Finite Group Databasebeta

← #13173053

Subgroups — #13173053

Order 1728.

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

normal subgroups only

This group has 626 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#11724469C212yesmaximal
2C32×#131⋊C3C211yesmaximal, characteristic
3#10593637—312nomaximal
4#10592842—31nomaximal
5C32×#41⋊C3C2211yescharacteristic
6C3⋊(C3⋊#226)—22no—
7#88853—41nomaximal
8#64265—324no—
9C32×#131D311yescharacteristic
10#64160—32no—
11C2×#3746—31no—
12C3×#131⋊C3D3112yes—
13#63651—32no—
14#63646—31no—
15#6308—42no—
16C32×SL(2,𝔽3)D411yescharacteristic
17C33×C8—41no—
18#5403—99no—
19#5143—94no—
20C32×#41C2×D311yescharacteristic
21C2×#634—32no—
22C3×#41⋊C3C2×D3112yes—
23C3⋊#226—624no—
24C32×M16—31no—
25C2×C32×C8—31no—
26#3756—62no—
27#3754—62no—
28#3748—32no—
29#3746—61no—
30#3746—61no—
31#3746—1212no—
32C33×C4—41no—
33C3⋊(C3⋊Dic3)—82no—
34#984—918no—
35C3×#131C3⋊D314yes—
36#951—98no—
37C2×Dic12—94no—
38#131⋊C3C3⋊D319yes—
39#824—98no—
40#819—94no—
41C32×Q8D1211yescharacteristic
42C32×D4—31no—
43C2×C32×C4—31no—
44C2×C3⋊Dic3—62no—
45#634—62no—
46#634—62no—
47#634—1224no—
48C3×SL(2,𝔽3)D12112yes—
49C32×C8—31no—
50C32×C8—412no—
51C32×C8S411yescharacteristic
52C3⋊(C3⋊C8)—62no—
53#473—271noSylow
54C2×C33—41no—
55C3×#41C2×C3⋊D314yes—
56C2×Dic6—98no—
57#41⋊C3C2×C3⋊D319yes—
58#226—1818no—
59C3×M16—34no—
60C2×C3×C8—34no—
61#216—188no—
62#214—188no—
63#208—98no—
64Dic12—184no—
65Dic12—184no—
66Dic12—369no—
67C22×C32—31no—
68C32×C4—31no—
69C32×C4—412no—
70C32×C4C2×S411yescharacteristic
71C3⋊Dic3—122no—
72C3⋊Dic3—2424no—
73#137—272no—
74C2×Q16—271no—
75#131C3⋊(C3⋊D3)11yescharacteristic
76#108—271no—
77#101—272no—
78C33—41noSylow
79C3×Q8C4⋊(C3⋊D3)14yes—
80C3×D4—34no—
81C2×C3×C4—34no—
82C2×Dic3—188no—
83Dic6—188no—
84Dic6—188no—
85Dic6—3618no—
86SL(2,𝔽3)C4⋊(C3⋊D3)19yes—
87C3×C8—34no—
88C3×C8—49no—
89C3×C8C3⋊S414yes—
90C3⋊C8—188no—
91C2×C32—412no—
92C2×C32—61no—
93C2×C32C4⋊S411yescharacteristic
94#41C2×C3⋊(C3⋊D3)11yescharacteristic
95C2×Q8—272no—
96Q16—541no—
97Q16—542no—
98Q16—541no—
99SD16—542no—
100M16—31no—
101M16—272no—
102C2×C8—31no—
103C22×C3—34no—
104C3×C4—34no—
105C3×C4—49no—
106C3×C4C2×C3⋊S414yes—
107Dic3—368no—
108Dic3—7218no—
109C32—412no—
110C32#540311yescharacteristic, complemented
111Q8—542no—
112Q8—542no—
113Q8C4⋊(C3⋊(C3⋊D3))11yescharacteristic
114D4—31no—
115C2×C4—31no—
116C2×C4—542no—
117C8—31no—
118C8—542no—
119C8C3⋊(C3⋊S4)11yescharacteristic
120C2×C3—49no—
121C2×C3—64no—
122C2×C3C4⋊(C3⋊S4)14yes—
123C22—31no—
124C4—31no—
125C4—1082no—
126C4C2×C3⋊(C3⋊S4)11yescharacteristic
127C3—49no—
128C3#1059363714yescomplemented
129C2—61no—
130C2C4⋊(C3⋊(C3⋊S4))11yescharacteristic
subgroupG/H
centerC2C4⋊(C3⋊(C3⋊S4))
commutator, prime radical, 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
110. C32#5403semidirect
128. C3#10593637semidirect

Maximal subgroups

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

subgroupindex
#117244692
C32×#131⋊C32
#105936373
#105928423
#888534