2#1020⋊C2
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22
Frattini, socleC23C2×S4
Fitting#393D3
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #5931C2semidirect
2. #393⋊C3C2semidirect
5. C22⋊SL(2,𝔽3)C22semidirect
8. #393D3semidirect
15. C22×Q8C2×D3semidirect
61. C23GL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
#59312
#393⋊C32
#16013
C22×S44