Finite Group Databasebeta

← C2×A4×AΣL(1,𝔽27) #14518386

Subgroups — C2×A4×AΣL(1,𝔽27) #14518386

Order 1944.

584 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 88 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
1A4×AΣL(1,𝔽27)C211yesmaximal, characteristic, direct product factor
2C2×C33×A4C311yesmaximal, characteristic, complemented
3C2×A4×3−1+2C312yesmaximal, complemented
4C2×A4×H(3,𝔽3)C311yesmaximal, characteristic, complemented
5C23×AΣL(1,𝔽27)C311yesmaximal, characteristic, complemented
6C2×#66368C312yesmaximal, complemented
7C2×#66366C314yesmaximal, complemented
8C2×#66363C312yesmaximal, complemented
9C2×C3×AΣL(1,𝔽27)—41nomaximal
10C33×A4C2×C311yescharacteristic, complemented
11A4×3−1+2C2×C312yescomplemented
12A4×H(3,𝔽3)C2×C311yescharacteristic, complemented
13C22×AΣL(1,𝔽27)—31no—
14C22×AΣL(1,𝔽27)—31no—
15C22×AΣL(1,𝔽27)C2×C311yescharacteristic, complemented
16#66368C2×C312yescomplemented
17#66366C2×C314yescomplemented
18#66363C2×C312yescomplemented
19C3×AΣL(1,𝔽27)—41noSylow
20C23×C33C3211yescharacteristic, complemented
21C2×C32×A4—32no—
22C2×C32×A4—31no—
23C2×C32×A4—39no—
24C2×C32×A4—31no—
25C2×C32×A4C3211yescharacteristic
26C2×C32×A4C3212yescomplemented
27C23×3−1+2C3212yescomplemented
28C23×H(3,𝔽3)C3211yescharacteristic, complemented
29C2×C22⋊H(3,𝔽3)—32no—
30C2×C22⋊H(3,𝔽3)C3212yescomplemented
31C2×#1094C3214yescomplemented
32C2×C9⋊A4—34no—
33C2×C9×A4—32no—
34C2×C34—41no—
35C2×C3×3−1+2—42no—
36C2×C3×H(3,𝔽3)—41no—
37C2×AΣL(1,𝔽27)—31no—
38C2×AΣL(1,𝔽27)—31no—
39C2×AΣL(1,𝔽27)—42no—
40C2×AΣL(1,𝔽27)—44no—
41C2×AΣL(1,𝔽27)—42no—
42C2×AΣL(1,𝔽27)A411yescharacteristic, direct product factor
43C22×C33—31no—
44C22×C33—31no—
45C22×C33C2×C3211yescharacteristic, complemented
46C32×A4—32no—
47C32×A4—31no—
48C32×A4—39no—
49C32×A4—31no—
50C32×A4C2×C3211yescharacteristic
51C32×A4C2×C3212yescomplemented
52C22×3−1+2—32no—
53C22×3−1+2—32no—
54C22×3−1+2C2×C3212yescomplemented
55C22×H(3,𝔽3)—31no—
56C22×H(3,𝔽3)—31no—
57C22×H(3,𝔽3)C2×C3211yescharacteristic, complemented
58C22⋊H(3,𝔽3)—32no—
59C22⋊H(3,𝔽3)C2×C3212yescomplemented
60#1094C2×C3214yescomplemented
61C9⋊A4—34no—
62C9×A4—32no—
63C34—41no—
64C3×3−1+2—42no—
65C3×H(3,𝔽3)—41no—
66AΣL(1,𝔽27)—42no—
67AΣL(1,𝔽27)—44no—
68AΣL(1,𝔽27)—42no—
69AΣL(1,𝔽27)C2×A411yescharacteristic, direct product factor
70C23×C32—31no—
71C23×C32—33no—
72C23×C32—31no—
73C23×C32C3311yescharacteristic
74C2×C3×A4—318no—
75C2×C3×A4—32no—
76C2×C3×A4—39no—
77C2×C3×A4—33no—
78C2×C3×A4—36no—
79C2×C3×A4—32no—
80C2×C3×A4—93no—
81C2×C3×A4H(3,𝔽3)12yes—
82C2×C3×A4H(3,𝔽3)11yescharacteristic
83C23×C9—32no—
84C2×C22⋊C9—34no—
85C2×C33—31no—
86C2×C33—31no—
87C2×C33—41no—
88C2×C33—42no—
89C2×C33—121no—
90C2×C33—122no—
91C2×C33—121no—
92C2×C33—129no—
93C2×C33C3×A411yescharacteristic, complemented
94C2×3−1+2—32no—
95C2×3−1+2—32no—
96C2×3−1+2—44no—
97C2×3−1+2—124no—
98C2×3−1+2C3×A412yescomplemented
99C2×H(3,𝔽3)—31no—
100C2×H(3,𝔽3)—31no—
