Finite Group Databasebeta

← C2×#10619681 #13047478

Subgroups — C2×#10619681 #13047478

Order 1296.

378 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C23×#731C211yesmaximal, characteristic, complemented
2C22×#4184C211yesmaximal, characteristic, complemented
3#10619681C214yesmaximal, direct product factor
4C2×#66331C211yesmaximal, characteristic, complemented
5C2×#611⋊C3—32nomaximal
6C2×#638⋊C3—31nomaximal
7C2×C3×#611—31nomaximal
8C22×#731—22no—
9C22×#731C2211yescharacteristic
10C22×#731C2212yescomplemented
11C2×#4184—22no—
12C2×#4184C2212yescomplemented
13#66331C2212yescomplemented
14C23×3−1+2D312yescomplemented
15C23×H(3,𝔽3)D311yescharacteristic
16C23×C3×C9D311yescharacteristic, complemented
17C22×AGL(1,ℤ/9)—32no—
18C22×(C3⋊D3)⋊C3—31no—
19C22×C3×D9—31no—
20#611⋊C3—38no—
21C2×3−1+2⋊C4—32no—
22#638⋊C3—34no—
23C2×(C3⋊Dic3)⋊C3—31no—
24C3×#611—34no—
25C2×C3×Dic9—31no—
26C2×#731—22no—
27C2×#731D412yescomplemented
28C2×#731C2311yescharacteristic
29#4184—22no—
30C2×#638—31no—
31C2×C3×#67—91no—
32C2×#611—31no—
33C2×#611—32no—
34C22×3−1+2—24no—
35C22×3−1+2C2×D312yes—
36C22×3−1+2C2×D314yescomplemented
37C22×H(3,𝔽3)—22no—
38C22×H(3,𝔽3)C2×D312yes—
39C22×H(3,𝔽3)C2×D311yescharacteristic
40C22×C3×C9—22no—
41C22×C3×C9C2×D311yescharacteristic
42C22×C3×C9C2×D312yescomplemented
43C2×AGL(1,ℤ/9)—34no—
44C2×AGL(1,ℤ/9)—64no—
45C2×(C3⋊D3)⋊C3—32no—
46C2×(C3⋊D3)⋊C3—62no—
47C2×C3×D9—32no—
48C2×C3×D9—62no—
493−1+2⋊C4—34no—
50(C3⋊Dic3)⋊C3—32no—
51C3×Dic9—32no—
52#731C2×D411yescharacteristic, complemented, Sylow
53C23×C32—31no—
54C23×C32C3⋊D311yescharacteristic
55C22×C3⋊D3—31no—
56C22×C3×D3—91no—
57#638—34no—
58C2×C3⋊Dic3—31no—
59C3×#67—94no—
60C2×C3×Dic3—91no—
61C23×C9—31no—
62C23×C9—32no—
63C22×D9—31no—
64C22×D9—32no—
65#611—34no—
66#611—38no—
67C2×Dic9—31no—
68C2×Dic9—32no—
69C2×3−1+2—24no—
70C2×3−1+2#6714yescomplemented
71C2×3−1+2C22×D312yes—
72C2×H(3,𝔽3)—22no—
73C2×H(3,𝔽3)#6712yes—
74C2×H(3,𝔽3)C22×D311yescharacteristic
75C2×C3×C9—22no—
76C2×C3×C9#6712yescomplemented
77C2×C3×C9C22×D311yescharacteristic
78AGL(1,ℤ/9)—64no—
79(C3⋊D3)⋊C3—62no—
80C3×D9—62no—
81C2×C3×D4—271no—
82C2×#67—91no—
83C2×#67—271no—
84C22×C32—22no—
85C22×C32—31no—
86C22×C32—32no—
87C22×C32—62no—
88C22×C32C2×C3⋊D312yes—
89C22×C32C2×C3⋊D311yescharacteristic
90C2×C3⋊D3—32no—
91C2×C3⋊D3—62no—
92C2×C3×D3—92no—
93C2×C3×D3—182no—
94C3⋊Dic3—32no—
95C3×Dic3—92no—
96C22×C9—31no—
97C22×C9—32no—
98C22×C9—32no—
99C22×C9—34no—
100C22×C9—62no—
101C22×C9—64no—
102C2×D9—32no—
103C2×D9—34no—
104C2×D9—62no—
