Finite Group Databasebeta

← #13136921

Subgroups — #13136921

Order 1728.

208 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 31 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#11721786C211yesmaximal, characteristic
2#11720779C211yesmaximal, characteristic, complemented
3C2×#88410C211yesmaximal, characteristic, complemented
4#88740C411yescharacteristic
5M16×H(3,𝔽3)C411yescharacteristic
6C2×#6180C2211yescharacteristic
7#88527—21no—
8C8×#272—21no—
9#88410—21no—
10#88410—21no—
11C3×#907—62no—
12C2×C4×H(3,𝔽3)C2×C411yescharacteristic
13#6235—21no—
14C4⋊#272C811yescharacteristic
15C4×#272—21no—
16#6229C811yescharacteristic
17C8×H(3,𝔽3)—21no—
18#6180D411yescharacteristic
19#6180D411yescharacteristic
20C3×#350—91nomaximal
21C3×#235—62no—
22C32×M16—22no—
23C2×C3×C3⋊C8—62no—
24C3×ΓL(1,𝔽25)—64no—
25C3×D3×C8—64no—
26C22×H(3,𝔽3)M1611yescharacteristic
27C2×#272—21no—
28C4×H(3,𝔽3)C22⋊C411yescharacteristic
29C4×H(3,𝔽3)C2×C811yescharacteristic
30#1084—21no—
31C3×#131—91no—
32#907—62no—
33C3×M32—91no—
34C2×C3×C16—91no—
35C2×C32×C4—22no—
36C3×#67—64no—
37C3×D12—62no—
38C3×C4×D3—64no—
39C3×Dic6—62no—
40C32×C8—24no—
41C3×C3⋊C8—62no—
42C3×C3⋊C8—62no—
43#350—91noSylow
44C2×H(3,𝔽3)—21no—
45C2×H(3,𝔽3)C22⋊C811yescharacteristic
46#272—41no—
47C3×#41—91no—
48#235—62no—
49C3×M16—62no—
50C3×M16—181no—
51C3×M16#15711yescharacteristic
52C2×C3×C8—91no—
53C2×C3×C8—181no—
54C2×C3⋊C8—62no—
55ΓL(1,𝔽25)—64no—
56D3×C8—64no—
57C3×C16—181no—
58C3×C16—181no—
59C22×C32—22no—
60C2×C3×D3—64no—
61C32×C4—22no—
62C32×C4—22no—
63C3×Dic3—64no—
64#131—91no—
65M32—91no—
66C2×C16—91no—
67H(3,𝔽3)#35011yescharacteristic, complemented, Sylow
68C3×Q8—91no—
69C3×D4—91no—
70C3×D4—181no—
71C2×C3×C4—62no—
72C2×C3×C4—181no—
73C2×C3×C4C2×#15711yescharacteristic
74#67—64no—
75D12—62no—
76C4×D3—64no—
77Dic6—62no—
78C3×C8—21no—
79C3×C8—64no—
80C3×C8—91no—
81C3×C8—91no—
82C3⋊C8—62no—
83C3⋊C8—62no—
84C2×C32—22no—
85C2×C32—42no—
86C3×D3—124no—
87#41—91no—
88M16—181no—
89M16#108811yescharacteristic
90C2×C8—91no—
91C2×C8—181no—
92C16—181no—
93C16—181no—
94C22×C3—62no—
95C22×C3—181no—
96C22×C3#378811yescharacteristic
97C2×D3—64no—
98C3×C4—62no—
99C3×C4—62no—
100C3×C4—181no—
101C3×C4#378711yescharacteristic
102C3×C4C22⋊#15711yescharacteristic
103Dic3—64no—
104C32—22no—
105Q8—91no—
106D4—91no—
107D4—181no—
108C2×C4—181no—
109C2×C4C2×#108811yescharacteristic
110C8—21no—
111C8—91no—
112C8—91no—
113C2×C3—21no—
114C2×C3—62no—
115C2×C3—122no—
116C2×C3—361no—
117C2×C3#6377811yescharacteristic
118D3—124no—
119C22—181no—
120C22#8862711yescharacteristic
121C4—181no—
122C4#8862611yescharacteristic
123C4C22⋊#108811yescharacteristic
124C3—62no—
125C3#1059018711yescharacteristic
126C2—21no—
127C2—361no—
128C2#1172078411yescharacteristic
subgroupG/H
centerC3×C4C22⋊#157
commutatorC4×H(3,𝔽3)C2×C8
FrattiniC2×C3×C4C2×#157
FittingM16×H(3,𝔽3)C4
socleC2×C3#63778
prime radical#88740C4
cosocleC2×#6180C22
hypercenterC3×M16#157
hypocenterH(3,𝔽3)#350
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
2. #11720779C2semidirect
3. C2×#88410C2semidirect
67. H(3,𝔽3)#350semidirect

Maximal subgroups

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

subgroupindex
#117217862
#117207792
C2×#884102
C3×#3509