Finite Group Databasebeta

← #126×AGL(1,ℤ/9) #13153246

Subgroups — #126×AGL(1,ℤ/9) #13153246

Order 1728.

768 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 222 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
13−1+2×#126C211yesmaximal, characteristic, complemented
2#63460⋊C3C213yesmaximal, complemented
3C4⋊C4×AGL(1,ℤ/9)C213yesmaximal, complemented
4C22⋊C4×AGL(1,ℤ/9)C213yesmaximal
5#63441⋊C3C213yesmaximal, complemented
6#11721396C211yesmaximal, characteristic, complemented
7C42×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
8C3×D3×#126—31nomaximal
9D9×#126C311yesmaximal, characteristic, complemented
10C23×AGL(1,ℤ/9)C2211yescharacteristic
11C2×C4×AGL(1,ℤ/9)C2213yes—
12C2×C4×AGL(1,ℤ/9)C2213yes—
13C4⋊C4×3−1+2C2213yescomplemented
14C22⋊C4×3−1+2C2213yes—
15C42×3−1+2C2211yescharacteristic, complemented
16#3676⋊C3C2213yes—
17(C2×AGL(1,ℤ/9))⋊C4C2213yes—
18(C2×AGL(1,ℤ/9))⋊C4C2213yes—
19C4⋊(3−1+2⋊C4)C2213yescomplemented
20(3−1+2⋊C4)⋊C4C2213yescomplemented
21(3−1+2⋊C4)⋊C4C2213yescomplemented
22C4×3−1+2⋊C4C2213yescomplemented
23C32×#126D311yescharacteristic
24C3×#898—33no—
25C3×D3×C4⋊C4—33no—
26C3×D3×C22⋊C4—33no—
27C3×#879—33no—
28C3×#876—31no—
29C3×C42×D3—31no—
30C9×#126—21no—
31C9×#126C2×C311yescharacteristic, complemented
32#63460C2×C313yescomplemented
33C4⋊C4×D9C2×C313yescomplemented
34C22⋊C4×D9C2×C313yes—
35#63441C2×C313yescomplemented
36#63438C2×C311yescharacteristic, complemented
37C42×D9C2×C311yescharacteristic, complemented
38C23×3−1+2C2311yescharacteristic
39C22×AGL(1,ℤ/9)—26no—
40C22×AGL(1,ℤ/9)C2311yescharacteristic
41C22×AGL(1,ℤ/9)C2311yescharacteristic
42C2×C4×3−1+2C2313yes—
43C2×C4×3−1+2C2313yes—
44C2×3−1+2⋊C4C2313yes—
45C2×3−1+2⋊C4C2313yes—
46C4×AGL(1,ℤ/9)—26no—
47C4×AGL(1,ℤ/9)—26no—
48C2×C3×#126—91no—
49D3×#126—31no—
50C23×C3×D3—31no—
51C2×C3×C4×D3—33no—
52C2×C3×C4×D3—33no—
53C23×D9C22×C311yescharacteristic
54C32×C4⋊C4C2×D313yes—
55C32×C22⋊C4C2×D313yes—
56C32×C42C2×D311yescharacteristic
57C3×#217—33no—
58C3×(C2×D3)⋊C4—33no—
59C3×(C2×D3)⋊C4—33no—
60C3×C4⋊Dic3—33no—
61C3×Dic3⋊C4—33no—
62C3×Dic3⋊C4—33no—
63C3×C4×Dic3—33no—
64C2×C4×D9C22×C313yes—
65C2×C4×D9C22×C313yes—
66C9×C4⋊C4—23no—
67C9×C4⋊C4C22×C313yescomplemented
68C9×C22⋊C4—23no—
69C9×C22⋊C4C22×C313yes—
70C42×C9—21no—
71C42×C9C22×C311yescharacteristic, complemented
72#3676C22×C313yes—
73(C2×D9)⋊C4C22×C313yes—
74(C2×D9)⋊C4C22×C313yes—
75C4⋊Dic9C22×C313yescomplemented
76Dic9⋊C4C22×C313yescomplemented
77Dic9⋊C4C22×C313yescomplemented
78C4×Dic9C22×C313yescomplemented
79C22×3−1+2—23no—
80C22×3−1+2C2411yescharacteristic
81C2×AGL(1,ℤ/9)—23no—
82C2×AGL(1,ℤ/9)—44no—
