Finite Group Databasebeta

← #10589509⋊C3 #13146859

Subgroups — #10589509⋊C3 #13146859

Order 1728.

237 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 33 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#63692⋊C3C212yesmaximal, complemented
2#11721359C211yesmaximal, characteristic, complemented
3C3×#5142—31nomaximal
4#10589509C311yesmaximal, characteristic, complemented
5#88691C2211yescharacteristic
6#3689⋊C3—22no—
7D72⋊C3—41nomaximal
8C32×#131D311yescharacteristic
9C3×#949—32no—
10C2×C3×D24—31no—
11#63692C2×C312yescomplemented
12C3×#823—32no—
13C3×#819—31no—
14#131⋊C9—21no—
15#131⋊C9C2×C311yescharacteristic, complemented
16C4⋊AGL(1,ℤ/9)—42no—
17#6204D411yescharacteristic
18C8×3−1+2—41no—
19C3×#472—91no—
20#5142—31no—
21C32×#41C2×D311yescharacteristic
22C2×C3×D12—32no—
23C32×M16—31no—
24C2×C32×C8—31no—
25C3×#215—62no—
26C3×#213—62no—
27C3×#208—32no—
28C3×D24—61no—
29C3×D24—61no—
30#41⋊C9—21no—
31#41⋊C9C22×C311yescharacteristic
32#3689—22no—
33D72—41no—
34C2×AGL(1,ℤ/9)—82no—
35C4×3−1+2—41no—
36C3×AGL(1,ℤ/8)—92no—
37C2×C3×D8—91no—
38C3×#131—31no—
39C3×#131C3×D311yescharacteristic
40#949—32no—
41C2×D24—31no—
42C3×#108—91no—
43C3×#100—92no—
44#823—32no—
45#819—31no—
46C22×C3×D3—62no—
47C32×Q8D1211yescharacteristic
48C32×D4—31no—
49C2×C32×C4—31no—
50C3×D12—62no—
51C3×D12—62no—
52C32×C8—31no—
53C32×C8S411yescharacteristic
54C3×C3⋊C8—62no—
55D36—42no—
56Q8⋊C9—21no—
57Q8⋊C9C3×D411yescharacteristic
58C8×C9—41no—
59C8×C9—81no—
60#472—91noSylow
61C2×3−1+2—41no—
62AGL(1,ℤ/9)—162no—
63C3×#41—31no—
64C3×#41C2×C3×D311yescharacteristic
65C2×C3×D4—92no—
66C2×D12—32no—
67C3×SD16—182no—
68C3×D8—181no—
69C3×D8—181no—
70C3×D8—182no—
71C3×M16—31no—
72C3×M16—31no—
73C3×M16—61no—
74C3×M16—92no—
75C2×C3×C8—31no—
76C2×C3×C8—31no—
77C2×C3×C8—61no—
78#215—62no—
79#213—62no—
80#208—32no—
81D24—61no—
82D24—61no—
83C22×C32—31no—
84C2×C3×D3—122no—
85C2×C3×D3—242no—
86C32×C4—31no—
87C32×C4C2×S411yescharacteristic
88C2×D9—82no—
89C4×C9—41no—
90C4×C9—81no—
91AGL(1,ℤ/8)—92no—
92C2×D8—91no—
93#131AGL(1,ℤ/9)11yescharacteristic, complemented
94#108—91no—
95#100—92no—
963−1+2—41noSylow
97C23×C3—182no—
98C22×D3—62no—
99C3×Q8—31no—
100C3×Q8C3×D1211yescharacteristic
101C3×D4—31no—
102C3×D4—31no—
103C3×D4—61no—
104C3×D4—182no—
105C3×D4—182no—
106C2×C3×C4—31no—
107C2×C3×C4—31no—
108C2×C3×C4—61no—
109D12—62no—
110D12—62no—
111C3×C8—31no—
112C3×C8—31no—
113C3×C8—31no—
114C3×C8—61no—
115C3×C8—182no—
116C3×C8C3×S411yescharacteristic
117C3⋊C8—62no—
118C2×C32—61no—
119C2×C32C4⋊S411yescharacteristic
120C3×D3—242no—
121C2×C9—41no—
122C2×C9—81no—
123D9—162no—
124#41C2×AGL(1,ℤ/9)11yescharacteristic
125C2×D4—92no—
126SD16—182no—
127D8—181no—
128D8—182no—
129D8—181no—
130M16—31no—
131M16—92no—
132C2×C8—31no—
133C22×C3—31no—
134C22×C3—31no—
135C22×C3—61no—
136C22×C3—362no—
137C22×C3—722no—
138C2×D3—122no—
139C2×D3—242no—
140C3×C4—31no—
141C3×C4—31no—
142C3×C4—31no—
143C3×C4—61no—
144C3×C4C2×C3×S411yescharacteristic
145C32#540711yescharacteristic
146C9—41no—
147C9—81no—
148C23—182no—
149Q8C4⋊AGL(1,ℤ/9)11yescharacteristic
150D4—31no—
151D4—182no—
152D4—182no—
153C2×C4—31no—
154C8—31no—
155C8—182no—
156C8((C22⋊C9)⋊C2)⋊C311yescharacteristic
157C2×C3—31no—
158C2×C3—61no—
159C2×C3—61no—
160C2×C3—121no—
161C2×C3—722no—
162C2×C3C3×C4⋊S411yescharacteristic
163D3—242no—
164C22—31no—
165C22—362no—
166C22—722no—
167C4—31no—
168C4C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
169C3—31no—
170C3C3×#540711yescharacteristic
171C2—61no—
172C2—722no—
173C2C4⋊(((C22⋊C9)⋊C2)⋊C3)11yescharacteristic
subgroupG/H
centerC2C4⋊(((C22⋊C9)⋊C2)⋊C3)
commutator, cosocle#41⋊C9C22×C3
FrattiniC3×C4C2×C3×S4
FittingC32×#131D3
socleC2×C3C3×C4⋊S4
hypercenterC8((C22⋊C9)⋊C2)⋊C3
hypocenterQ8⋊C9C3×D4
LCSG ▷ #41⋊C9 ▷ Q8⋊C9
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #63692⋊C3C2semidirect
2. #11721359C2semidirect
4. #10589509C3semidirect
11. #63692C2×C3semidirect
15. #131⋊C9C2×C3semidirect
93. #131AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
#63692⋊C32
#117213592
C3×#51423
#105895093
D72⋊C34