Finite Group Databasebeta

← #72268947

Subgroups — #72268947

Order 2592.

525 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C2×A4⋊AGL(1,ℤ/9)C212yesmaximal, complemented
2C4×A4×3−1+2C211yesmaximal, characteristic, complemented
3C3×C4⋊(C3⋊S4)—31nomaximal
4C4⋊(C9⋊S4)C311yesmaximal, characteristic, complemented
5C4⋊(((C22⋊C9)⋊C2)⋊C3)—32nomaximal
6#63492⋊C3—31nomaximal
7C2×A4×3−1+2C2211yescharacteristic
8A4⋊AGL(1,ℤ/9)—22no—
9C4⋊(C3⋊AGL(1,ℤ/9))—41nomaximal
10C2×C3×C3⋊S4—32no—
11C32×C4×A4D311yescharacteristic
12C2×C9⋊S4C2×C312yescomplemented
13C2×((C22⋊C9)⋊C2)⋊C3—34no—
14C22×C4×3−1+2D311yescharacteristic, complemented
15C2×#611⋊C3—32no—
16C2×C4⋊AGL(1,ℤ/9)—31no—
17C4×#1094D312yescomplemented
18C4×C9⋊A4—21no—
19C4×C9⋊A4—22no—
20C4×C9×A4—21no—
21C4×C9×A4C2×C311yescharacteristic, complemented
22(C2×AGL(1,ℤ/9))⋊C4—32no—
23C4⋊(3−1+2⋊C4)—31no—
24C2×C3⋊AGL(1,ℤ/9)—42no—
25A4×3−1+2D411yescharacteristic, complemented
26C3×C4×3−1+2—41no—
27C4⋊(C3⋊S4)—31no—
28C3×C4⋊S4—93no—
29C3×#930—91no—
30C4⋊((C22⋊C9)⋊C2)—32no—
31#63492—31no—
32C2×C32×A4C2×D311yescharacteristic
33C3×C3⋊S4—62no—
34C3×C4⋊(C3⋊D3)—121no—
35C23×3−1+2C2×D311yescharacteristic
36C22×AGL(1,ℤ/9)—32no—
37C2×#1094C2×D312yes—
38C2×C9⋊A4—21no—
39C2×C9⋊A4—22no—
40C2×C9×A4—21no—
41C2×C9×A4C22×C311yescharacteristic
42C9⋊S4—22no—
43((C22⋊C9)⋊C2)⋊C3—64no—
44C2×C4×3−1+2—31no—
45C2×C4×3−1+2—31no—
46C4⋊(C3⋊D9)—41no—
47#611⋊C3—62no—
48#611⋊C3—62no—
49C2×3−1+2⋊C4—32no—
50C4⋊AGL(1,ℤ/9)—61no—
51C4⋊AGL(1,ℤ/9)—61no—
52C4⋊AGL(1,ℤ/9)—122no—
53C2×C3×3−1+2—41no—
54C3⋊AGL(1,ℤ/9)—82no—
55C2×C3⋊S4—32no—
56C2×C3×S4—96no—
57C22×C32×C4C3⋊D311yescharacteristic
58C2×C3×#67—92no—
59C2×C3×D12—91no—
60C3×C4×A4—21no—
61C3×C4×A4—33no—
62C3×C4×A4C3×D311yescharacteristic
63C2×(C22⋊C9)⋊C2—34no—
64C3×(C2×D3)⋊C4—92no—
65C3×C4⋊Dic3—91no—
66C22×C4×C9—21no—
67C22×C4×C9C3×D311yescharacteristic, complemented
68C2×#611—32no—
69C2×D36—31no—
70C4×C22⋊C9—22no—
71C4×C22⋊C9C3×D312yescomplemented
72(C2×D9)⋊C4—32no—
73C4⋊Dic9—31no—
74C2×C3×C3⋊D3—122no—
75C32×A4D1211yescharacteristic
76C33×C4—41no—
77C22×3−1+2—31no—
78C22×3−1+2—31no—
79C22×3−1+2D1211yescharacteristic, complemented
80C2×C3⋊D9—42no—
81C2×AGL(1,ℤ/9)—62no—
82C2×AGL(1,ℤ/9)—62no—
83C2×AGL(1,ℤ/9)—62no—
84C2×AGL(1,ℤ/9)—124no—
85#1094D1212yescomplemented
86C9⋊A4—21no—
87C9⋊A4—22no—
88C9×A4—21no—
89C9×A4C3×D411yescharacteristic, complemented
90C4×3−1+2—31no—
91C4×3−1+2—42no—
92C4×3−1+2—82no—
93C4×3−1+2—81no—
94C4×3−1+2S411yescharacteristic, complemented
95C3×C4×C9—41no—
96C3×C4×C9—81no—
973−1+2⋊C4—62no—
98C4⋊S4—93no—
99C3×#121—271no—
100#930—91no—
101C3×3−1+2—41noSylow
102C23×C32C2×C3⋊D311yescharacteristic
103C22×C3×D3—92no—
