Finite Group Databasebeta

← C9×#4732 #13141592

Subgroups — C9×#4732 #13141592

Order 1728.

644 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C9×D12C211yesmaximal, characteristic, complemented
2C9×#930C211yesmaximal, characteristic, complemented
3C2×C9×D24C212yesmaximal, complemented
4C3×C9×C22⋊C8C211yesmaximal, characteristic, complemented
5C9×#821C212yesmaximal, complemented
6C3×#4732C311yesmaximal, characteristic
7C9×#447—31nomaximal
8C23×D3×C9—21no—
9C22×C3×C4×C9C2211yescharacteristic
10C2×C9×#67—21no—
11C2×C9×D12—21no—
12C2×C9×D12—24no—
13C2×C9×D12C2212yes—
14C2×C9×D12C2211yescharacteristic, complemented
15C2×C3×C8×C9C2212yescomplemented
16C9×(C2×D3)⋊C4—21no—
17C9×C4⋊Dic3C2211yescharacteristic, complemented
18C9×D24—24no—
19C22×C3×D12C2×C311yescharacteristic
20C3×#930C2×C311yescharacteristic
21C2×C3×D24C2×C312yes—
22C22×D4×C9—31no—
23C32×C22⋊C8C2×C311yescharacteristic
24C3×#821C2×C312yes—
25C2×C9×D8—32no—
26C9×#121—31no—
27C9×#102—32no—
28C9×C22⋊C8—21no—
29C9×C22⋊C8D311yescharacteristic, complemented
30C23×C3×C9D411yescharacteristic
31C22×D3×C9—24no—
32C22×D3×C9—22no—
33C22×D3×C9—21no—
34C22×D3×C9—42no—
35C22×D3×C9—42no—
36C2×C3×C4×C9—22no—
37C2×C3×C4×C9D411yescharacteristic
38C2×C3×C4×C9C2311yescharacteristic
39C9×#67—42no—
40C2×C9×Dic3—21no—
41C9×D12—24no—
42C9×D12—22no—
43C9×D12—22no—
44C9×D12D414yescomplemented
45C3×C8×C9—22no—
46C3×#447—31no—
47#4732C911yescharacteristic, direct product factor
48C23×C3×D3—21no—
49C22×C32×C4C22×C311yescharacteristic
50C2×C3×#67—21no—
51C2×C3×D12—21no—
52C2×C3×D12—24no—
53C2×C3×D12C22×C311yescharacteristic
54C2×C3×D12C22×C312yes—
55C24×C9—61no—
56C2×C32×C8C22×C312yes—
57C3×(C2×D3)⋊C4—21no—
58C3×C4⋊Dic3C22×C311yescharacteristic
59C3×D24—24no—
60C2×D4×C9—32no—
61C2×D4×C9—31no—
62C2×D4×C9—64no—
63C2×D4×C9—61no—
64C2×D4×C9—61no—
65C22×C4×C9—21no—
66C22×C4×C9C2×D311yescharacteristic
67C9×D8—64no—
68C2×C8×C9—22no—
69C2×C8×C9C2×D312yescomplemented
70C9×C4⋊C4—31no—
71C9×C22⋊C4—61no—
72C22×C3×C9—21no—
73C22×C3×C9—21no—
74C22×C3×C9D812yescomplemented
75C22×C3×C9C2×D411yescharacteristic
76C2×D3×C9—24no—
77C2×D3×C9—48no—
78C2×D3×C9—42no—
79C2×D3×C9—42no—
80C2×D3×C9—41no—
81C2×D3×C9—41no—
82C2×D3×C9—41no—
83C3×C4×C9—21no—
84C3×C4×C9C2×D412yes—
85C9×Dic3—41no—
86C22×C3×D4—31no—
87C22×D12C2×C911yescharacteristic, complemented
88C2×C3×D8—32no—
89C3×#121—31no—
90#930C2×C911yescharacteristic, complemented
91C2×D24C2×C912yescomplemented
92C3×#102—32no—
93C3×C22⋊C8—21no—
94C3×C22⋊C8C2×C911yescharacteristic, complemented
95C3×C22⋊C8C3×D311yescharacteristic
96#821C2×C912yescomplemented
97C23×C32C3×D411yescharacteristic
98C22×C3×D3—24no—
99C22×C3×D3—22no—
100C22×C3×D3—21no—
101C22×C3×D3—42no—
102C22×C3×D3—42no—
103C2×C32×C4—22no—
104C2×C32×C4C3×D411yescharacteristic
105C2×C32×C4C23×C311yescharacteristic
106C3×#67—42no—
107C2×C3×Dic3—21no—
108C3×D12—24no—
109C3×D12—22no—
110C3×D12—22no—
111C3×D12C3×D414yes—
112C23×C9—21no—
113C23×C9—64no—
114C23×C9—62no—
115C23×C9—61no—
116C23×C9—122no—
117C23×C9—122no—
118C23×C9D1211yescharacteristic
119C32×C8—22no—
120D4×C9—34no—
121D4×C9—64no—
