Finite Group Databasebeta

← C3×A4×AΣL(1,𝔽27) #72228395

Subgroups — C3×A4×AΣL(1,𝔽27) #72228395

Order 2916.

2798 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 2798 conjugacy classes of subgroups — too many to draw. Only its 292 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
1C34×A4C311yesmaximal, characteristic, complemented
2C3×A4×3−1+2C312yesmaximal, complemented
3C3×A4×H(3,𝔽3)C311yesmaximal, characteristic, complemented
4C22×C3×AΣL(1,𝔽27)C311yesmaximal, characteristic, complemented
5C3×#66368C312yesmaximal, complemented
6C3×#66366C314yesmaximal, complemented
7C3×#66363C312yesmaximal, complemented
8A4×AΣL(1,𝔽27)C3127yesmaximal, direct product factor
9C32×AΣL(1,𝔽27)—41nomaximal, Sylow
10C2×C3×AΣL(1,𝔽27)—31no—
11C22×C34C3211yescharacteristic, complemented
12C33×A4—32no—
13C33×A4—36no—
14C33×A4—31no—
15C33×A4—327no—
16C33×A4—31no—
17C33×A4C3211yescharacteristic
18C33×A4C3219yescomplemented
19C33×A4C3212yescomplemented
20C22×C3×3−1+2C3212yescomplemented
21C22×C3×H(3,𝔽3)C3211yescharacteristic, complemented
22C3×C22⋊H(3,𝔽3)—32no—
23C3×C22⋊H(3,𝔽3)C3212yescomplemented
24C3×#1094C3214yescomplemented
25A4×3−1+2—312no—
26A4×3−1+2C32118yescomplemented
27A4×H(3,𝔽3)—36no—
28A4×H(3,𝔽3)C3219yescomplemented
29C3×C9⋊A4—34no—
30C3×C9×A4—32no—
31C22×AΣL(1,𝔽27)C3219yescomplemented
32#66368C32118yescomplemented
33#66366C32136yescomplemented
34#66363C32118yescomplemented
35C35—41no—
36C32×3−1+2—42no—
37C32×H(3,𝔽3)—41no—
38C3×AΣL(1,𝔽27)—42no—
39C3×AΣL(1,𝔽27)—44no—
40C3×AΣL(1,𝔽27)—42no—
41C3×AΣL(1,𝔽27)—427no—
42C3×AΣL(1,𝔽27)A411yescharacteristic, direct product factor
43C2×C34—31no—
44C2×C3×3−1+2—32no—
45C2×C3×H(3,𝔽3)—31no—
46C2×AΣL(1,𝔽27)—39no—
47C22×C33—39no—
48C22×C33—32no—
49C22×C33—31no—
50C22×C33—31no—
51C22×C33C3311yescharacteristic
52C22×C33C3313yescomplemented
53C32×A4—354no—
54C32×A4—3162no—
55C32×A4—34no—
56C32×A4—32no—
57C32×A4—327no—
58C32×A4—39no—
59C32×A4—318no—
60C32×A4—39no—
61C32×A4—318no—
62C32×A4—381no—
63C32×A4—32no—
64C32×A4—39no—
65C32×A4—99no—
66C32×A4H(3,𝔽3)12yes—
67C32×A4H(3,𝔽3)16yes—
68C32×A4H(3,𝔽3)11yescharacteristic
69C32×A4C3313yes—
70C32×A4C3316yescomplemented
71C22×3−1+2—34no—
72C22×3−1+2C3316yescomplemented
73C22×H(3,𝔽3)—32no—
74C22×H(3,𝔽3)C3313yescomplemented
75C22×C3×C9—32no—
76C22⋊H(3,𝔽3)—34no—
77C22⋊H(3,𝔽3)—318no—
78C22⋊H(3,𝔽3)C3316yescomplemented
79#1094—38no—
80#1094C33112yescomplemented
81C3×C22⋊C9—34no—
82C9⋊A4—336no—
83C9×A4—318no—
84C34—41no—
85C34—42no—
86C34—49no—
87C34—126no—
88C34—121no—
89C34—122no—
90C34—121no—
91C34—1227no—
92C34C3×A411yescharacteristic, complemented
93C3×3−1+2—44no—
94C3×3−1+2—418no—
95C3×3−1+2—124no—
96C3×3−1+2—1212no—
97C3×3−1+2C3×A412yescomplemented
98C3×H(3,𝔽3)—42no—
99C3×H(3,𝔽3)—49no—
100C3×H(3,𝔽3)—126no—
101C3×H(3,𝔽3)—122no—
102C3×H(3,𝔽3)C3×A411yescharacteristic, complemented
103C32×C9—122no—
104AΣL(1,𝔽27)—418no—
105AΣL(1,𝔽27)—436no—
