Finite Group Databasebeta

← #10589595⋊C3 #13152383

Subgroups — #10589595⋊C3 #13152383

Order 1728.

1104 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 252 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
1C22×#611⋊C3C212yesmaximal, complemented
2C2×C22⋊C4×3−1+2C211yesmaximal, characteristic, complemented
3#63492⋊C3C214yesmaximal, complemented
4#63448⋊C3C214yesmaximal, complemented
5#63444⋊C3C214yesmaximal, complemented
6C3×#5493—31nomaximal
7#10589595C311yesmaximal, characteristic, complemented
8C24×3−1+2C2211yescharacteristic
9C23×AGL(1,ℤ/9)C2212yescomplemented
10C22×C4×3−1+2C2212yescomplemented
11C2×#611⋊C3—28no—
12C2×#611⋊C3C2214yes—
13C2×#611⋊C3C2214yescomplemented
14C22×3−1+2⋊C4C2212yescomplemented
15C2×C4⋊AGL(1,ℤ/9)C2214yescomplemented
16C22⋊C4×3−1+2C2214yescomplemented
17(C2×AGL(1,ℤ/9))⋊C4C2218yescomplemented
18C4⋊(3−1+2⋊C4)C2214yescomplemented
19C22×C3×#67—32no—
20C2×C32×C22⋊C4D311yescharacteristic
21C3×#930—34no—
22C3×#886—34no—
23C3×#882—34no—
24C22×#611C2×C312yescomplemented
25C2×C9×C22⋊C4—21no—
26C2×C9×C22⋊C4C2×C311yescharacteristic, complemented
27#63492C2×C314yescomplemented
28#63448C2×C314yescomplemented
29#63444C2×C314yescomplemented
30C23×3−1+2—42no—
31C23×3−1+2D414yescomplemented
32C23×3−1+2C2312yes—
33C23×3−1+2C2311yescharacteristic
34C22×AGL(1,ℤ/9)—24no—
35C22×AGL(1,ℤ/9)—44no—
36C22×AGL(1,ℤ/9)C2314yes—
37C2×C4×3−1+2—42no—
38C2×C4×3−1+2C2314yes—
39#611⋊C3—28no—
40#611⋊C3—28no—
41C2×3−1+2⋊C4—24no—
42C2×3−1+2⋊C4C2314yes—
43C4⋊AGL(1,ℤ/9)—44no—
44C3×#534—91no—
45#5493—31no—
46C24×C32C2×D311yescharacteristic
47C23×C3×D3—32no—
48C22×C32×C4C2×D312yes—
49C2×C3×#67—34no—
50C2×C3×#67—34no—
51C2×C3×#67—68no—
52C22×C3×Dic3—32no—
53C2×C3×D12—34no—
54C24×C9—21no—
55C24×C9C22×C311yescharacteristic
56C23×D9C22×C312yescomplemented
57C32×C22⋊C4C2×D314yes—
58C3×(C2×D3)⋊C4—38no—
59C3×C4⋊Dic3—34no—
60C22×C4×C9—22no—
61C22×C4×C9C22×C312yescomplemented
62C2×#611—28no—
63C2×#611C22×C314yes—
64C2×#611C22×C314yescomplemented
65C22×Dic9C22×C312yescomplemented
66C2×D36C22×C314yescomplemented
67C9×C22⋊C4—24no—
68C9×C22⋊C4C22×C314yescomplemented
69(C2×D9)⋊C4C22×C318yescomplemented
70C4⋊Dic9C22×C314yescomplemented
71C22×3−1+2—22no—
72C22×3−1+2—24no—
73C22×3−1+2—44no—
74C22×3−1+2C2×D414yes—
75C22×3−1+2C2×D412yes—
76C22×3−1+2C2411yescharacteristic
77C2×AGL(1,ℤ/9)—24no—
78C2×AGL(1,ℤ/9)—28no—
79C2×AGL(1,ℤ/9)—48no—
80C4×3−1+2—24no—
813−1+2⋊C4—24no—
82C22×C3×D4—92no—
83C22×#67—32no—
84C3×#123—94no—
85C3×#121—94no—
86C3×#120—94no—
87C2×C3×C22⋊C4—31no—
88C2×C3×C22⋊C4C3×D311yescharacteristic
89#930—34no—
90#886—34no—
91#882—34no—
92C23×C32—42no—
93C23×C32D1214yes—
94C23×C32C22×D312yes—
95C23×C32C22×D311yescharacteristic
96C22×C3×D3—34no—
97C22×C3×D3—64no—
98C22×C3×D3—124no—
99C2×C32×C4—42no—
