Finite Group Databasebeta

← C2×C4×#4183 #13047426

Subgroups — C2×C4×#4183 #13047426

Order 1296.

540 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C22×#4183C211yesmaximal, characteristic
2C2×C4×AΣL(1,𝔽27)C211yesmaximal, characteristic, complemented
3C2×#66330C211yesmaximal, characteristic, complemented
4C4×#4183C214yesmaximal, direct product factor
5C2×C3×C4×C3⋊D3—31nomaximal
6C2×C4×AGL(1,ℤ/9)—32nomaximal
7C2×C4×(C3⋊D3)⋊C3—31nomaximal
8C22×AΣL(1,𝔽27)C2211yescharacteristic
9C2×#4183C414yesdirect product factor
10C2×#4183C2212yes—
11C4×AΣL(1,𝔽27)C2212yescomplemented
12#66330C2212yescomplemented
13C22×C3×C3⋊D3—31no—
14C2×C33×C4D311yescharacteristic, complemented
15C2×C3×C3⋊Dic3—31no—
16C3×C4×C3⋊D3—34no—
17C22×AGL(1,ℤ/9)—32no—
18C22×(C3⋊D3)⋊C3—31no—
19C2×C4×3−1+2D312yescomplemented
20C2×C4×H(3,𝔽3)D311yescharacteristic, complemented
21C2×3−1+2⋊C4—32no—
22C2×(C3⋊Dic3)⋊C3—31no—
23C4×AGL(1,ℤ/9)—38no—
24C4×(C3⋊D3)⋊C3—34no—
25C2×AΣL(1,𝔽27)C2×C412yescomplemented
26C2×AΣL(1,𝔽27)C2311yescharacteristic
27#4183C2×C414yesdirect product factor
28C2×C4×C3⋊D3—31no—
29C2×C4×C3⋊D3—31no—
30C2×C3×C4×D3—93no—
31C2×C3×C4×D3—91no—
32C2×C4×D9—32no—
33C22×C33C2×D311yescharacteristic
34C2×C3×C3⋊D3—34no—
35C2×C3×C3⋊D3—32no—
36C33×C4C2×D312yescomplemented
37C3×C3⋊Dic3—32no—
38C22×3−1+2C2×D312yes—
39C22×H(3,𝔽3)C2×D311yescharacteristic
40C2×AGL(1,ℤ/9)—38no—
41C2×AGL(1,ℤ/9)—34no—
42C2×(C3⋊D3)⋊C3—34no—
43C2×(C3⋊D3)⋊C3—32no—
44C4×3−1+2C2×D314yescomplemented
45C4×H(3,𝔽3)C2×D312yescomplemented
463−1+2⋊C4—34no—
47(C3⋊Dic3)⋊C3—32no—
48AΣL(1,𝔽27)C22×C411yescharacteristic, complemented, Sylow
49C22×C3⋊D3—31no—
50C22×C3⋊D3—31no—
51C22×C3×D3—93no—
52C22×C3×D3—91no—
53C2×C32×C4—31no—
54C2×C32×C4—33no—
55C2×C32×C4—31no—
56C2×C32×C4C3⋊D311yescharacteristic
57C2×C3⋊Dic3—31no—
58C2×C3⋊Dic3—31no—
59C4×C3⋊D3—34no—
60C4×C3⋊D3—34no—
61C2×C3×Dic3—93no—
62C2×C3×Dic3—91no—
63C3×C4×D3—912no—
64C3×C4×D3—94no—
65C22×D9—32no—
66C2×C4×C9—32no—
67C2×Dic9—32no—
68C4×D9—38no—
69C2×C33C4×D312yescomplemented
70C2×C33C22×D311yescharacteristic
71C3×C3⋊D3—34no—
72C2×3−1+2C4×D314yescomplemented
73C2×3−1+2C22×D312yes—
74C2×H(3,𝔽3)C4×D312yescomplemented
75C2×H(3,𝔽3)C22×D311yescharacteristic
76AGL(1,ℤ/9)—38no—
77(C3⋊D3)⋊C3—34no—
78C22×C3×C4—271no—
79C2×C4×D3—93no—
80C2×C4×D3—91no—
81C2×C4×D3—271no—
82C22×C32—31no—
83C22×C32—33no—
84C22×C32—31no—
85C22×C32C2×C3⋊D311yescharacteristic
86C2×C3⋊D3—32no—
87C2×C3⋊D3—34no—
88C2×C3⋊D3—32no—
89C2×C3⋊D3—34no—
90C2×C3×D3—96no—
91C2×C3×D3—912no—
92C2×C3×D3—94no—
93C2×C3×D3—92no—
94C32×C4—32no—
95C32×C4—36no—
96C32×C4—32no—
97C32×C4C2×C3⋊D312yes—
98C3⋊Dic3—32no—
99C3⋊Dic3—32no—
100C3×Dic3—96no—
101C3×Dic3—92no—
102C22×C9—32no—
103C2×D9—34no—
104C2×D9—38no—
105C4×C9—34no—
106Dic9—34no—
107C33C2×C4×D311yescharacteristic, complemented