104C2×C3×A4—21no—
105C2×C3×A4—33no—
106C2×C3×A4C2×C3×D311yescharacteristic
107C3⋊S4—62no—
108C3×S4—186no—
109C2×C32×C4—31no—
110C2×C32×C4—31no—
111C4⋊(C3⋊D3)—121no—
112C3×#67—182no—
113C3×#67—182no—
114C2×C3×Dic3—92no—
115C3×D12—181no—
116C3×D12—181no—
117C3×D12—363no—
118C23×C9—21no—
119C23×C9C2×C3×D311yescharacteristic
120C22×D9—32no—
121C2×C22⋊C9—22no—
122C2×C22⋊C9C2×C3×D312yes—
123(C22⋊C9)⋊C2—64no—
124C2×C4×C9—31no—
125C2×C4×C9—31no—
126C2×C4×C9—61no—
127C2×C4×C9—61no—
128#611—62no—
129#611—62no—
130C2×Dic9—32no—
131D36—61no—
132D36—61no—
133D36—122no—
134C2×C33—41no—
135C3×C3⋊D3—242no—
136C2×3−1+2—31no—
137C2×3−1+2—31no—
138C2×3−1+2—42no—
139C2×3−1+2—82no—
140C2×3−1+2—81no—
141C2×3−1+2C2×S411yescharacteristic
142C2×C3×C9—41no—
143C2×C3×C9—81no—
144C3⋊D9—82no—
145AGL(1,ℤ/9)—122no—
146AGL(1,ℤ/9)—244no—
147C2×S4—96no—
148C2×C3×D4—271no—
149C2×C3×D4—272no—
150C22×C3×C4—31no—
151C22×C3×C4C3×C3⋊D311yescharacteristic
152C2×#67—92no—
153C2×D12—91no—
154C4×A4—61no—
155C4×A4AGL(1,ℤ/9)13yescomplemented
156C3×C4⋊C4—271no—
157C3×C22⋊C4—272no—
158(C2×D3)⋊C4—92no—
159C4⋊Dic3—91no—
160C22×C32—31no—
161C22×C32—31no—
162C22×C32C4⋊(C3⋊D3)11yescharacteristic
163C2×C3⋊D3—122no—
164C2×C3×D3—182no—
165C2×C3×D3—182no—
166C2×C3×D3—182no—
167C2×C3×D3—366no—
168C3×A4—21no—
169C3×A4—33no—
170C3×A4C3×D1211yescharacteristic
171C32×C4—31no—
172C32×C4—41no—
173C32×C4—81no—
174C32×C4—123no—
175C32×C4C3⋊S411yescharacteristic
176C3×Dic3—182no—
177C22×C9—21no—
178C22×C9—31no—
179C22×C9—31no—
180C22×C9—61no—
181C22×C9—61no—
182C22×C9C3×D1211yescharacteristic, complemented
183C2×D9—62no—
184C2×D9—62no—
185C2×D9—62no—
186C2×D9—124no—
187C22⋊C9—22no—
188C22⋊C9C3×D1212yescomplemented
189C4×C9—21no—
190C4×C9—31no—
191C4×C9—42no—
192C4×C9—61no—
193C4×C9—82no—
194C4×C9C3×S411yescharacteristic, complemented
195Dic9—62no—
196#121—271noSylow
197C33—41no—
1983−1+2—42no—
1993−1+2—82no—
2003−1+2—81no—
2013−1+2C4⋊S411yescharacteristic, complemented
202C3×C9—41no—
203C3×C9—81no—
204C23×C3—31no—
205C23×C3—272no—
206C23×C3C2×C3×C3⋊D311yescharacteristic
207C22×D3—92no—
208C2×A4—61no—
209C2×A4C2×AGL(1,ℤ/9)13yes—
210S4—186no—
211C3×D4—541no—
212C3×D4—541no—
213C3×D4—542no—
214C3×D4—542no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—91no—
218C2×C3×C4—91no—
219C2×C3×C4—272no—
220#67—182no—
221#67—182no—
222C2×Dic3—92no—
223D12—181no—
224D12—181no—
225D12—363no—
226C2×C32—31no—
227C2×C32—31no—
228C2×C32—41no—
229C2×C32—81no—
230C2×C32—123no—
231C2×C32C2×C3⋊S411yescharacteristic
232C3⋊D3—242no—
233C3×D3—362no—
234C3×D3—726no—
235C2×C9—21no—
236C2×C9—31no—
237C2×C9—31no—
238C2×C9—42no—
239C2×C9—61no—
240C2×C9—61no—
241C2×C9—82no—
