Finite Group Databasebeta

← #13147358

Subgroups — #13147358

Order 1728.

250 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 57 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#11720771C212yesmaximal, complemented
5#11720770C212yesmaximal, complemented
6#88740C411yescharacteristic
7M16×H(3,𝔽3)C411yescharacteristic
8C2×#6180C2211yescharacteristic
9#88527C412yes—
10C8×#272C2212yes—
11#88410C2211yescharacteristic
12#88410C2212yes—
13#88410C2211yescharacteristic
14C3×#907—62no—
15C2×C4×H(3,𝔽3)C2×C411yescharacteristic
16#6235C812yes—
17C4⋊#272C811yescharacteristic
18C4×#272C2×C412yes—
19#6229C811yescharacteristic
20C8×H(3,𝔽3)C2×C412yes—
21#6180C2×C411yescharacteristic
22#6180C2311yescharacteristic
23C3×#504—91nomaximal
24C3×#235—62no—
25C32×M16—22no—
26C2×C3×C3⋊C8—62no—
27C3×ΓL(1,𝔽25)—64no—
28C3×D3×C8—64no—
29C22×H(3,𝔽3)C2×C811yescharacteristic
30C2×#272C2×C812yes—
31C4×H(3,𝔽3)C2×C811yescharacteristic
32C4×H(3,𝔽3)C22×C411yescharacteristic
33#1084C2×C812yes—
34C3×#131—91no—
35#907—62no—
36C3×M32—92no—
37C3×M32—91no—
38C2×C3×C16—92no—
39C2×C3×C16—91no—
40C2×C32×C4—22no—
41C3×#67—64no—
42C3×D12—62no—
43C3×C4×D3—64no—
44C3×Dic6—62no—
45C32×C8—24no—
46C3×C3⋊C8—62no—
47C3×C3⋊C8—62no—
48#504—91noSylow
49C2×H(3,𝔽3)—21no—
50C2×H(3,𝔽3)C22×C811yescharacteristic
51#272—22no—
52C3×#41—91no—
53#235—62no—
54C3×M16—62no—
55C3×M16—92no—
56C3×M16#15711yescharacteristic
57C2×C3×C8—91no—
58C2×C3×C8—92no—
59C2×C3⋊C8—62no—
60ΓL(1,𝔽25)—64no—
61D3×C8—64no—
62C3×C16—91no—
63C3×C16—92no—
64C3×C16—91no—
65C22×C32—22no—
66C2×C3×D3—64no—
67C32×C4—22no—
68C32×C4—22no—
69C3×Dic3—64no—
70#131—91no—
71M32—91no—
72M32—92no—
73C2×C16—91no—
74C2×C16—92no—
75H(3,𝔽3)#50411yescharacteristic, complemented, Sylow
76C3×Q8—91no—
77C3×D4—92no—
78C3×D4—91no—
79C2×C3×C4—62no—
80C2×C3×C4—92no—
81C2×C3×C4C2×#15711yescharacteristic
82#67—64no—
83D12—62no—
84C4×D3—64no—
85Dic6—62no—
86C3×C8—64no—
87C3×C8—91no—
88C3×C8—91no—
89C3×C8C2×#15712yes—
90C3⋊C8—62no—
91C3⋊C8—62no—
92C2×C32—22no—
93C2×C32—42no—
94C3×D3—124no—
95#41—91no—
96M16—92no—
97M16#108811yescharacteristic
98C2×C8—91no—
99C2×C8—92no—
100C16—91no—
101C16—91no—
102C16—92no—
103C22×C3—62no—
104C22×C3—92no—
105C22×C3#378711yescharacteristic
106C2×D3—64no—
107C3×C4—62no—
108C3×C4—62no—
109C3×C4—92no—
110C3×C4#378711yescharacteristic
111C3×C4C22×#15711yescharacteristic
112Dic3—64no—
113C32—22no—
114Q8—91no—
115D4—92no—
116D4—91no—
117C2×C4—92no—
118C2×C4C2×#108811yescharacteristic
119C8—91no—
120C8—91no—
121C8C2×#108812yes—
122C2×C3—21no—
123C2×C3—62no—
124C2×C3—122no—
125C2×C3—182no—
126C2×C3C2×#378711yescharacteristic
127D3—124no—
128C22—92no—
129C22#8862611yescharacteristic
130C4—92no—
131C4#8862611yescharacteristic
132C4C22×#108811yescharacteristic
133C3—62no—
134C3#1059377211yescharacteristic
135C2—21no—
136C2—182no—
137C2C2×#8862611yescharacteristic
subgroupG/H
center, FrattiniC3×C4C22×#157
commutatorC2×H(3,𝔽3)C22×C8
FittingM16×H(3,𝔽3)C4
socleC2×C3C2×#3787
prime radical#88740C4
cosocle#6180C23
hypercenterC3×M16#157
hypocenterH(3,𝔽3)#504
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
2. #11720779C2semidirect
3. C2×#88410C2semidirect
4. #11720771C2semidirect
5. #11720770C2semidirect
75. H(3,𝔽3)#504semidirect

Maximal subgroups

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

subgroupindex
#117217862
#117207792
C2×#884102
#117207712
#117207702
C3×#5049