Finite Group Databasebeta

← C22×C22⋊SL(2,𝔽3) #87541

Subgroups — C22×C22⋊SL(2,𝔽3) #87541

Order 384.

1139 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 1139 conjugacy classes of subgroups — too many to draw. Only its 62 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×C22⋊SL(2,𝔽3)C213yesmaximal, direct product factor
2C24×Q8C311yesmaximal, characteristic, complemented, Sylow
3C23×A4—41nomaximal
4C22⋊SL(2,𝔽3)C2211yescharacteristic, direct product factor
5C22×SL(2,𝔽3)—44nomaximal
6C23×Q8—312no—
7C23×Q8—34no—
8C23×Q8—33no—
9C23×Q8C2×C313yescomplemented
10C24×C4—31no—
11C22×A4—44no—
12C22×A4—43no—
13C2×SL(2,𝔽3)—412no—
14C25A411yescharacteristic
15C22×Q8—312no—
16C22×Q8—324no—
17C22×Q8—336no—
18C22×Q8—336no—
19C22×Q8—336no—
20C22×Q8—38no—
21C22×Q8—32no—
22C22×Q8—34no—
23C22×Q8—312no—
24C22×Q8—312no—
25C22×Q8—33no—
26C22×Q8A414yescomplemented
27C22×Q8C22×C311yescharacteristic, complemented
28C23×C4—318no—
29C23×C4—36no—
30C23×C4—33no—
31C23×C4—33no—
32C23×C3—161no—
33C2×A4—46no—
34C2×A4—41no—
35SL(2,𝔽3)—44no—
36C24—33no—
37C24—34no—
38C24—31no—
39C24SL(2,𝔽3)14yescomplemented
40C24C2×A413yes—
41C2×Q8—336no—
42C2×Q8—336no—
43C2×Q8—372no—
44C2×Q8—372no—
45C2×Q8—324no—
46C2×Q8—336no—
47C2×Q8—312no—
48C2×Q8—312no—
49C2×Q8—34no—
50C2×Q8—312no—
51C2×Q8C2×A4112yescomplemented
52C22×C4—318no—
53C22×C4—318no—
54C22×C4—324no—
55C22×C4—318no—
56C22×C4—336no—
57C22×C4—318no—
58C22×C4—33no—
59C22×C4—31no—
60C22×C4—34no—
61C22×C3—164no—
62C22×C3—163no—
63A4—41no—
64C23—33no—
65C23—312no—
66C23—36no—
67C23—32no—
68C23—312no—
69C23—34no—
70C23—34no—
71C23—33no—
72C23—33no—
73C23C2×SL(2,𝔽3)16yescomplemented
74C23C22×A411yescharacteristic
75C23C22⋊A411yescharacteristic
76Q8—312no—
77Q8—324no—
78Q8—38no—
79Q8—336no—
80Q8—34no—
81Q8C22×A414yescomplemented
82C2×C4—336no—
83C2×C4—318no—
84C2×C4—336no—
85C2×C4—36no—
86C2×C4—312no—
87C2×C4—312no—
88C2×C3—166no—
89C2×C3—161no—
90C22—36no—
91C22—31no—
92C22—36no—
93C22—38no—
94C22—312no—
95C22—36no—
96C22—36no—
97C22—33no—
98C22—31no—
99C22C22×SL(2,𝔽3)11yescharacteristic, complemented
100C22C22⋊SL(2,𝔽3)14yesdirect product factor
101C22C2×C22⋊A413yes—
102C4—312no—
103C4—34no—
104C3—161noSylow
105C2—36no—
106C2—31no—
107C2—31no—
108C2C2×C22⋊SL(2,𝔽3)16yesdirect product factor
109C2C22×C22⋊A411yescharacteristic
subgroupG/H
center, hypercenterC23C22⋊A4
commutator, cosocle, hypocenterC22×Q8C22×C3
FrattiniC2C22×C22⋊A4
FittingC24×Q8C3
socleC25A4
LCSG ▷ C22×Q8
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊SL(2,𝔽3)C2direct (N × H)
2. C24×Q8C3semidirect
4. C22⋊SL(2,𝔽3)C22direct (N × H)
9. C23×Q8C2×C3semidirect
26. C22×Q8A4semidirect
27. C22×Q8C22×C3semidirect
39. C24SL(2,𝔽3)semidirect
51. C2×Q8C2×A4semidirect
73. C23C2×SL(2,𝔽3)semidirect
81. Q8C22×A4semidirect
99. C22C22×SL(2,𝔽3)semidirect
100. C22C22⋊SL(2,𝔽3)direct (N × H)
108. C2C2×C22⋊SL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊SL(2,𝔽3)2
C24×Q83
C23×A44
C22×SL(2,𝔽3)4