242C2×C9C2×C3×S411yescharacteristic
243D9—122no—
244D9—244no—
245C2×D4—271no—
246C2×D4—272no—
247C22×C4C3⋊AGL(1,ℤ/9)11yescharacteristic, complemented
248C4⋊C4—271no—
249C22⋊C4—272no—
250C22×C3—31no—
251C22×C3—31no—
252C22×C3—31no—
253C22×C3—91no—
254C22×C3—91no—
255C22×C3—542no—
256C22×C3—542no—
257C22×C3—542no—
258C22×C3C3×C4⋊(C3⋊D3)11yescharacteristic
259C2×D3—182no—
260C2×D3—182no—
261C2×D3—182no—
262C2×D3—366no—
263A4—61no—
264A4C4⋊AGL(1,ℤ/9)13yescomplemented
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—43no—
268C3×C4—91no—
269C3×C4—241no—
270C3×C4—542no—
271C3×C4C3×C3⋊S411yescharacteristic
272Dic3—182no—
273C32—41no—
274C32—81no—
275C32—123no—
276C32C4⋊(C3⋊S4)11yescharacteristic
277C9—21no—
278C9—42no—
279C9—82no—
280C9C3×C4⋊S411yescharacteristic, complemented
281C23—272no—
282C23C2×C3⋊AGL(1,ℤ/9)11yescharacteristic
283D4—541no—
284D4—542no—
285D4—542no—
286D4—541no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—272no—
290C2×C3—31no—
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—43no—
294C2×C3—91no—
295C2×C3—91no—
296C2×C3—241no—
297C2×C3—1082no—
298C2×C3C2×C3×C3⋊S411yescharacteristic
299D3—362no—
300D3—726no—
301C22—31no—
302C22—31no—
303C22—542no—
304C22—542no—
305C22—542no—
306C22C4⋊(C3⋊AGL(1,ℤ/9))11yescharacteristic, complemented
307C4—31no—
308C4—542no—
309C4A4⋊AGL(1,ℤ/9)11yescharacteristic, complemented
310C3—31no—
311C3—43no—
312C3—241no—
313C3C3×C4⋊(C3⋊S4)11yescharacteristic
314C2—31no—
315C2—31no—
316C2—1082no—
317C2C2×A4⋊AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC2C2×A4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C9×A4C22×C3
FrattiniC2×C3C2×C3×C3⋊S4
FittingC22×C4×3−1+2D3
socleC23×C3C2×C3×C3⋊D3
hypercenterC4A4⋊AGL(1,ℤ/9)
hypocenterC9×A4C3×D4
LCSG ▷ C2×C9×A4 ▷ C9×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×A4⋊AGL(1,ℤ/9)C2semidirect
2. C4×A4×3−1+2C2semidirect
4. C4⋊(C9⋊S4)C3semidirect
12. C2×C9⋊S4C2×C3semidirect
14. C22×C4×3−1+2D3semidirect
17. C4×#1094D3semidirect
21. C4×C9×A4C2×C3semidirect
25. A4×3−1+2D4semidirect
67. C22×C4×C9C3×D3semidirect
71. C4×C22⋊C9C3×D3semidirect
79. C22×3−1+2D12semidirect
85. #1094D12semidirect
89. C9×A4C3×D4semidirect
94. C4×3−1+2S4semidirect
155. C4×A4AGL(1,ℤ/9)semidirect
182. C22×C9C3×D12semidirect
188. C22⋊C9C3×D12semidirect
194. C4×C9C3×S4semidirect
201. 3−1+2C4⋊S4semidirect
247. C22×C4C3⋊AGL(1,ℤ/9)semidirect
264. A4C4⋊AGL(1,ℤ/9)semidirect
280. C9C3×C4⋊S4semidirect
306. C22C4⋊(C3⋊AGL(1,ℤ/9))semidirect
309. C4A4⋊AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C2×A4⋊AGL(1,ℤ/9)2
C4×A4×3−1+22
C3×C4⋊(C3⋊S4)3
C4⋊(C9⋊S4)3
C4⋊(((C22⋊C9)⋊C2)⋊C3)3
#63492⋊C33
C4⋊(C3⋊AGL(1,ℤ/9))4