1083−1+2C2×C4×D312yescomplemented
109H(3,𝔽3)C2×C4×D311yescharacteristic, complemented
110C23×C3—271no—
111C22×D3—93no—
112C22×D3—91no—
113C22×D3—271no—
114C2×C3×C4—31no—
115C2×C3×C4—33no—
116C2×C3×C4—91no—
117C2×C3×C4—274no—
118C2×C3×C4—271no—
119C2×C3×C4#27211yescharacteristic
120C2×Dic3—93no—
121C2×Dic3—91no—
122C2×Dic3—271no—
123C4×D3—912no—
124C4×D3—94no—
125C4×D3—274no—
126C2×C32—32no—
127C2×C32—31no—
128C2×C32—36no—
129C2×C32—33no—
130C2×C32—32no—
131C2×C32—31no—
132C2×C32C4×C3⋊D312yes—
133C2×C32C22×C3⋊D311yescharacteristic
134C3⋊D3—34no—
135C3⋊D3—34no—
136C3×D3—912no—
137C3×D3—94no—
138C2×C9—34no—
139C2×C9—32no—
140D9—38no—
141C22×C4—271noSylow
142C22×C3—33no—
143C22×C3—31no—
144C22×C3—91no—
145C22×C3—272no—
146C22×C3—274no—
147C22×C3C2×#27211yescharacteristic
148C2×D3—96no—
149C2×D3—912no—
150C2×D3—94no—
151C2×D3—92no—
152C2×D3—272no—
153C2×D3—274no—
154C3×C4—36no—
155C3×C4—32no—
156C3×C4—92no—
157C3×C4—272no—
158C3×C4C2×#27212yes—
159Dic3—96no—
160Dic3—92no—
161Dic3—272no—
162C32—33no—
163C32—31no—
164C32—31no—
165C32C2×C4×C3⋊D311yescharacteristic
166C9—32no—
167C23—271no—
168C2×C4—271no—
169C2×C4—274no—
170C2×C4#418311yescharacteristic, direct product factor
171C2×C3—36no—
172C2×C3—32no—
173C2×C3—33no—
174C2×C3—31no—
175C2×C3—92no—
176C2×C3—91no—
177C2×C3—274no—
178C2×C3C4×#27212yes—
179C2×C3C22×#27211yescharacteristic
180D3—912no—
181D3—94no—
182D3—274no—
183C22—274no—
184C22—272no—
185C22C2×#418311yescharacteristic
186C4—272no—
187C4C2×#418312yesdirect product factor
188C3—33no—
189C3—31no—
190C3—91no—
191C3C2×C4×#27211yescharacteristic
192C2—274no—
193C2C4×#418312yesdirect product factor
194C2C22×#418311yescharacteristic
subgroupG/H
center, hypercenterC2×C4#4183
commutator, hypocenterAΣL(1,𝔽27)C22×C4
FrattiniC2×C32C22×C3⋊D3
FittingC2×C4×AΣL(1,𝔽27)C2
socleC22×C3C2×#272
prime radicalC22×#4183C2
cosocleC2×AΣL(1,𝔽27)C23
LCSG ▷ AΣL(1,𝔽27)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×AΣL(1,𝔽27)C2semidirect
3. C2×#66330C2semidirect
4. C4×#4183C2direct (N × H)
9. C2×#4183C4direct (N × H)
11. C4×AΣL(1,𝔽27)C22semidirect
12. #66330C22semidirect
14. C2×C33×C4D3semidirect
19. C2×C4×3−1+2D3semidirect
20. C2×C4×H(3,𝔽3)D3semidirect
25. C2×AΣL(1,𝔽27)C2×C4semidirect
27. #4183C2×C4direct (N × H)
36. C33×C4C2×D3semidirect
44. C4×3−1+2C2×D3semidirect
45. C4×H(3,𝔽3)C2×D3semidirect
48. AΣL(1,𝔽27)C22×C4semidirect
69. C2×C33C4×D3semidirect
72. C2×3−1+2C4×D3semidirect
74. C2×H(3,𝔽3)C4×D3semidirect
107. C33C2×C4×D3semidirect
108. 3−1+2C2×C4×D3semidirect
109. H(3,𝔽3)C2×C4×D3semidirect
170. C2×C4#4183direct (N × H)
187. C4C2×#4183direct (N × H)
193. C2C4×#4183direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#41832
C2×C4×AΣL(1,𝔽27)2
C2×#663302
C4×#41832
C2×C3×C4×C3⋊D33
C2×C4×AGL(1,ℤ/9)3
C2×C4×(C3⋊D3)⋊C33