101C2×H(3,𝔽3)—42no—
102C2×H(3,𝔽3)—122no—
103C2×H(3,𝔽3)C3×A411yescharacteristic, complemented
104C2×C3×C9—122no—
105C22×C32—31no—
106C22×C32—33no—
107C22×C32—31no—
108C22×C32—31no—
109C22×C32—31no—
110C22×C32—91no—
111C22×C32—91no—
112C22×C32—91no—
113C22×C32—91no—
114C22×C32—93no—
115C22×C32—93no—
116C22×C32C2×C3311yescharacteristic
117C3×A4—318no—
118C3×A4—32no—
119C3×A4—33no—
120C3×A4—39no—
121C3×A4—36no—
122C3×A4—32no—
123C3×A4—93no—
124C3×A4C2×H(3,𝔽3)12yes—
125C3×A4C2×H(3,𝔽3)11yescharacteristic
126C22×C9—32no—
127C22×C9—92no—
128C22×C9—92no—
129C22⋊C9—34no—
130C33—41no—
131C33—42no—
132C33—121no—
133C33—122no—
134C33—121no—
135C33—129no—
136C33C2×C3×A411yescharacteristic, complemented
1373−1+2—44no—
1383−1+2—124no—
1393−1+2C2×C3×A412yescomplemented
140H(3,𝔽3)—42no—
141H(3,𝔽3)—122no—
142H(3,𝔽3)C2×C3×A411yescharacteristic, complemented
143C3×C9—122no—
144C23×C3—33no—
145C23×C3—31no—
146C23×C3—91no—
147C23×C3C3×H(3,𝔽3)11yescharacteristic
148C2×A4—36no—
149C2×A4—32no—
150C2×A4—92no—
151C2×A4AΣL(1,𝔽27)13yesdirect product factor
152C2×C32—31no—
153C2×C32—33no—
154C2×C32—31no—
155C2×C32—31no—
156C2×C32—31no—
157C2×C32—41no—
158C2×C32—42no—
159C2×C32—91no—
160C2×C32—91no—
161C2×C32—91no—
162C2×C32—91no—
163C2×C32—93no—
164C2×C32—93no—
165C2×C32—123no—
166C2×C32—129no—
167C2×C32—1218no—
168C2×C32—126no—
169C2×C32—122no—
170C2×C32—122no—
171C2×C32—363no—
172C2×C32C32×A411yescharacteristic
173C2×C9—32no—
174C2×C9—92no—
175C2×C9—92no—
176C2×C9—124no—
177C22×C3—31no—
178C22×C3—31no—
179C22×C3—33no—
180C22×C3—31no—
181C22×C3—91no—
182C22×C3—93no—
183C22×C3—91no—
184C22×C3—93no—
185C22×C3—91no—
186C22×C3—271no—
187C22×C3—271no—
188C22×C3C2×C3×H(3,𝔽3)11yescharacteristic
189A4—36no—
190A4—32no—
191A4—92no—
192A4C2×AΣL(1,𝔽27)13yesdirect product factor
193C32—33no—
194C32—31no—
195C32—31no—
196C32—41no—
197C32—42no—
198C32—123no—
199C32—129no—
200C32—126no—
201C32—1218no—
202C32—122no—
203C32—122no—
204C32—363no—
205C32C2×C32×A411yescharacteristic
206C9—32no—
207C9—124no—
208C23C3×AΣL(1,𝔽27)11yescharacteristic, complemented, Sylow
209C2×C3—31no—
210C2×C3—31no—
211C2×C3—33no—
212C2×C3—31no—
213C2×C3—43no—
214C2×C3—91no—
215C2×C3—93no—
216C2×C3—91no—
217C2×C3—93no—
218C2×C3—91no—
219C2×C3—122no—
220C2×C3—126no—
221C2×C3—271no—
222C2×C3—271no—
223C2×C3—362no—
224C2×C3A4×H(3,𝔽3)11yescharacteristic
225C22—31no—
226C22—31no—
227C22C2×C3×AΣL(1,𝔽27)11yescharacteristic, complemented
228C3—33no—
229C3—31no—
230C3—43no—
231C3—91no—
232C3—122no—
233C3—126no—
234C3—362no—
235C3C2×A4×H(3,𝔽3)11yescharacteristic
236C2—31no—
237C2—31no—
238C2A4×AΣL(1,𝔽27)11yescharacteristic, direct product factor
subgroupG/H
centerC2×C3A4×H(3,𝔽3)
commutator, cosocleC22×C32C2×C33
FrattiniC32C2×C32×A4
FittingC23×AΣL(1,𝔽27)C3
socleC23×C3C3×H(3,𝔽3)
hypercenterC2×AΣL(1,𝔽27)A4
hypocenterC22C2×C3×AΣL(1,𝔽27)
LCSG ▷ C22×C32 ▷ C22×C3 ▷ C22
UCSC1 ◁ C2×C3 ◁ C2×C32 ◁ C2×AΣL(1,𝔽27)

