Finite Group Databasebeta

← C4×C22⋊SL(2,𝔽3) #85643

Subgroups — C4×C22⋊SL(2,𝔽3) #85643

Order 384.

329 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 34 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)C211yesmaximal, characteristic
2C22×C4×Q8C311yesmaximal, characteristic, complemented, Sylow
3C22⋊SL(2,𝔽3)C411yescharacteristic, direct product factor
4C2×C4×A4—41nomaximal
5C4×SL(2,𝔽3)—44nomaximal
6C23×Q8C2×C311yescharacteristic
7C2×C4×Q8—34no—
8C2×C4×Q8—34no—
9C22×C4⋊C4—31no—
10C22×C42—31no—
11C22×A4—41no—
12C2×SL(2,𝔽3)—44no—
13C4×A4—42no—
14C22×Q8—34no—
15C22×Q8—34no—
16C22×Q8—31no—
17C22×Q8C3×C411yescharacteristic, complemented
18C23×C4—31no—
19C23×C4—31no—
20C23×C4A411yescharacteristic
21C4×Q8—312no—
22C4×Q8—34no—
23C4×Q8—34no—
24C4×Q8A414yescomplemented
25C2×C4⋊C4—312no—
26C2×C42—36no—
27C2×C42—36no—
28C2×A4—41no—
29C2×A4—41no—
30C2×A4—41no—
31C2×C3×C4—161no—
32SL(2,𝔽3)—44no—
33C24C2×A411yescharacteristic
34C2×Q8—34no—
35C2×Q8—312no—
36C2×Q8—312no—
37C2×Q8—34no—
38C2×Q8—34no—
39C2×Q8C2×A414yes—
40C22×C4—36no—
41C22×C4—36no—
42C22×C4—31no—
43C22×C4—31no—
44C22×C4—31no—
45C22×C4—31no—
46C22×C4—32no—
47C22×C4—36no—
48C22×C4—63no—
49C22×C4—61no—
50C22×C4SL(2,𝔽3)12yescomplemented
51C4⋊C4—34no—
52C4⋊C4—312no—
53C42—312no—
54C42—34no—
55C22×C3—161no—
56A4—41no—
57C3×C4—162no—
58C23—31no—
59C23—31no—
60C23—31no—
61C23—31no—
62C23C4×A411yescharacteristic
63C23C2×SL(2,𝔽3)11yescharacteristic
64C23#41⋊C311yescharacteristic
65Q8—34no—
66Q8—312no—
67Q8—34no—
68Q8C4×A414yescomplemented
69C2×C4—32no—
70C2×C4—36no—
71C2×C4—312no—
72C2×C4—36no—
73C2×C4—31no—
74C2×C4—34no—
75C2×C4—32no—
76C2×C4—32no—
77C2×C4—32no—
78C2×C4—34no—
79C2×C4—66no—
80C2×C4—66no—
81C2×C4C22⋊A411yescharacteristic
82C2×C3—161no—
83C2×C3—161no—
84C2×C3—161no—
85C22—31no—
86C22—31no—
87C22—31no—
88C22—32no—
89C22—31no—
90C22—31no—
91C22—31no—
92C22—31no—
93C22—31no—
94C22—31no—
95C22C4×SL(2,𝔽3)11yescharacteristic, complemented
96C22C2×C22⋊A411yescharacteristic
97C4—34no—
98C4—34no—
99C4—32no—
100C4—64no—
101C4C22⋊SL(2,𝔽3)12yesdirect product factor
102C3—161noSylow
103C2—31no—
104C2—31no—
105C2—31no—
106C2—31no—
107C2C4×C22⋊A411yescharacteristic
108C2C2×C22⋊SL(2,𝔽3)11yescharacteristic
109C2#594811yescharacteristic
subgroupG/H
center, hypercenterC2×C4C22⋊A4
commutator, hypocenterC22×Q8C3×C4
FrattiniC22C2×C22⋊A4
FittingC22×C4×Q8C3
socleC24C2×A4
prime radicalC2×C22⋊SL(2,𝔽3)C2
cosocleC23×Q8C2×C3
LCSG ▷ C22×Q8
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×Q8C3semidirect
3. C22⋊SL(2,𝔽3)C4direct (N × H)
17. C22×Q8C3×C4semidirect
24. C4×Q8A4semidirect
50. C22×C4SL(2,𝔽3)semidirect
68. Q8C4×A4semidirect
95. C22C4×SL(2,𝔽3)semidirect
101. C4C22⋊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
C22×C4×Q83
C2×C4×A44
C4×SL(2,𝔽3)4