Finite Group Databasebeta

← #13173057

Subgroups — #13173057

Order 1728.

694 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 111 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#11724471C212yesmaximal, complemented
2C32×#131⋊C3C211yesmaximal, characteristic, complemented
3#10593641—312nomaximal
4#10592841—31nomaximal
5C32×#41⋊C3C2211yescharacteristic
6C3⋊(C3⋊GL(2,𝔽3))—22no—
7#88852—41nomaximal
8#64267—324no—
9C32×#131D311yescharacteristic, complemented
10#64158—32no—
11C2×#3745—31no—
12C3×#131⋊C3D3112yescomplemented
13#63650—32no—
14#63646—31no—
15C4⋊(C3⋊(C3⋊D3))—42no—
16C32×SL(2,𝔽3)D411yescharacteristic
17C33×C8—41no—
18#5407—99no—
19#5142—94no—
20C32×#41C2×D311yescharacteristic
21C2×C4⋊(C3⋊D3)—32no—
22C3×#41⋊C3C2×D3112yes—
23C3⋊GL(2,𝔽3)—624no—
24C32×M16—31no—
25C2×C32×C8—31no—
26#3755—62no—
27#3753—62no—
28#3748—32no—
29#3745—61no—
30#3745—61no—
31#3745—1212no—
32C2×C3⋊(C3⋊D3)—82no—
33C33×C4—41no—
34#986—918no—
35C3×#131C3⋊D314yescomplemented
36#949—98no—
37C2×D24—94no—
38#131⋊C3C3⋊D319yescomplemented
39#823—98no—
40#819—94no—
41C22×C3⋊D3—62no—
42C32×Q8D1211yescharacteristic
43C32×D4—31no—
44C2×C32×C4—31no—
45C4⋊(C3⋊D3)—62no—
46C4⋊(C3⋊D3)—62no—
47C4⋊(C3⋊D3)—1224no—
48C3×SL(2,𝔽3)D12112yes—
49C32×C8—31no—
50C32×C8—412no—
51C32×C8S411yescharacteristic
52C3⋊(C3⋊C8)—62no—
53#472—271noSylow
54C2×C33—41no—
55C3⋊(C3⋊D3)—162no—
56C3×#41C2×C3⋊D314yes—
57C2×D12—98no—
58#41⋊C3C2×C3⋊D319yes—
59GL(2,𝔽3)—1818no—
60C3×M16—34no—
61C2×C3×C8—34no—
62#215—188no—
63#213—188no—
64#208—98no—
65D24—184no—
66D24—184no—
67D24—369no—
68C22×C32—31no—
69C2×C3⋊D3—122no—
70C2×C3⋊D3—2424no—
71C2×C3⋊D3—242no—
72C32×C4—31no—
73C32×C4—412no—
74C32×C4C2×S411yescharacteristic
75AGL(1,ℤ/8)—272no—
76C2×D8—271no—
77#131C3⋊(C3⋊D3)11yescharacteristic, complemented
78#108—271no—
79#100—272no—
80C33—41noSylow
81C22×D3—188no—
82C3×Q8C4⋊(C3⋊D3)14yes—
83C3×D4—34no—
84C2×C3×C4—34no—
85D12—188no—
86D12—188no—
87D12—3618no—
88SL(2,𝔽3)C4⋊(C3⋊D3)19yes—
89C3×C8—34no—
90C3×C8—49no—
91C3×C8C3⋊S414yes—
92C3⋊C8—188no—
93C2×C32—412no—
94C2×C32—61no—
95C2×C32C4⋊S411yescharacteristic
96C3⋊D3—242no—
97C3⋊D3—4824no—
98#41C2×C3⋊(C3⋊D3)11yescharacteristic
99C2×D4—272no—
100SD16—542no—
101D8—541no—
102D8—542no—
103D8—541no—
104M16—31no—
105M16—272no—
106C2×C8—31no—
107C22×C3—34no—
108C2×D3—368no—
109C2×D3—7218no—
110C2×D3—728no—
111C3×C4—34no—
112C3×C4—49no—
113C3×C4C2×C3⋊S414yes—
114C32—412no—
115C32#540711yescharacteristic, complemented
116C23—542no—
117Q8C4⋊(C3⋊(C3⋊D3))11yescharacteristic
118D4—31no—
119D4—542no—
120D4—542no—
121C2×C4—31no—
122C8—31no—
123C8—542no—
124C8C3⋊(C3⋊S4)11yescharacteristic
125C2×C3—49no—
126C2×C3—64no—
127C2×C3C4⋊(C3⋊S4)14yes—
128D3—728no—
129D3—14418no—
130C22—31no—
131C22—1082no—
132C22—2162no—
133C4—31no—
134C4C2×C3⋊(C3⋊S4)11yescharacteristic
135C3—49no—
136C3#1059364114yescomplemented
137C2—61no—
138C2—2162no—
139C2C4⋊(C3⋊(C3⋊S4))11yescharacteristic
subgroupG/H
centerC2C4⋊(C3⋊(C3⋊S4))
commutator, cosocleC32×#41⋊C3C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC32×#131D3
socleC2×C32C4⋊S4
hypercenterC8C3⋊(C3⋊S4)
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C32×#41⋊C3 ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11724471C2semidirect
2. C32×#131⋊C3C2semidirect
9. C32×#131D3semidirect
12. C3×#131⋊C3D3semidirect
35. C3×#131C3⋊D3semidirect
38. #131⋊C3C3⋊D3semidirect
77. #131C3⋊(C3⋊D3)semidirect
115. C32#5407semidirect
136. C3#10593641semidirect

Maximal subgroups

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

subgroupindex
#117244712
C32×#131⋊C32
#105936413
#105928413
#888524