100C2×C32×C4C22×D314yes—
101C3×#67—68no—
102C3×#67—68no—
103C2×C3×Dic3—34no—
104C2×C3×Dic3—64no—
105C3×D12—124no—
106C23×C9—24no—
107C23×C9—21no—
108C23×C9—22no—
109C23×C9—42no—
110C23×C9—44no—
111C23×C9C3×D414yescomplemented
112C23×C9C23×C312yes—
113C23×C9C23×C311yescharacteristic
114C22×D9—24no—
115C22×D9—44no—
116C22×D9C23×C314yes—
117C2×C4×C9—24no—
118C2×C4×C9—42no—
119C2×C4×C9—44no—
120C2×C4×C9C23×C314yes—
121#611—28no—
122#611—28no—
123C2×Dic9—24no—
124C2×Dic9C23×C314yes—
125D36—44no—
126#534—91noSylow
127C2×3−1+2—22no—
128C2×3−1+2—24no—
129C2×3−1+2C22×D411yescharacteristic
130C2×3−1+2H(5,𝔽2)12yes—
131AGL(1,ℤ/9)—44no—
132C24×C3—31no—
133C24×C3—92no—
134C24×C3C2×C3×D311yescharacteristic
135C23×D3—32no—
136C2×C3×D4—94no—
137C2×C3×D4—94no—
138C2×C3×D4—94no—
139C2×C3×D4—188no—
140C22×C3×C4—32no—
141C22×C3×C4—92no—
142C22×C3×C4C2×C3×D312yes—
143C2×#67—34no—
144C2×#67—34no—
145C2×#67—68no—
146C22×Dic3—32no—
147C2×D12—34no—
148C3×C4⋊C4—94no—
149C3×C22⋊C4—34no—
150C3×C22⋊C4—98no—
151C3×C22⋊C4C2×C3×D314yes—
152(C2×D3)⋊C4—38no—
153C4⋊Dic3—34no—
154C22×C32—22no—
155C22×C32—24no—
156C22×C32—44no—
157C22×C32C2×D1214yes—
158C22×C32C2×D1212yes—
159C22×C32C23×D311yescharacteristic
160C2×C3×D3—64no—
161C2×C3×D3—68no—
162C2×C3×D3—128no—
163C32×C4—24no—
164C3×Dic3—64no—
165C22×C9—24no—
166C22×C9—22no—
167C22×C9—24no—
168C22×C9—21no—
169C22×C9—22no—
170C22×C9—24no—
171C22×C9—44no—
172C22×C9—44no—
173C22×C9—48no—
174C22×C9C2×C3×D414yes—
175C22×C9C2×C3×D412yes—
176C22×C9C24×C311yescharacteristic
177C2×D9—24no—
178C2×D9—28no—
179C2×D9—48no—
180C4×C9—24no—
181C4×C9—44no—
182Dic9—24no—
183C22×D4—92no—
184#123—94no—
185#121—94no—
186#120—94no—
187C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic, complemented
1883−1+2#53411yescharacteristic, complemented, Sylow
189C23×C3—34no—
190C23×C3—32no—
191C23×C3—31no—
192C23×C3—42no—
193C23×C3—94no—
194C23×C3—122no—
195C23×C3—184no—
196C23×C3—364no—
197C23×C3C3×D1214yes—
198C23×C3C22×C3×D312yes—
199C23×C3C22×C3×D311yescharacteristic
200C22×D3—34no—
201C22×D3—64no—
202C22×D3—124no—
203C3×D4—188no—
204C3×D4—188no—
205C3×D4—364no—
206C2×C3×C4—34no—
207C2×C3×C4—42no—
208C2×C3×C4—94no—
209C2×C3×C4—122no—
210C2×C3×C4—184no—
211C2×C3×C4C22×C3×D314yes—
212#67—68no—
213#67—68no—
214C2×Dic3—34no—
215C2×Dic3—64no—
216D12—124no—
217C2×C32—24no—
218C2×C32—22no—
219C2×C32C22×D1211yescharacteristic
220C2×C32#100412yes—
221C3×D3—124no—
222C2×C9—24no—
223C2×C9—24no—
224C2×C9—22no—
225C2×C9—21no—
226C2×C9—22no—
227C2×C9—44no—
228C2×C9C22×C3×D411yescharacteristic
229C2×C9C3×H(5,𝔽2)12yes—
230D9—44no—
231C24—92no—
232C24C2×AGL(1,ℤ/9)11yescharacteristic
233C2×D4—94no—
234C2×D4—94no—
235C2×D4—94no—
236C2×D4—188no—
237C22×C4—92no—
238C22×C4C2×AGL(1,ℤ/9)12yescomplemented
