Finite Group Databasebeta

← A4×#6909 #72228401

Subgroups — A4×#6909 #72228401

Order 2916.

1270 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 271 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
1C3×A4×3−1+2C312yesmaximal, complemented
2C3×A4×H(3,𝔽3)C311yesmaximal, characteristic, complemented
3C3×A4×H(3,𝔽3)C311yesmaximal, characteristic, complemented
4C22×#6909C311yesmaximal, characteristic, complemented
5#11757851C314yesmaximal, complemented
6#11757841C312yesmaximal, complemented
7#11757836C312yesmaximal, complemented
8#11757801C3118yesmaximal, complemented
9A4×AΣL(1,𝔽27)C319yesmaximal, complemented
10C3×#6909—41nomaximal, Sylow
11C2×#6909—31no—
12C33×A4—31no—
13C33×A4C3211yescharacteristic
14C33×A4C3213yescomplemented
15C22×C3×3−1+2C3212yescomplemented
16C22×C3×H(3,𝔽3)C3211yescharacteristic, complemented
17C22×C3×H(3,𝔽3)C3211yescharacteristic, complemented
18C3×C22⋊H(3,𝔽3)—32no—
19C3×C22⋊H(3,𝔽3)C3216yescomplemented
20C3×C22⋊H(3,𝔽3)C3212yescomplemented
21C3×C22⋊H(3,𝔽3)C3212yescomplemented
22C3×#1094C3214yescomplemented
23A4×3−1+2—32no—
24A4×3−1+2—32no—
25A4×3−1+2—34no—
26A4×3−1+2—34no—
27A4×3−1+2C3216yescomplemented
28A4×3−1+2C32112yescomplemented
29A4×H(3,𝔽3)—32no—
30A4×H(3,𝔽3)—31no—
31A4×H(3,𝔽3)—34no—
32A4×H(3,𝔽3)—32no—
33A4×H(3,𝔽3)—34no—
34A4×H(3,𝔽3)—32no—
35A4×H(3,𝔽3)C3213yes—
36A4×H(3,𝔽3)C3216yescomplemented
37C3×C9⋊A4—34no—
38C3×C9×A4—32no—
39C22×AΣL(1,𝔽27)C3219yescomplemented
40#66368C32118yescomplemented
41#66366C32136yescomplemented
42#66363C32118yescomplemented
43C32×3−1+2—42no—
44C32×H(3,𝔽3)—41no—
45C32×H(3,𝔽3)—41no—
46#6909—44no—
47#6909—42no—
48#6909—42no—
49#6909—418no—
50#6909A411yescharacteristic, direct product factor
51C3×AΣL(1,𝔽27)—49no—
52C2×C3×3−1+2—32no—
53C2×C3×H(3,𝔽3)—31no—
54C2×C3×H(3,𝔽3)—31no—
55C2×AΣL(1,𝔽27)—39no—
56C22×C33—31no—
57C22×C33C3311yescharacteristic
58C22×C33C3313yescomplemented
59C32×A4—32no—
60C32×A4—36no—
61C32×A4—33no—
62C32×A4—33no—
63C32×A4—36no—
64C32×A4—93no—
65C32×A4—99no—
66C32×A4—99no—
67C32×A4H(3,𝔽3)12yes—
68C32×A4H(3,𝔽3)11yescharacteristic
69C32×A4H(3,𝔽3)14yes—
70C32×A4H(3,𝔽3)12yes—
71C32×A4C3311yescharacteristic
72C32×A4C3312yes—
73C32×A4C3316yescomplemented
74C22×3−1+2—32no—
75C22×3−1+2—32no—
76C22×3−1+2C3316yescomplemented
77C22×H(3,𝔽3)—32no—
78C22×H(3,𝔽3)—31no—
79C22×H(3,𝔽3)—32no—
80C22×H(3,𝔽3)C3313yes—
81C22×C3×C9—32no—
82C22⋊H(3,𝔽3)—34no—
83C22⋊H(3,𝔽3)—32no—
84C22⋊H(3,𝔽3)—34no—
85C22⋊H(3,𝔽3)—312no—
86C22⋊H(3,𝔽3)—36no—
87C22⋊H(3,𝔽3)—36no—
88C22⋊H(3,𝔽3)—312no—
89C22⋊H(3,𝔽3)C3316yes—
90#1094—34no—
91#1094—34no—
92#1094C33112yescomplemented
93C3×C22⋊C9—34no—
94C9⋊A4—312no—
95C9⋊A4—312no—
96C9⋊A4—312no—
97C9×A4—312no—
98C9×A4—36no—
99C34—41no—
100C34—43no—
101C34—121no—
102C3×3−1+2—44no—
103C3×3−1+2—46no—
104C3×3−1+2—412no—
105C3×3−1+2—124no—
106C3×3−1+2—122no—
107C3×3−1+2—122no—
108C3×3−1+2—124no—
109C3×3−1+2—124no—
110C3×3−1+2C3×A412yescomplemented
111C3×H(3,𝔽3)—42no—
112C3×H(3,𝔽3)—43no—
113C3×H(3,𝔽3)—46no—
114C3×H(3,𝔽3)—42no—
115C3×H(3,𝔽3)—46no—
116C3×H(3,𝔽3)—122no—
117C3×H(3,𝔽3)—122no—
118C3×H(3,𝔽3)—121no—
119C3×H(3,𝔽3)—124no—
120C3×H(3,𝔽3)—124no—