122D4×C9—62no—
123D4×C9—62no—
124D4×C9—122no—
125C2×C4×C9—22no—
126C2×C4×C9—21no—
127C2×C4×C9—21no—
128C2×C4×C9—42no—
129C2×C4×C9—61no—
130C2×C4×C9D1211yescharacteristic
131C2×C4×C9C22×D311yescharacteristic
132C8×C9—22no—
133C8×C9—42no—
134#447—31noSylow
135C2×C3×C9—22no—
136C2×C3×C9#12011yescharacteristic
137C2×C3×C9C2×D811yescharacteristic
138C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
139D3×C9—44no—
140D3×C9—81no—
141C24×C3—61no—
142C23×D3—21no—
143C2×C3×D4—32no—
144C2×C3×D4—31no—
145C2×C3×D4—64no—
146C2×C3×D4—61no—
147C2×C3×D4—61no—
148C22×C3×C4—21no—
149C22×C3×C4C22×C911yescharacteristic
150C22×C3×C4C2×C3×D311yescharacteristic
151C2×#67—21no—
152C2×D12—21no—
153C2×D12—24no—
154C2×D12C22×C912yes—
155C2×D12C22×C911yescharacteristic, complemented
156C3×D8—64no—
157C2×C3×C8—22no—
158C2×C3×C8C22×C912yescomplemented
159C2×C3×C8C2×C3×D312yes—
160C3×C4⋊C4—31no—
161C3×C22⋊C4—61no—
162(C2×D3)⋊C4—21no—
163C4⋊Dic3C22×C911yescharacteristic, complemented
164D24—24no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32C3×D812yes—
168C22×C32C2×C3×D411yescharacteristic
169C2×C3×D3—24no—
170C2×C3×D3—48no—
171C2×C3×D3—42no—
172C2×C3×D3—42no—
173C2×C3×D3—41no—
174C2×C3×D3—41no—
175C2×C3×D3—41no—
176C32×C4—21no—
177C32×C4C2×C3×D412yes—
178C3×Dic3—41no—
179C22×C9—21no—
180C22×C9—21no—
181C22×C9—22no—
182C22×C9—21no—
183C22×C9—41no—
184C22×C9—41no—
185C22×C9—64no—
186C22×C9—128no—
187C22×C9—122no—
188C22×C9—122no—
189C22×C9—121no—
190C22×C9—121no—
191C22×C9—121no—
192C22×C9D2412yescomplemented
193C22×C9C2×D1211yescharacteristic
194C4×C9—21no—
195C4×C9—22no—
196C4×C9—41no—
197C4×C9—121no—
198C4×C9D3×D412yes—
199C22×D4—31no—
200C2×D8—32no—
201#121—31no—
202#102—32no—
203C22⋊C8D3×C911yescharacteristic, complemented
204C3×C9#44711yescharacteristic, complemented, Sylow
205C23×C3—21no—
206C23×C3—64no—
207C23×C3—62no—
208C23×C3—61no—
209C23×C3—122no—
210C23×C3—122no—
211C23×C3D4×C911yescharacteristic
212C23×C3C3×D1211yescharacteristic
213C22×D3—24no—
214C22×D3—22no—
215C22×D3—21no—
216C22×D3—42no—
217C22×D3—42no—
218C3×D4—34no—
219C3×D4—64no—
220C3×D4—62no—
221C3×D4—62no—
222C3×D4—122no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—42no—
228C2×C3×C4—61no—
229C2×C3×C4D4×C911yescharacteristic
230C2×C3×C4C23×C911yescharacteristic
231C2×C3×C4C3×D1211yescharacteristic
232C2×C3×C4C22×C3×D311yescharacteristic
233#67—42no—
234C2×Dic3—21no—
235D12—24no—
236D12—22no—
237D12—22no—
238D12D4×C914yescomplemented
239C3×C8—22no—
240C3×C8—22no—
241C3×C8—42no—
242C2×C32—22no—
243C2×C32C3×#12011yescharacteristic
244C2×C32C2×C3×D811yescharacteristic
245C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
246C3×D3—44no—
247C3×D3—81no—
248C2×C9—22no—
249C2×C9—21no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—42no—
253C2×C9—124no—
254C2×C9—241no—
255C2×C9#88211yescharacteristic
256C2×C9C2×D2411yescharacteristic
257C2×C9#90811yescharacteristic
258C24—61no—
259C2×D4—32no—
260C2×D4—31no—
261C2×D4—64no—
262C2×D4—61no—
263C2×D4—61no—
264C22×C4C2×D3×C911yescharacteristic
265D8—64no—
266C2×C8C2×D3×C912yescomplemented
