Finite Group Databasebeta

← #13136661

Subgroups — #13136661

Order 1728.

167 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 23 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#11720833C211yesmaximal, characteristic
2#11720777C211yesmaximal, characteristic
3#11720252C211yesmaximal, characteristic
4#88651—21no—
5#88627—21no—
6C8×#272C2211yescharacteristic
7#3746⋊C3—21no—
8#88433C412yes—
9(C3⋊(C3⋊C16))⋊C3—21no—
10D3×Dic12—61no—
11#63543—61no—
12C4×#272D411yescharacteristic
13#634⋊C3—22no—
14#6199—41no—
15#6196—41no—
16#6188—41no—
17C8×H(3,𝔽3)D411yescharacteristic
18#6180C2×C411yescharacteristic
19#4496—91nomaximal
20D3×Dic6—62no—
21#3746—21no—
22C3×Dic12—61no—
23C3×D3×C8—61no—
24C3×D3×C8—61no—
25#3719—62no—
26C3⋊(C3⋊C16)—21no—
27C3×C3⋊C16—61no—
28C2×#272SD1611yescharacteristic
29C4×H(3,𝔽3)C22⋊C411yescharacteristic
30#1084D811yescharacteristic
31(C3⋊Dic3)⋊C3—42no—
32D3×Q16—181no—
33C2×Dic12—91no—
34C3×#108—91no—
35#812—91no—
36#798—181no—
37#634—22no—
38C3×C4×D3—61no—
39C3×C4×D3—61no—
40C3×Dic6—62no—
41#627—122no—
42D3×Dic3—122no—
43C32×C8—21no—
44C32×C8—21no—
45C3×C3⋊C8—61no—
46C3×C3⋊C8—61no—
47#362—271noSylow
48C2×H(3,𝔽3)#10211yescharacteristic
49#272—21no—
50D3×Q8—182no—
51C2×Dic6—181no—
52C3×Q16—181no—
53C3×M16—181no—
54C2×C3×C8—91no—
55#216—182no—
56Dic12—61no—
57Dic12—92no—
58Dic12—181no—
59D3×C8—61no—
60D3×C8—61no—
61C3×C16—181no—
62C3⋊C16—61no—
63C3⋊C16—181no—
64C2×C3×D3—61no—
65C2×C3×D3—61no—
66C32×C4—21no—
67C32×C4—21no—
68C3⋊Dic3—42no—
69C3×Dic3—61no—
70C3×Dic3—61no—
71C3×Dic3—122no—
72C2×Q16—271no—
73M32—271no—
74#108—91no—
75H(3,𝔽3)#36211yescharacteristic, complemented, Sylow
76C3×Q8—182no—
77C2×C3×C4—91no—
78C2×Dic3—361no—
79C4×D3—61no—
80C4×D3—61no—
81C4×D3—362no—
82Dic6—62no—
83Dic6—182no—
84Dic6—362no—
85Dic6—361no—
86C3×C8—61no—
87C3×C8—61no—
88C3×C8—91no—
89C3×C8—361no—
90C3×C8GO+(4,𝔽2)11yescharacteristic
91C3⋊C8—61no—
92C3⋊C8—61no—
93C2×C32—21no—
94C2×C32—21no—
95C3×D3—62no—
96C3×D3—121no—
97C2×Q8—541no—
98Q16—181no—
99Q16—272no—
100M16—181no—
101C2×C8—91no—
102C16—181no—
103C22×C3—91no—
104C2×D3—61no—
105C2×D3—61no—
106C3×C4—61no—
107C3×C4—61no—
108C3×C4—91no—
109C3×C4—362no—
110C3×C4D32⋊C411yescharacteristic
111Dic3—61no—
112Dic3—61no—
113Dic3—122no—
114Dic3—362no—
115C32—21no—
116C32—21no—
117Q8—182no—
118Q8—1081no—
119C2×C4—91no—
120C2×C4—1081no—
121C8—91no—
122C8—361no—
123C8#1088⋊C211yescharacteristic
124C2×C3—61no—
125C2×C3—61no—
126C2×C3—181no—
127C2×C3#6372911yescharacteristic
128D3—62no—
129D3—121no—
130C22—91no—
131C4—91no—
132C4—362no—
133C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
134C3—61no—
135C3—61no—
136C3#1059002111yescharacteristic
137C2—181no—
138C2#1172069511yescharacteristic
subgroupG/H
centerC2#11720695
commutator#6180C2×C4
FrattiniC3×C8GO+(4,𝔽2)
FittingC8×H(3,𝔽3)D4
socleC2×C3#63729
prime radicalC2×#272SD16
cosocleC8×#272C22
hypercenterC8#1088⋊C2
hypocenterH(3,𝔽3)#362
LCSG ▷ #6180 ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
75. H(3,𝔽3)#362semidirect

Maximal subgroups

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

subgroupindex
#117208332
#117207772
#117202522
#44969