83C2×AGL(1,ℤ/9)#4116yes—
84C4×3−1+2—23no—
85C4×3−1+2—23no—
863−1+2⋊C4—23no—
873−1+2⋊C4—23no—
88C3×#126—31no—
89C3×#126—91no—
90C3×#126—93no—
91C3×#126—93no—
92C3×#126C3×D311yescharacteristic
93C2×C3×C4⋊C4—93no—
94C2×C3×C22⋊C4—93no—
95C2×C3×C42—91no—
96#898—33no—
97D3×C4⋊C4—33no—
98D3×C22⋊C4—33no—
99#879—33no—
100#876—31no—
101C42×D3—31no—
102C23×C32C22×D311yescharacteristic
103C22×C3×D3—31no—
104C22×C3×D3—31no—
105C22×C3×D3—66no—
106C2×C32×C4C22×D313yes—
107C2×C32×C4C22×D313yes—
108C2×C3×Dic3—33no—
109C2×C3×Dic3—33no—
110C3×C4×D3—66no—
111C3×C4×D3—66no—
112C23×C9—21no—
113C23×C9C23×C311yescharacteristic
114C22×D9—26no—
115C22×D9C23×C311yescharacteristic
116C22×D9C23×C311yescharacteristic
117C2×C4×C9—23no—
118C2×C4×C9—23no—
119C2×C4×C9C23×C313yes—
120C2×C4×C9C23×C313yes—
121C2×Dic9C23×C313yes—
122C2×Dic9C23×C313yes—
123C4×D9—26no—
124C4×D9—26no—
125C2×#126—91noSylow
126C2×3−1+2—41no—
127C2×3−1+2C2×#4113yes—
128AGL(1,ℤ/9)—41no—
129AGL(1,ℤ/9)#12614yesdirect product factor
130C24×C3—91no—
131C23×D3—31no—
132C22×C3×C4—93no—
133C22×C3×C4—93no—
134C2×C4×D3—33no—
135C2×C4×D3—33no—
136C3×C4⋊C4—33no—
137C3×C4⋊C4—93no—
138C3×C4⋊C4—93no—
139C3×C4⋊C4—93no—
140C3×C4⋊C4C2×C3×D313yes—
141C3×C22⋊C4—33no—
142C3×C22⋊C4—93no—
143C3×C22⋊C4—93no—
144C3×C22⋊C4—93no—
145C3×C22⋊C4C2×C3×D313yes—
146C3×C42—31no—
147C3×C42—93no—
148C3×C42C2×C3×D311yescharacteristic
149#217—33no—
150(C2×D3)⋊C4—33no—
151(C2×D3)⋊C4—33no—
152C4⋊Dic3—33no—
153Dic3⋊C4—33no—
154Dic3⋊C4—33no—
155C4×Dic3—33no—
156C22×C32—23no—
157C22×C32C23×D311yescharacteristic
158C2×C3×D3—36no—
159C2×C3×D3—63no—
160C2×C3×D3—124no—
161C32×C4—23no—
162C32×C4—23no—
163C3×Dic3—63no—
164C3×Dic3—63no—
165C22×C9—23no—
166C22×C9—21no—
167C22×C9—43no—
168C22×C9C24×C311yescharacteristic
169C2×D9—23no—
170C2×D9—44no—
171C2×D9C3×#4116yes—
172C4×C9—23no—
173C4×C9—23no—
174C4×C9—43no—
175C4×C9—43no—
176Dic9—23no—
177Dic9—23no—
178#126—93no—
179#126—93no—
180#126—91no—
181#126AGL(1,ℤ/9)11yescharacteristic, direct product factor
182C2×C4⋊C4—93no—
183C2×C22⋊C4—93no—
184C2×C42—91no—
1853−1+2C2×#12611yescharacteristic, complemented, Sylow
186C23×C3—31no—
187C23×C3—91no—
188C23×C3—91no—
189C23×C3—186no—
190C23×C3C22×C3×D311yescharacteristic
191C22×D3—31no—
192C22×D3—31no—
193C22×D3—66no—
194C2×C3×C4—33no—
195C2×C3×C4—33no—
196C2×C3×C4—93no—
197C2×C3×C4—93no—
198C2×C3×C4—186no—
199C2×C3×C4—186no—
200C2×C3×C4C22×C3×D313yes—
201C2×C3×C4C22×C3×D313yes—
202C2×Dic3—33no—
203C2×Dic3—33no—
204C4×D3—66no—
205C4×D3—66no—
206C2×C32—41no—
207C2×C32D3×#4113yes—
208C3×D3—34no—
209C3×D3—121no—
210C2×C9—23no—
211C2×C9—41no—
