Finite Group Databasebeta

← #85430

Subgroups — #85430

Order 384.

271 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 42 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
1Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C4⋊GL(2,𝔽3)C213yesmaximal, complemented
3C4×GL(2,𝔽3)C213yesmaximal, complemented
4#3362—31nomaximal, Sylow
5C2×#239—41nomaximal
6C2×GL(2,𝔽3)C2213yes—
7C4×SL(2,𝔽3)C2213yes—
8#859C2211yescharacteristic
9Q82D311yescharacteristic, complemented
10#550—31no—
11#486—33no—
12#476—33no—
13#460—33no—
14C8×Q8—31no—
15C4×SD16—33no—
16C2×C3×Q8—41no—
17#239—48no—
18C2×D12—43no—
19C2×C4×D3—43no—
20C2×SL(2,𝔽3)C2311yescharacteristic
21GL(2,𝔽3)—23no—
22C2×#41—61no—
23C2×SD16—33no—
24C4⋊Q8—33no—
25#127—33no—
26#121—63no—
27C4×Q8—31no—
28C4×Q8C2×D313yes—
29C4×D4—33no—
30#106—33no—
31C4⋊C8—33no—
32#103—31no—
33#102—33no—
34#102—36no—
35C4×C8—33no—
36C22×D3—43no—
37C3×Q8—44no—
38C2×C3×C4—43no—
39C2×Dic3—41no—
40D12—412no—
41C4×D3—412no—
42SL(2,𝔽3)#4111yescharacteristic, complemented
43#41—124no—
44C2×Q8—61no—
45C2×Q8S411yescharacteristic
46C2×Q8C22×D311yescharacteristic
47C2×D4—33no—
48C2×D4—63no—
49C2×D4—63no—
50C22×C4—63no—
51SD16—63no—
52SD16—63no—
53C2×C8—31no—
54C2×C8—33no—
55C4⋊C4—33no—
56C4⋊C4—33no—
57C4⋊C4—31no—
58C22⋊C4—63no—
59C42—33no—
60C22×C3—41no—
61C2×D3—46no—
62C2×D3—83no—
63C2×D3—83no—
64C3×C4—46no—
65Dic3—42no—
66C23—63no—
67Q8—31no—
68Q8—241no—
69Q8GL(2,𝔽3)14yescomplemented
70Q8#23911yescharacteristic, complemented
71D4—63no—
72D4—63no—
73D4—126no—
74D4—243no—
75C2×C4—31no—
76C2×C4—33no—
77C2×C4—61no—
78C2×C4—126no—
79C2×C4C2×S413yes—
80C8—61no—
81C8—63no—
82C2×C3—41no—
83C2×C3—41no—
84C2×C3—41no—
85D3—86no—
86C22—123no—
87C22—123no—
88C22—123no—
89C22C22×S411yescharacteristic
90C4—31no—
91C4—31no—
92C4—63no—
93C4—121no—
94C4C2×GL(2,𝔽3)16yes—
95C3—41noSylow
96C2—243no—
97C2C22×GL(2,𝔽3)11yescharacteristic
98C2#591911yescharacteristic
99C2#592511yescharacteristic
subgroupG/H
center, Frattini, socleC22C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23
FittingQ82D3
hypercenterC2×Q8S4
hypocenterSL(2,𝔽3)#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×SL(2,𝔽3)C2semidirect
2. C4⋊GL(2,𝔽3)C2semidirect
3. C4×GL(2,𝔽3)C2semidirect
9. Q82D3semidirect
42. SL(2,𝔽3)#41semidirect
69. Q8GL(2,𝔽3)semidirect
70. Q8#239semidirect

Maximal subgroups

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

subgroupindex
Q8×SL(2,𝔽3)2
C4⋊GL(2,𝔽3)2
C4×GL(2,𝔽3)2
#33623
C2×#2394