121C3×H(3,𝔽3)—122no—
122C3×H(3,𝔽3)—122no—
123C3×H(3,𝔽3)C3×A411yescharacteristic, complemented
124C3×H(3,𝔽3)C3×A411yescharacteristic, complemented
125C32×C9—122no—
126AΣL(1,𝔽27)—418no—
127AΣL(1,𝔽27)—436no—
128AΣL(1,𝔽27)—418no—
129AΣL(1,𝔽27)C3×A419yescomplemented
130C2×C33—31no—
131C2×C33—33no—
132C2×C33—91no—
133C2×3−1+2—36no—
134C2×3−1+2—92no—
135C2×3−1+2—92no—
136C2×H(3,𝔽3)—33no—
137C2×H(3,𝔽3)—92no—
138C2×H(3,𝔽3)—92no—
139C2×H(3,𝔽3)—91no—
140C2×C3×C9—92no—
141C22×C32—33no—
142C22×C32—33no—
143C22×C32—91no—
144C22×C32—93no—
145C22×C32—93no—
146C22×C32C3×H(3,𝔽3)12yes—
147C22×C32C3×H(3,𝔽3)11yescharacteristic
148C22×C32C3411yescharacteristic
149C3×A4—33no—
150C3×A4—33no—
151C3×A4—36no—
152C3×A4—36no—
153C3×A4—36no—
154C3×A4—94no—
155C3×A4—92no—
156C3×A4—92no—
157C3×A4—918no—
158C3×A4—918no—
159C3×A4—96no—
160C3×A4—99no—
161C3×A4—99no—
162C3×A4—96no—
163C3×A4C3×H(3,𝔽3)14yes—
164C3×A4C3×H(3,𝔽3)12yes—
165C3×A4C3×H(3,𝔽3)12yes—
166C3×A4C3×H(3,𝔽3)11yescharacteristic
167C22×C9—36no—
168C22⋊C9—312no—
169C33—31no—
170C33—42no—
171C33—41no—
172C33—41no—
173C33—42no—
174C33—44no—
175C33—42no—
176C33—46no—
177C33—123no—
178C33—126no—
179C33—123no—
180C33—126no—
181C33—122no—
182C33—363no—
183C33—369no—
184C33—369no—
185C33C32×A411yescharacteristic
186C33C32×A413yescomplemented
1873−1+2—32no—
1883−1+2—32no—
1893−1+2—412no—
1903−1+2—1212no—
1913−1+2—1212no—
1923−1+2—1212no—
1933−1+2—124no—
1943−1+2—124no—
1953−1+2C32×A416yescomplemented
196H(3,𝔽3)—32no—
197H(3,𝔽3)—31no—
198H(3,𝔽3)—32no—
199H(3,𝔽3)—46no—
200H(3,𝔽3)—1212no—
201H(3,𝔽3)—126no—
202H(3,𝔽3)—1212no—
203H(3,𝔽3)—126no—
204H(3,𝔽3)—124no—
205H(3,𝔽3)—124no—
206H(3,𝔽3)—122no—
207H(3,𝔽3)C32×A413yes—
208C3×C9—32no—
209C3×C9—126no—
210C3×C9—1212no—
211C3×C9—124no—
212C2×C32—32no—
213C2×C32—31no—
214C2×C32—31no—
215C2×C32—93no—
216C2×C32—93no—
217C2×C32—271no—
218C2×C32—273no—
219C2×C32—273no—
220C2×C9—96no—
221C22×C3—31no—
222C22×C3—31no—
223C22×C3—32no—
224C22×C3—93no—
225C22×C3—93no—
226C22×C3C32×H(3,𝔽3)11yescharacteristic
227A4—32no—
228A4—32no—
229A4—34no—
230A4—96no—
231A4—96no—
232A4#690913yesdirect product factor
233C32—33no—
234C32—33no—
235C32—41no—
236C32—44no—
237C32—42no—
238C32—42no—
239C32—91no—
240C32—93no—
241C32—93no—
242C32—123no—
243C32—123no—
244C32—126no—
245C32—126no—
246C32—126no—
247C32—362no—
248C32—362no—
249C32—364no—
250C32—366no—
251C32—366no—
252C32—369no—
253C32—369no—
254C32—3618no—
255C32—3618no—
256C32A4×H(3,𝔽3)12yes—
257C32A4×H(3,𝔽3)11yescharacteristic
258C32C33×A411yescharacteristic
259C9—36no—
260C9—1212no—
261C2×C3—31no—
262C2×C3—92no—
263C2×C3—91no—
264C2×C3—91no—
265C2×C3—273no—
266C2×C3—273no—
267C22C3×#690911yescharacteristic, complemented, Sylow
268C3—31no—
269C3—31no—
270C3—32no—
271C3—43no—
272C3—93no—
273C3—93no—
274C3—122no—
275C3—122no—
276C3—124no—
277C3—366no—
278C3—366no—
279C3C3×A4×H(3,𝔽3)11yescharacteristic
280C2—31no—
subgroupG/H
centerC3C3×A4×H(3,𝔽3)
commutator, cosocleC22×C32C34
FrattiniC32C33×A4
FittingC22×#6909C3
socleC22×C3C32×H(3,𝔽3)
hypercenter#6909A4
hypocenterC22C3×#6909
LCSG ▷ C22×C32 ▷ C22×C3 ▷ C22
UCSC1 ◁ C3 ◁ C33 ◁ #6909