267C4⋊C4—31no—
268C22⋊C4—61no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—22no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—64no—
278C22×C3—128no—
279C22×C3—122no—
280C22×C3—122no—
281C22×C3—121no—
282C22×C3—121no—
283C22×C3—121no—
284C22×C3C9×D812yescomplemented
285C22×C3C2×D4×C911yescharacteristic
286C22×C3C3×D2412yes—
287C22×C3C2×C3×D1211yescharacteristic
288C2×D3—24no—
289C2×D3—48no—
290C2×D3—42no—
291C2×D3—42no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—41no—
299C3×C4—121no—
300C3×C4C2×D4×C912yes—
301C3×C4C3×D3×D412yes—
302Dic3—41no—
303C32C3×#44711yescharacteristic
304C9—21no—
305C9#473211yescharacteristic, direct product factor
306C23—64no—
307C23—62no—
308C23—61no—
309C23—122no—
310C23—122no—
311C23C9×D1211yescharacteristic
312D4—34no—
313D4—64no—
314D4—62no—
315D4—62no—
316D4—122no—
317C2×C4—22no—
318C2×C4—61no—
319C2×C4C9×D1211yescharacteristic
320C2×C4C22×D3×C911yescharacteristic
321C8—22no—
322C2×C3—22no—
323C2×C3—22no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—42no—
328C2×C3—124no—
329C2×C3—241no—
330C2×C3C9×#12011yescharacteristic
331C2×C3C2×C9×D811yescharacteristic
332C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
333C2×C3C3×#88211yescharacteristic
334C2×C3C2×C3×D2411yescharacteristic
335C2×C3C3×#90811yescharacteristic
336D3—44no—
337D3—81no—
338C22—21no—
339C22—21no—
340C22—64no—
341C22—128no—
342C22—122no—
343C22—122no—
344C22—121no—
345C22—121no—
346C22—121no—
347C22C9×D2412yescomplemented
348C22C2×C9×D1211yescharacteristic
349C4—21no—
350C4—121no—
351C4D3×D4×C912yes—
352C3—21no—
353C3C9×#44711yescharacteristic, complemented
354C3C3×#473211yescharacteristic
355C2—22no—
356C2—124no—
357C2—241no—
358C2C9×#88211yescharacteristic
359C2C2×C9×D2411yescharacteristic
360C2C9×#90811yescharacteristic
subgroupG/H
centerC22×C9C2×D12
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×C22⋊C8C2
socleC22×C32C2×C3×D4
prime radicalC3×#4732C3
cosocleC2×C32×C4C23×C3
hypercenterC9×C22⋊C8D3
hypocenterC3C9×#447
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C22×C4×C9 ◁ C9×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×C9×D12C2semidirect
2. C9×#930C2semidirect
3. C2×C9×D24C2semidirect
4. C3×C9×C22⋊C8C2semidirect
5. C9×#821C2semidirect
14. C2×C9×D12C22semidirect
15. C2×C3×C8×C9C22semidirect
17. C9×C4⋊Dic3C22semidirect
29. C9×C22⋊C8D3semidirect
44. C9×D12D4semidirect
47. #4732C9direct (N × H)
69. C2×C8×C9C2×D3semidirect
74. C22×C3×C9D8semidirect
87. C22×D12C2×C9semidirect
90. #930C2×C9semidirect
91. C2×D24C2×C9semidirect
94. C3×C22⋊C8C2×C9semidirect
96. #821C2×C9semidirect
155. C2×D12C22×C9semidirect
158. C2×C3×C8C22×C9semidirect
163. C4⋊Dic3C22×C9semidirect
192. C22×C9D24semidirect
203. C22⋊C8D3×C9semidirect
204. C3×C9#447semidirect
238. D12D4×C9semidirect
266. C2×C8C2×D3×C9semidirect
284. C22×C3C9×D8semidirect
305. C9#4732direct (N × H)
347. C22C9×D24semidirect
353. C3C9×#447semidirect

Maximal subgroups

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

subgroupindex
C22×C9×D122
C9×#9302
C2×C9×D242
C3×C9×C22⋊C82
C9×#8212
C3×#47323
C9×#4473