Complements

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

NH ≅ G/Ntype
1. A4×AΣL(1,𝔽27)C2direct (N × H)
2. C2×C33×A4C3semidirect
3. C2×A4×3−1+2C3semidirect
4. C2×A4×H(3,𝔽3)C3semidirect
5. C23×AΣL(1,𝔽27)C3semidirect
6. C2×#66368C3semidirect
7. C2×#66366C3semidirect
8. C2×#66363C3semidirect
10. C33×A4C2×C3semidirect
11. A4×3−1+2C2×C3semidirect
12. A4×H(3,𝔽3)C2×C3semidirect
15. C22×AΣL(1,𝔽27)C2×C3semidirect
16. #66368C2×C3semidirect
17. #66366C2×C3semidirect
18. #66363C2×C3semidirect
20. C23×C33C32semidirect
26. C2×C32×A4C32semidirect
27. C23×3−1+2C32semidirect
28. C23×H(3,𝔽3)C32semidirect
30. C2×C22⋊H(3,𝔽3)C32semidirect
31. C2×#1094C32semidirect
42. C2×AΣL(1,𝔽27)A4direct (N × H)
45. C22×C33C2×C32semidirect
51. C32×A4C2×C32semidirect
54. C22×3−1+2C2×C32semidirect
57. C22×H(3,𝔽3)C2×C32semidirect
59. C22⋊H(3,𝔽3)C2×C32semidirect
60. #1094C2×C32semidirect
69. AΣL(1,𝔽27)C2×A4direct (N × H)
93. C2×C33C3×A4semidirect
98. C2×3−1+2C3×A4semidirect
103. C2×H(3,𝔽3)C3×A4semidirect
136. C33C2×C3×A4semidirect
139. 3−1+2C2×C3×A4semidirect
142. H(3,𝔽3)C2×C3×A4semidirect
151. C2×A4AΣL(1,𝔽27)direct (N × H)
192. A4C2×AΣL(1,𝔽27)direct (N × H)
208. C23C3×AΣL(1,𝔽27)semidirect
227. C22C2×C3×AΣL(1,𝔽27)semidirect
238. C2A4×AΣL(1,𝔽27)direct (N × H)

Maximal subgroups

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

subgroupindex
A4×AΣL(1,𝔽27)2
C2×C33×A43
C2×A4×3−1+23
C2×A4×H(3,𝔽3)3
C23×AΣL(1,𝔽27)3
C2×#663683
C2×#663663
C2×#663633
C2×C3×AΣL(1,𝔽27)4