Finite Group Databasebeta

← #13136660

Subgroups — #13136660

Order 1728.

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

normal subgroups only

This group has 195 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#11720838C211yesmaximal, characteristic
2#11720777C211yesmaximal, characteristic, complemented
3#11720252C211yesmaximal, characteristic, complemented
4#88655—21no—
5#88627—21no—
6C8×#272C2211yescharacteristic
7#3745⋊C3—21no—
8#88432C412yes—
9(C3⋊(C3⋊C16))⋊C3—21no—
10D3×D24—61no—
11#63543—61no—
12C2×((C3⋊D3)⋊C3)⋊C2—41no—
13C4×#272D411yescharacteristic
14C4⋊((C3⋊D3)⋊C3)—22no—
15#6198—41no—
16#6188—41no—
17C8×H(3,𝔽3)D411yescharacteristic
18#6180C2×C411yescharacteristic
19#4495—91nomaximal
20D3×D12—62no—
21#3745—21no—
22C3×D24—61no—
23C3×D3×C8—61no—
24C3×D3×C8—61no—
25#3714—62no—
26C3⋊(C3⋊C16)—21no—
27C3×C3⋊C16—61no—
28C2×#272SD1611yescharacteristic
29C2×(C3⋊D3)⋊C3—42no—
30((C3⋊D3)⋊C3)⋊C2—82no—
31C4×H(3,𝔽3)C22⋊C411yescharacteristic
32#1084D811yescharacteristic
33D3×D8—181no—
34C2×D24—91no—
35C3×#108—91no—
36#812—91no—
37#798—181no—
38C2×D32—122no—
39C4⋊(C3⋊D3)—22no—
40C3×D12—62no—
41C3×C4×D3—61no—
42C3×C4×D3—61no—
43#626—122no—
44C32×C8—21no—
45C32×C8—21no—
46C3×C3⋊C8—61no—
47C3×C3⋊C8—61no—
48#361—271noSylow
49C2×H(3,𝔽3)#10211yescharacteristic
50#272—21no—
51(C3⋊D3)⋊C3—82no—
52D3×D4—182no—
53C2×D12—181no—
54C3×D8—181no—
55C3×M16—181no—
56C2×C3×C8—91no—
57#213—182no—
58D24—61no—
59D24—92no—
60D24—181no—
61D3×C8—61no—
62D3×C8—61no—
63C3×C16—181no—
64C3⋊C16—61no—
65C3⋊C16—181no—
66C2×C3⋊D3—42no—
67C2×C3×D3—61no—
68C2×C3×D3—61no—
69C2×C3×D3—122no—
70D32—244no—
71C32×C4—21no—
72C32×C4—21no—
73C3×Dic3—61no—
74C3×Dic3—61no—
75C2×D8—271no—
76M32—271no—
77#108—91no—
78H(3,𝔽3)#36111yescharacteristic, complemented, Sylow
79C22×D3—362no—
80C22×D3—361no—
81C3×D4—182no—
82C2×C3×C4—91no—
83#67—362no—
84D12—62no—
85D12—182no—
86D12—361no—
87C4×D3—61no—
88C4×D3—61no—
89C3×C8—61no—
90C3×C8—61no—
91C3×C8—91no—
92C3×C8—361no—
93C3×C8GO+(4,𝔽2)11yescharacteristic
94C3⋊C8—61no—
95C3⋊C8—61no—
96C2×C32—21no—
97C2×C32—21no—
98C3⋊D3—82no—
99C3×D3—62no—
100C3×D3—121no—
101C3×D3—242no—
102C2×D4—541no—
103D8—181no—
104D8—272no—
105M16—181no—
106C2×C8—91no—
107C16—181no—
108C22×C3—91no—
109C22×C3—362no—
110C2×D3—61no—
111C2×D3—61no—
112C2×D3—122no—
113C2×D3—362no—
114C2×D3—724no—
115C2×D3—722no—
116C3×C4—61no—
117C3×C4—61no—
118C3×C4—91no—
119C3×C4D32⋊C411yescharacteristic
120Dic3—61no—
121Dic3—61no—
122C32—21no—
123C32—21no—
124C23—1081no—
125D4—182no—
126D4—1081no—
127C2×C4—91no—
128C8—91no—
129C8—361no—
130C8#1088⋊C211yescharacteristic
131C2×C3—61no—
132C2×C3—61no—
133C2×C3—181no—
134C2×C3—722no—
135C2×C3#6372911yescharacteristic
136D3—62no—
137D3—121no—
138D3—242no—
139D3—722no—
140C22—91no—
141C22—362no—
142C22—2162no—
143C4—91no—
144C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
145C3—61no—
146C3—61no—
147C3#1059002011yescharacteristic
148C2—181no—
149C2—722no—
150C2#1172069511yescharacteristic
subgroupG/H
centerC2#11720695
commutator#6180C2×C4
FrattiniC3×C8GO+(4,𝔽2)
FittingC8×H(3,𝔽3)D4
socleC2×C3#63729
prime radical#11720838C2
cosocleC8×#272C22
hypercenterC8#1088⋊C2
hypocenterH(3,𝔽3)#361
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
2. #11720777C2semidirect
3. #11720252C2semidirect
78. H(3,𝔽3)#361semidirect

Maximal subgroups

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

subgroupindex
#117208382
#117207772
#117202522
#44959