106AΣL(1,𝔽27)—418no—
107AΣL(1,𝔽27)C3×A419yesdirect product factor
108C2×C33—31no—
109C2×C33—33no—
110C2×C33—91no—
111C2×C33—92no—
112C2×C33—91no—
113C2×C33—99no—
114C2×3−1+2—36no—
115C2×3−1+2—94no—
116C2×H(3,𝔽3)—33no—
117C2×H(3,𝔽3)—92no—
118C2×C3×C9—92no—
119C22×C32—318no—
120C22×C32—33no—
121C22×C32—39no—
122C22×C32—33no—
123C22×C32—39no—
124C22×C32—33no—
125C22×C32—93no—
126C22×C32C3×H(3,𝔽3)12yes—
127C22×C32C3×H(3,𝔽3)11yescharacteristic
128C22×C32C3411yescharacteristic
129C3×A4—336no—
130C3×A4—3162no—
131C3×A4—36no—
132C3×A4—318no—
133C3×A4—36no—
134C3×A4—381no—
135C3×A4—318no—
136C3×A4—39no—
137C3×A4—318no—
138C3×A4—36no—
139C3×A4—96no—
140C3×A4—927no—
141C3×A4AΣL(1,𝔽27)19yesdirect product factor
142C3×A4C3×H(3,𝔽3)16yes—
143C3×A4C3×H(3,𝔽3)13yes—
144C22×C9—36no—
145C22⋊C9—312no—
146C33—39no—
147C33—32no—
148C33—31no—
149C33—31no—
150C33—41no—
151C33—46no—
152C33—42no—
153C33—43no—
154C33—46no—
155C33—129no—
156C33—1218no—
157C33—129no—
158C33—1227no—
159C33—129no—
160C33—1281no—
161C33—1218no—
162C33—12162no—
163C33—1254no—
164C33—124no—
165C33—122no—
166C33—122no—
167C33—369no—
168C33C32×A411yescharacteristic
169C33C32×A413yescomplemented
1703−1+2—34no—
1713−1+2—412no—
1723−1+2—1236no—
1733−1+2—128no—
1743−1+2C32×A416yescomplemented
175H(3,𝔽3)—32no—
176H(3,𝔽3)—46no—
177H(3,𝔽3)—1218no—
178H(3,𝔽3)—124no—
179H(3,𝔽3)C32×A413yescomplemented
180C3×C9—32no—
181C3×C9—1218no—
182C3×C9—124no—
183C2×C32—31no—
184C2×C32—32no—
185C2×C32—31no—
186C2×C32—93no—
187C2×C32—93no—
188C2×C32—99no—
189C2×C32—93no—
190C2×C32—918no—
191C2×C32—99no—
192C2×C32—273no—
193C2×C9—96no—
194C22×C3—39no—
195C22×C3—32no—
196C22×C3—31no—
197C22×C3—93no—
198C22×C3C3×AΣL(1,𝔽27)13yescomplemented
199C22×C3C32×H(3,𝔽3)11yescharacteristic
200A4—318no—
201A4—36no—
202A4—96no—
203A4C3×AΣL(1,𝔽27)19yesdirect product factor
204C32—39no—
205C32—318no—
206C32—33no—
207C32—39no—
208C32—33no—
209C32—33no—
210C32—43no—
211C32—49no—
212C32—46no—
213C32—93no—
214C32—1218no—
215C32—129no—
216C32—1281no—
217C32—1218no—
218C32—12162no—
219C32—1236no—
220C32—126no—
221C32—126no—
222C32—1218no—
223C32—126no—
224C32—3627no—
225C32—366no—
226C32A4×H(3,𝔽3)12yes—
227C32A4×H(3,𝔽3)11yescharacteristic
228C32C33×A411yescharacteristic
229C9—36no—
230C9—1212no—
231C2×C3—31no—
232C2×C3—33no—
233C2×C3—92no—
234C2×C3—91no—
235C2×C3—99no—
236C2×C3—273no—
237C22C32×AΣL(1,𝔽27)11yescharacteristic, complemented, Sylow
238C3—39no—
239C3—32no—
240C3—31no—
241C3—49no—
242C3—93no—
243C3—126no—
244C3—1218no—
245C3—366no—
246C3A4×AΣL(1,𝔽27)13yesdirect product factor
247C3C3×A4×H(3,𝔽3)11yescharacteristic
248C2—31no—
subgroupG/H
centerC32A4×H(3,𝔽3)
commutator, cosocleC22×C32C34
FrattiniC32C33×A4
FittingC22×C3×AΣL(1,𝔽27)C3
socleC22×C32C3×H(3,𝔽3)
hypercenterC3×AΣL(1,𝔽27)A4
hypocenterC22C32×AΣL(1,𝔽27)
LCSG ▷ C22×C32 ▷ C22×C3 ▷ C22
UCSC1 ◁ C32 ◁ C33 ◁ C3×AΣL(1,𝔽27)

