Finite Group Databasebeta

← #13047754

Subgroups — #13047754

Order 1296.

152 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C4×#4207C211yesmaximal, characteristic
2#10619725C211yesmaximal, characteristic, complemented
3#10619724C211yesmaximal, characteristic, complemented
4#88430—31nomaximal
5#88414—31nomaximal
6#88413—31nomaximal
7C2×#4207C411yescharacteristic
8C4×#737C2211yescharacteristic
9#66409C411yescharacteristic
10C4×C3⋊D9D311yescharacteristic
11C4×AGL(1,ℤ/9)—61no—
12C4×(C3⋊D3)⋊C3—31no—
13C3×C4×D9—31no—
14#6180D311yescharacteristic
15C3⋊(C9⋊C8)D311yescharacteristic, complemented
163−1+2⋊C8—61no—
17(C3⋊(C3⋊C8))⋊C3—31no—
18C9×C3⋊C8—31no—
19C9×C3⋊C8—31no—
20C9×C3⋊C8—61no—
21C3×C9⋊C8—31no—
22C2×#737C2×C411yescharacteristic
23#4207—21no—
24#3712—91no—
25#3711—91no—
26#3663—91no—
27C2×C3⋊D9Dic311yescharacteristic
28C2×AGL(1,ℤ/9)—61no—
29C2×(C3⋊D3)⋊C3—31no—
30C2×C3×D9—31no—
31C4×3−1+2—21no—
32C4×3−1+2—21no—
33C4×3−1+2—41no—
34C4×H(3,𝔽3)C2×D311yescharacteristic
35C3×C4×C9—21no—
36C3×C4×C9C2×D311yescharacteristic
37C3×C4×C9C2×D311yescharacteristic
38C3⋊Dic9Dic311yescharacteristic
393−1+2⋊C4—61no—
40(C3⋊Dic3)⋊C3—31no—
41C3×Dic9—31no—
42#737M1611yescharacteristic, complemented, Sylow
43C4×C3⋊D3—31no—
44C3×C4×D3—91no—
45C3⋊(C3⋊C8)—31no—
46C3×C3⋊C8—31no—
47C3×C3⋊C8—31no—
48C3×C3⋊C8—61no—
49C3×C3⋊C8—91no—
50C4×D9—31no—
51C4×D9—61no—
52C8×C9—91no—
53C9⋊C8—31no—
54C9⋊C8—61no—
55C2×3−1+2—21no—
56C2×3−1+2—21no—
57C2×3−1+2—41no—
58C2×H(3,𝔽3)C4×D311yescharacteristic
59C2×C3×C9—21no—
60C2×C3×C9C4×D311yescharacteristic
61C2×C3×C9C2×Dic311yescharacteristic
62C3⋊D9—21no—
63AGL(1,ℤ/9)—62no—
64(C3⋊D3)⋊C3—61no—
65C3×D9—61no—
66#208—271no—
67ΓL(1,𝔽25)—271no—
68ΓL(1,𝔽25)—271no—
69C2×C3⋊D3—31no—
70C2×C3×D3—91no—
71C32×C4—21no—
72C32×C4D3211yescharacteristic
73C32×C4D3211yescharacteristic
74C3⋊Dic3—31no—
75C3×Dic3—91no—
76C2×D9—31no—
77C2×D9—61no—
78C4×C9—21no—
79C4×C9—21no—
80C4×C9—41no—
81C4×C9D3211yescharacteristic
82Dic9—31no—
83Dic9—61no—
843−1+2—21no—
853−1+2—21no—
863−1+2—41no—
87H(3,𝔽3)ΓL(1,𝔽25)11yescharacteristic
88C3×C9—21no—
89C3×C9ΓL(1,𝔽25)11yescharacteristic, complemented
90C3×C9#20811yescharacteristic, complemented
91C2×C3×C4—271no—
92C4×D3—91no—
93C4×D3—271no—
94C3×C8—91no—
95C3×C8—271no—
96C3⋊C8—31no—
97C3⋊C8—31no—
98C3⋊C8—61no—
99C3⋊C8—91no—
100C3⋊C8—271no—
101C2×C32—21no—
102C2×C32D3×Dic311yescharacteristic
103C2×C32#62411yescharacteristic
104C3⋊D3—61no—
105C3×D3—181no—
106C2×C9—21no—
107C2×C9—21no—
108C2×C9—41no—
109C2×C9D3×Dic311yescharacteristic
110D9—62no—
111D9—61no—
112M16—811noSylow
113C22×C3—271no—
114C2×D3—91no—
115C2×D3—271no—
116C3×C4—31no—
117C3×C4—31no—
118C3×C4—61no—
119C3×C4—271no—
120C3×C4#111311yescharacteristic
121Dic3—91no—
122Dic3—271no—
123C32—21no—
124C32#371111yescharacteristic
125C32#371211yescharacteristic
126C9—21no—
127C9—21no—
128C9—41no—
129C9#371111yescharacteristic
130C2×C4—271no—
131C8—91no—
132C8—271no—
133C2×C3—31no—
134C2×C3—31no—
135C2×C3—61no—
136C2×C3—541no—
137C2×C3#629411yescharacteristic
138D3—181no—
139D3—541no—
140C22—271no—
141C4—271no—
142C4#4207⋊C211yescharacteristic
143C3—31no—
144C3—31no—
145C3—61no—
146C3#8880911yescharacteristic
147C2—541no—
148C2#4207⋊C411yescharacteristic
subgroupG/H
center, hypercenterC4#4207⋊C2
commutatorC2×#737C2×C4
FrattiniC3×C4#1113
Fitting, cosocleC4×#737C22
socleC2×C3#6294
prime radicalC2×#4207C4
hypocenter#737M16
LCSG ▷ C2×#737 ▷ #737
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #10619725C2semidirect
3. #10619724C2semidirect
15. C3⋊(C9⋊C8)D3semidirect
42. #737M16semidirect
89. C3×C9ΓL(1,𝔽25)semidirect
90. C3×C9#208semidirect

Maximal subgroups

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

subgroupindex
C4×#42072
#106197252
#106197242
#884303
#884143
#884133