Finite Group Databasebeta

← AGL(1,𝔽9)⋊(C3⋊C8) #13165130

Subgroups — AGL(1,𝔽9)⋊(C3⋊C8) #13165130

Order 1728.

138 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C3×C4×AGL(1,𝔽9)C211yesmaximal, characteristic
2#11722804C211yesmaximal, characteristic
3#11722786C211yesmaximal, characteristic
4AGL(1,𝔽9)⋊C8—31nomaximal
5C2×C3×AGL(1,𝔽9)C411yescharacteristic
6C2×C3×AGL(1,𝔽9)C411yescharacteristic
7#88988—21no—
8C3×C4×#157C2211yescharacteristic
9#88808—21no—
10C4×AGL(1,𝔽9)D311yescharacteristic
11#63744—31no—
12#63732—31no—
13C2×C3×#157C2×C411yescharacteristic
14C2×C3×#157Q811yescharacteristic
15C3×AGL(1,𝔽9)—21no—
16C3×AGL(1,𝔽9)C812yescomplemented
17C3×C4×C3⋊D3D411yescharacteristic
18C3⋊#622—41no—
19C3×C3⋊(C3⋊C8)—41no—
20#4457—91nomaximal
21C2×AGL(1,𝔽9)Dic311yescharacteristic
22C2×AGL(1,𝔽9)Dic311yescharacteristic
23C4×#157C2×D311yescharacteristic
24#3787—61no—
25#3710—61no—
26D3×C3⋊C8—121no—
27C2×C3×C3⋊D3C4⋊C411yescharacteristic
28C3×#157—21no—
29C3×#157C2×C811yescharacteristic
30C3×#157M1611yescharacteristic
31C33×C4SD1611yescharacteristic
32C3×C3⋊Dic3SD1611yescharacteristic
33C3×C4×C8—91no—
34C4⋊(C3⋊C8)—91no—
35C4⋊(C3⋊C8)—91no—
36C2×#157Dic611yescharacteristic
37C2×#157C2×Dic311yescharacteristic
38AGL(1,𝔽9)—21no—
39AGL(1,𝔽9)C3⋊C812yescomplemented
40C4×C3⋊D3D1211yescharacteristic
41C3×C4×D3—121no—
42#622—121no—
43C3⋊(C3⋊C8)—41no—
44C3×C3⋊C8—121no—
45C3×C3⋊C8—121no—
46C3×C3⋊C8—241no—
47#334—271noSylow
48C2×C33#10611yescharacteristic
49C3×C3⋊D3C4⋊C811yescharacteristic
50C3×C3⋊D3#10811yescharacteristic
51C2×C3×C8—91no—
52C2×C3×C8—91no—
53C3×C42—91no—
54C2×C3⋊C8—181no—
55C2×C3⋊C8—181no—
56D3×C8—361no—
57C2×C3⋊D3C4⋊Dic311yescharacteristic
58C2×C3×D3—121no—
59#157—21no—
60#157C2×C3⋊C811yescharacteristic
61#157#20811yescharacteristic
62C32×C4—41no—
63C32×C4—81no—
64C32×C4#20411yescharacteristic
65C3⋊Dic3#20411yescharacteristic
66C3×Dic3—121no—
67C4⋊C8—271no—
68C4⋊C8—271no—
69C4×C8—91no—
70C33#33411yescharacteristic, complemented, Sylow
71C2×C3×C4—91no—
72C2×C3×C4—91no—
73C2×C3×C4—91no—
74C4×D3—121no—
75C3×C8—92no—
76C3×C8—181no—
77C3×C8—361no—
78C3⋊C8—121no—
79C3⋊C8—121no—
80C3⋊C8—241no—
81C3⋊C8—361no—
82C2×C32—41no—
83C2×C32—81no—
84C2×C32#81711yescharacteristic
85C3⋊D3C4⋊(C3⋊C8)11yescharacteristic
86C3⋊D3#81911yescharacteristic
87C3×D3—121no—
88C3×D3—121no—
89C2×C8—91no—
90C2×C8—91no—
91C2×C8—541no—
92C2×C8—541no—
93C42—91no—
94C22×C3—91no—
95C2×D3—121no—
96C3×C4—41no—
97C3×C4—81no—
98C3×C4—91no—
99C3×C4—91no—
100C3×C4—91no—
101C3×C4—181no—
102C3×C4AΓL(1,𝔽9)11yescharacteristic
103Dic3—121no—
104C32—41no—
105C32—81no—
106C32#445711yescharacteristic, complemented
107C2×C4—91no—
108C2×C4—91no—
109C2×C4—91no—
110C8—92no—
111C8—181no—
112C8—361no—
113C8—1081no—
114C2×C3—41no—
115C2×C3—81no—
116C2×C3—91no—
117C2×C3—91no—
118C2×C3AGL(1,𝔽9)⋊C411yescharacteristic
119D3—121no—
120D3—121no—
121C22—91no—
122C4—91no—
123C4—91no—
124C4—91no—
125C4—181no—
126C4AGL(1,𝔽9)⋊D311yescharacteristic
127C3—41no—
128C3—81no—
129C3AGL(1,𝔽9)⋊C811yescharacteristic, complemented
130C2—91no—
131C2—91no—
132C2AGL(1,𝔽9)⋊Dic311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4AGL(1,𝔽9)⋊D3
commutatorC3×#157C2×C8
FittingC33×C4SD16
socleC2×C33#106
prime radicalC2×C3×C3⋊D3C4⋊C4
cosocleC3×C4×#157C22
hypocenterC33#334
LCSG ▷ C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
16. C3×AGL(1,𝔽9)C8semidirect
39. AGL(1,𝔽9)C3⋊C8semidirect
70. C33#334semidirect
106. C32#4457semidirect
129. C3AGL(1,𝔽9)⋊C8semidirect

Maximal subgroups

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

subgroupindex
C3×C4×AGL(1,𝔽9)2
#117228042
#117227862
AGL(1,𝔽9)⋊C83
#44579