Complements

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

NH ≅ G/Ntype
1. C3×A4×3−1+2C3semidirect
2. C3×A4×H(3,𝔽3)C3semidirect
3. C3×A4×H(3,𝔽3)C3semidirect
4. C22×#6909C3semidirect
5. #11757851C3semidirect
6. #11757841C3semidirect
7. #11757836C3semidirect
8. #11757801C3semidirect
9. A4×AΣL(1,𝔽27)C3semidirect
14. C33×A4C32semidirect
15. C22×C3×3−1+2C32semidirect
16. C22×C3×H(3,𝔽3)C32semidirect
17. C22×C3×H(3,𝔽3)C32semidirect
19. C3×C22⋊H(3,𝔽3)C32semidirect
20. C3×C22⋊H(3,𝔽3)C32semidirect
21. C3×C22⋊H(3,𝔽3)C32semidirect
22. C3×#1094C32semidirect
27. A4×3−1+2C32semidirect
28. A4×3−1+2C32semidirect
36. A4×H(3,𝔽3)C32semidirect
39. C22×AΣL(1,𝔽27)C32semidirect
40. #66368C32semidirect
41. #66366C32semidirect
42. #66363C32semidirect
50. #6909A4direct (N × H)
58. C22×C33C33semidirect
73. C32×A4C33semidirect
76. C22×3−1+2C33semidirect
92. #1094C33semidirect
110. C3×3−1+2C3×A4semidirect
123. C3×H(3,𝔽3)C3×A4semidirect
124. C3×H(3,𝔽3)C3×A4semidirect
129. AΣL(1,𝔽27)C3×A4semidirect
186. C33C32×A4semidirect
195. 3−1+2C32×A4semidirect
232. A4#6909direct (N × H)
267. C22C3×#6909semidirect

Maximal subgroups

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

subgroupindex
C3×A4×3−1+23
C3×A4×H(3,𝔽3)3
C22×#69093
#117578513
#117578413
#117578363
#117578013
A4×AΣL(1,𝔽27)3
C3×#69094