239C4⋊C4—94no—
240C22⋊C4—98no—
241C22⋊C4C2×AGL(1,ℤ/9)14yescomplemented
242C22×C3—22no—
243C22×C3—24no—
244C22×C3—31no—
245C22×C3—34no—
246C22×C3—32no—
247C22×C3—44no—
248C22×C3—62no—
249C22×C3—64no—
250C22×C3—124no—
251C22×C3—184no—
252C22×C3—188no—
253C22×C3—368no—
254C22×C3C2×C3×D1214yes—
255C22×C3C2×C3×D1212yes—
256C22×C3C23×C3×D311yescharacteristic
257C2×D3—64no—
258C2×D3—68no—
259C2×D3—128no—
260C3×C4—24no—
261C3×C4—64no—
262C3×C4—184no—
263Dic3—64no—
264C32#549311yescharacteristic
265C9—21no—
266C9C3×#53411yescharacteristic, complemented
267C23—42no—
268C23—94no—
269C23—184no—
270C23—364no—
271C23C4⋊AGL(1,ℤ/9)14yescomplemented
272C23C22×AGL(1,ℤ/9)12yes—
273C23C22×AGL(1,ℤ/9)11yescharacteristic
274D4—188no—
275D4—188no—
276D4—364no—
277C2×C4—42no—
278C2×C4—94no—
279C2×C4—184no—
280C2×C4C22×AGL(1,ℤ/9)14yes—
281C2×C3—24no—
282C2×C3—22no—
283C2×C3—31no—
284C2×C3—32no—
285C2×C3—64no—
286C2×C3—62no—
287C2×C3—364no—
288C2×C3C22×C3×D1211yescharacteristic
289C2×C3C3×#100412yes—
290D3—124no—
291C22—22no—
292C22—24no—
293C22—44no—
294C22—184no—
295C22—188no—
296C22—368no—
297C22C2×C4⋊AGL(1,ℤ/9)14yes—
298C22C2×C4⋊AGL(1,ℤ/9)12yes—
299C22C23×AGL(1,ℤ/9)11yescharacteristic
300C4—24no—
301C4—184no—
302C3—31no—
303C3C3×#549311yescharacteristic
304C2—24no—
305C2—22no—
306C2—364no—
307C2C22×C4⋊AGL(1,ℤ/9)11yescharacteristic
308C2#63710⋊C312yes—
subgroupG/H
centerC22C23×AGL(1,ℤ/9)
commutator, cosocleC22×C9C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×C22⋊C4×3−1+2C2
hypercenterC2×C22⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#534
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#611⋊C3C2semidirect
2. C2×C22⋊C4×3−1+2C2semidirect
3. #63492⋊C3C2semidirect
4. #63448⋊C3C2semidirect
5. #63444⋊C3C2semidirect
7. #10589595C3semidirect
9. C23×AGL(1,ℤ/9)C22semidirect
10. C22×C4×3−1+2C22semidirect
13. C2×#611⋊C3C22semidirect
14. C22×3−1+2⋊C4C22semidirect
15. C2×C4⋊AGL(1,ℤ/9)C22semidirect
16. C22⋊C4×3−1+2C22semidirect
17. (C2×AGL(1,ℤ/9))⋊C4C22semidirect
18. C4⋊(3−1+2⋊C4)C22semidirect
24. C22×#611C2×C3semidirect
26. C2×C9×C22⋊C4C2×C3semidirect
27. #63492C2×C3semidirect
28. #63448C2×C3semidirect
29. #63444C2×C3semidirect
31. C23×3−1+2D4semidirect
56. C23×D9C22×C3semidirect
61. C22×C4×C9C22×C3semidirect
64. C2×#611C22×C3semidirect
65. C22×Dic9C22×C3semidirect
66. C2×D36C22×C3semidirect
68. C9×C22⋊C4C22×C3semidirect
69. (C2×D9)⋊C4C22×C3semidirect
70. C4⋊Dic9C22×C3semidirect
111. C23×C9C3×D4semidirect
187. C2×C22⋊C4AGL(1,ℤ/9)semidirect
188. 3−1+2#534semidirect
238. C22×C4C2×AGL(1,ℤ/9)semidirect
241. C22⋊C4C2×AGL(1,ℤ/9)semidirect
266. C9C3×#534semidirect
271. C23C4⋊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
C22×#611⋊C32
C2×C22⋊C4×3−1+22
#63492⋊C32
#63448⋊C32
#63444⋊C32
C3×#54933
#105895953