212C2×C9—81no—
213C2×C9C2×C3×#4113yes—
214D9—41no—
215D9C3×#12614yescomplemented
216C24—91no—
217C22×C4—93no—
218C22×C4—93no—
219C4⋊C4—93no—
220C4⋊C4—93no—
221C4⋊C4—93no—
222C4⋊C4C2×AGL(1,ℤ/9)13yescomplemented
223C22⋊C4—93no—
224C22⋊C4—93no—
225C22⋊C4—93no—
226C22⋊C4C2×AGL(1,ℤ/9)13yes—
227C42—93no—
228C42C2×AGL(1,ℤ/9)11yescharacteristic, complemented
229C22×C3—23no—
230C22×C3—31no—
231C22×C3—63no—
232C22×C3—96no—
233C22×C3—183no—
234C22×C3—364no—
235C22×C3C23×C3×D311yescharacteristic
236C2×D3—36no—
237C2×D3—63no—
238C2×D3—124no—
239C3×C4—23no—
240C3×C4—23no—
241C3×C4—63no—
242C3×C4—63no—
243C3×C4—183no—
244C3×C4—183no—
245Dic3—63no—
246Dic3—63no—
247C32D3×#12611yescharacteristic
248C9—21no—
249C9C2×C3×#12611yescharacteristic, complemented
250C23—91no—
251C23—91no—
252C23—186no—
253C23C22×AGL(1,ℤ/9)11yescharacteristic
254C2×C4—93no—
255C2×C4—93no—
256C2×C4—186no—
257C2×C4—186no—
258C2×C4C22×AGL(1,ℤ/9)13yes—
259C2×C4C22×AGL(1,ℤ/9)13yes—
260C2×C3—33no—
261C2×C3—41no—
262C2×C3—94no—
263C2×C3—121no—
264C2×C3—361no—
265C2×C3C3×D3×#4113yes—
266D3—34no—
267D3—121no—
268C22—23no—
269C22—96no—
270C22—183no—
271C22—364no—
272C22C23×AGL(1,ℤ/9)11yescharacteristic
273C4—23no—
274C4—23no—
275C4—183no—
276C4—183no—
277C3—31no—
278C3C3×D3×#12611yescharacteristic
279C2—41no—
280C2—94no—
281C2—361no—
282C2#41×AGL(1,ℤ/9)13yes—
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting3−1+2×#126C2
prime radicalC23×AGL(1,ℤ/9)C22
hypercenter#126AGL(1,ℤ/9)
hypocenterC9C2×C3×#126
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#126C2semidirect
2. #63460⋊C3C2semidirect
3. C4⋊C4×AGL(1,ℤ/9)C2semidirect
5. #63441⋊C3C2semidirect
6. #11721396C2semidirect
7. C42×AGL(1,ℤ/9)C2semidirect
9. D9×#126C3semidirect
13. C4⋊C4×3−1+2C22semidirect
15. C42×3−1+2C22semidirect
19. C4⋊(3−1+2⋊C4)C22semidirect
20. (3−1+2⋊C4)⋊C4C22semidirect
21. (3−1+2⋊C4)⋊C4C22semidirect
22. C4×3−1+2⋊C4C22semidirect
31. C9×#126C2×C3semidirect
32. #63460C2×C3semidirect
33. C4⋊C4×D9C2×C3semidirect
35. #63441C2×C3semidirect
36. #63438C2×C3semidirect
37. C42×D9C2×C3semidirect
67. C9×C4⋊C4C22×C3semidirect
71. C42×C9C22×C3semidirect
75. C4⋊Dic9C22×C3semidirect
76. Dic9⋊C4C22×C3semidirect
77. Dic9⋊C4C22×C3semidirect
78. C4×Dic9C22×C3semidirect
129. AGL(1,ℤ/9)#126direct (N × H)
181. #126AGL(1,ℤ/9)direct (N × H)
185. 3−1+2C2×#126semidirect
215. D9C3×#126semidirect
222. C4⋊C4C2×AGL(1,ℤ/9)semidirect
228. C42C2×AGL(1,ℤ/9)semidirect
249. C9C2×C3×#126semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1262
#63460⋊C32
C4⋊C4×AGL(1,ℤ/9)2
C22⋊C4×AGL(1,ℤ/9)2
#63441⋊C32
#117213962
C42×AGL(1,ℤ/9)2
C3×D3×#1263
D9×#1263