Finite Group Databasebeta

← #11724490

Subgroups — #11724490

Order 864.

332 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 51 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×C32×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
4#64364C311yesmaximal, characteristic, complemented
5#1007⋊C3—34nomaximal
6C3×#638—41nomaximal
7C32×SL(2,𝔽3)C2211yescharacteristic
8C2×C32×Q8C2×C311yescharacteristic, complemented
9#3832—31no—
10#3832C2×C311yescharacteristic, complemented
11Q8×C3⋊D3—31no—
12Q8×C3⋊D3C2×C311yescharacteristic, complemented
13#3828—31no—
14#3828—31no—
15C2×C3×SL(2,𝔽3)—24no—
16C2×C3×SL(2,𝔽3)D314yescomplemented
17D3×SL(2,𝔽3)—34no—
18C3⋊(#41⋊C3)—34no—
19C22×C33—41no—
20C2×C3×C3⋊D3—41no—
21C3×C3⋊Dic3—41no—
22#1007—34no—
232−1+4⋊C3—91no—
24C32×Q8—31no—
25C32×Q8C22×C311yescharacteristic
26C2×C32×C4—31no—
27#638—31no—
28#638A411yescharacteristic
29C4⋊(C3⋊D3)—31no—
30C4⋊(C3⋊D3)—31no—
31C4×C3⋊D3—32no—
32C4×C3⋊D3—31no—
33C4×C3⋊D3—31no—
34#634—31no—
35#634—31no—
36C3×#67—124no—
37C3×SL(2,𝔽3)—24no—
38C3×SL(2,𝔽3)C2×D314yes—
39C2×C33—41no—
40C2×C33—81no—
41C3×C3⋊D3—81no—
42C2×C3×Q8C3×D314yescomplemented
43#239—34no—
44#239—94no—
45D3×Q8—34no—
46D3×Q8—94no—
47#235—94no—
48#235—94no—
49#41⋊C3—91no—
50C2×SL(2,𝔽3)—24no—
51C2×SL(2,𝔽3)—91no—
52C2×SL(2,𝔽3)C3⋊D311yescharacteristic, complemented
53C22×C32—44no—
54C22×C32—84no—
55C22×C32C2×A411yescharacteristic
56C2×C3⋊D3—31no—
57C2×C3⋊D3C2×A411yescharacteristic
58C2×C3×D3—124no—
59C32×C4—31no—
60C32×C4—31no—
61C3⋊Dic3—31no—
62C3⋊Dic3C2×A411yescharacteristic
63C3×Dic3—124no—
642−1+4—91noSylow
65C33—41noSylow
66C3×Q8—34no—
67C3×Q8C2×C3×D314yes—
68C3×D4—361no—
69C2×C3×C4—34no—
70#67—34no—
71#67—94no—
72D12—94no—
73D12—94no—
74C4×D3—94no—
75C4×D3—94no—
76C4×D3—98no—
77Dic6—94no—
78Dic6—94no—
79SL(2,𝔽3)—24no—
80SL(2,𝔽3)C2×C3⋊D311yescharacteristic
81C2×C32—21no—
82C2×C32—44no—
83C2×C32—84no—
84C2×C32—88no—
85C2×C32—84no—
86C2×C32C22×A411yescharacteristic
87C3⋊D3—21no—
88C3⋊D3—61no—
89C3×D3—244no—
90#41—91no—
91#41—271no—
92#41—271no—
93#41—271no—
94C2×Q8—91no—
95C2×Q8—271no—
96C2×Q8C3×C3⋊D311yescharacteristic, complemented
97C22×C3—41no—
98C22×C3—84no—
99C22×C3—361no—
100C22×C3D3×A414yes—
101C2×D3—34no—
102C2×D3—94no—
103C3×C4—34no—
104C3×C4—34no—
105C3×C4—361no—
106Dic3—34no—
107Dic3—94no—
108C32—44no—
109C32—84no—
110C322−1+4⋊C311yescharacteristic, complemented
111Q8—31no—
112Q8—271no—
113Q8—271no—
114Q8C2×C3×C3⋊D311yescharacteristic
115D4—91no—
116D4—271no—
117D4—271no—
118D4—271no—
119C2×C4—31no—
120C2×C4—271no—
121C2×C4—271no—
122C2×C4—272no—
123C2×C3—24no—
124C2×C3—41no—
125C2×C3—81no—
126C2×C3—88no—
127C2×C3—84no—
128C2×C3—721no—
129C2×C3C2×D3×A414yes—
130D3—64no—
131D3—184no—
132C22—91no—
133C22—271no—
134C22A4×C3⋊D311yescharacteristic
135C4—31no—
136C4—31no—
137C4—91no—
138C4—271no—
139C3—41no—
140C3—84no—
141C3#1007⋊C314yescomplemented
142C2—21no—
143C2—181no—
144C2—541no—
145C2C2×A4×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×A4×C3⋊D3
commutator, cosocle, hypocenterC32×Q8C22×C3
FittingC2×C32×Q8C2×C3
socleC2×C32C22×A4
hypercenterC22A4×C3⋊D3
LCSG ▷ C32×Q8
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C32×SL(2,𝔽3)C2semidirect
2. C3⋊D3×SL(2,𝔽3)C2semidirect
3. C3⋊(C3⋊(#41⋊C3))C2semidirect
4. #64364C3semidirect
8. C2×C32×Q8C2×C3semidirect
10. #3832C2×C3semidirect
12. Q8×C3⋊D3C2×C3semidirect
16. C2×C3×SL(2,𝔽3)D3semidirect
42. C2×C3×Q8C3×D3semidirect
52. C2×SL(2,𝔽3)C3⋊D3semidirect
96. C2×Q8C3×C3⋊D3semidirect
110. C322−1+4⋊C3semidirect
141. C3#1007⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C32×SL(2,𝔽3)2
C3⋊D3×SL(2,𝔽3)2
C3⋊(C3⋊(#41⋊C3))2
#643643
#1007⋊C33
C3×#6384