105C2×D9—64no—
106Dic9—32no—
107Dic9—34no—
1083−1+2C2×#6712yescomplemented
109H(3,𝔽3)C2×#6711yescharacteristic
110C3×C9C2×#6711yescharacteristic, complemented
111C23×C3—31no—
112C23×C3—91no—
113C23×C3—271no—
114C23×C3#27211yescharacteristic
115C22×D3—91no—
116C22×D3—271no—
117C3×D4—274no—
118C2×C3×C4—271no—
119#67—94no—
120#67—274no—
121C2×Dic3—91no—
122C2×Dic3—271no—
123C2×C32—22no—
124C2×C32—32no—
125C2×C32—31no—
126C2×C32—62no—
127C2×C32#63812yes—
128C2×C32C22×C3⋊D311yescharacteristic
129C3⋊D3—62no—
130C3×D3—182no—
131C2×C9—32no—
132C2×C9—34no—
133C2×C9—31no—
134C2×C9—32no—
135C2×C9—62no—
136C2×C9—64no—
137D9—62no—
138D9—64no—
139C2×D4—271noSylow
140C22×C3—22no—
141C22×C3—31no—
142C22×C3—32no—
143C22×C3—62no—
144C22×C3—91no—
145C22×C3—92no—
146C22×C3—182no—
147C22×C3—272no—
148C22×C3—542no—
149C22×C3C2×#27211yescharacteristic
150C22×C3C2×#27212yes—
151C2×D3—92no—
152C2×D3—182no—
153C2×D3—272no—
154C2×D3—542no—
155C3×C4—272no—
156Dic3—92no—
157Dic3—272no—
158C32—31no—
159C32C2×#63811yescharacteristic
160C9—31no—
161C9—32no—
162C23—271no—
163C23#418411yescharacteristic, complemented
164D4—274no—
165C2×C4—271no—
166C2×C3—22no—
167C2×C3—32no—
168C2×C3—31no—
169C2×C3—62no—
170C2×C3—92no—
171C2×C3—91no—
172C2×C3—182no—
173C2×C3—542no—
174C2×C3#623512yes—
175C2×C3C22×#27211yescharacteristic
176D3—182no—
177D3—542no—
178C22—22no—
179C22—272no—
180C22—542no—
181C22C2×#418411yescharacteristic
182C22C2×#418412yescomplemented
183C4—272no—
184C3—31no—
185C3—91no—
186C3C2×#623511yescharacteristic
187C2—22no—
188C2—542no—
189C2#1061968112yesdirect product factor
190C2C22×#418411yescharacteristic
subgroupG/H
centerC22C2×#4184
commutator, cosocleC2×#731C23
FrattiniC2×C32C22×C3⋊D3
FittingC23×#731C2
socleC22×C3C2×#272
hypercenterC23#4184
hypocenter#731C2×D4
LCSG ▷ C2×#731 ▷ #731
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×#731C2semidirect
2. C22×#4184C2semidirect
3. #10619681C2direct (N × H)
4. C2×#66331C2semidirect
10. C22×#731C22semidirect
12. C2×#4184C22semidirect
13. #66331C22semidirect
14. C23×3−1+2D3semidirect
16. C23×C3×C9D3semidirect
27. C2×#731D4semidirect
36. C22×3−1+2C2×D3semidirect
42. C22×C3×C9C2×D3semidirect
52. #731C2×D4semidirect
70. C2×3−1+2#67semidirect
76. C2×C3×C9#67semidirect
108. 3−1+2C2×#67semidirect
110. C3×C9C2×#67semidirect
163. C23#4184semidirect
182. C22C2×#4184semidirect
189. C2#10619681direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#7312
C22×#41842
#106196812
C2×#663312
C2×#611⋊C33
C2×#638⋊C33
C2×C3×#6113