Complements

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

NH ≅ G/Ntype
1. C34×A4C3semidirect
2. C3×A4×3−1+2C3semidirect
3. C3×A4×H(3,𝔽3)C3semidirect
4. C22×C3×AΣL(1,𝔽27)C3semidirect
5. C3×#66368C3semidirect
6. C3×#66366C3semidirect
7. C3×#66363C3semidirect
8. A4×AΣL(1,𝔽27)C3direct (N × H)
11. C22×C34C32semidirect
18. C33×A4C32semidirect
19. C33×A4C32semidirect
20. C22×C3×3−1+2C32semidirect
21. C22×C3×H(3,𝔽3)C32semidirect
23. C3×C22⋊H(3,𝔽3)C32semidirect
24. C3×#1094C32semidirect
26. A4×3−1+2C32semidirect
28. A4×H(3,𝔽3)C32semidirect
31. C22×AΣL(1,𝔽27)C32semidirect
32. #66368C32semidirect
33. #66366C32semidirect
34. #66363C32semidirect
42. C3×AΣL(1,𝔽27)A4direct (N × H)
52. C22×C33C33semidirect
70. C32×A4C33semidirect
72. C22×3−1+2C33semidirect
74. C22×H(3,𝔽3)C33semidirect
78. C22⋊H(3,𝔽3)C33semidirect
80. #1094C33semidirect
92. C34C3×A4semidirect
97. C3×3−1+2C3×A4semidirect
102. C3×H(3,𝔽3)C3×A4semidirect
107. AΣL(1,𝔽27)C3×A4direct (N × H)
141. C3×A4AΣL(1,𝔽27)direct (N × H)
169. C33C32×A4semidirect
174. 3−1+2C32×A4semidirect
179. H(3,𝔽3)C32×A4semidirect
198. C22×C3C3×AΣL(1,𝔽27)semidirect
203. A4C3×AΣL(1,𝔽27)direct (N × H)
237. C22C32×AΣL(1,𝔽27)semidirect
246. C3A4×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
C34×A43
C3×A4×3−1+23
C3×A4×H(3,𝔽3)3
C22×C3×AΣL(1,𝔽27)3
C3×#663683
C3×#663663
C3×#663633
A4×AΣL(1,𝔽27)3
C32×AΣL(1,𝔽27)4