Finite Group Databasebeta

← C9×#4786 #13141721

Subgroups — C9×#4786 #13141721

Order 1728.

470 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#940C211yesmaximal, characteristic, complemented
2C2×C9×#213C211yesmaximal, characteristic, complemented
3C2×C9×D24C211yesmaximal, characteristic, complemented
4C9×#893C211yesmaximal, characteristic, complemented
5C3×C9×#102C211yesmaximal, characteristic, complemented
6C9×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C9×#809C211yesmaximal, characteristic, complemented
8C3×#4786C311yesmaximal, characteristic
9C9×#459—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C9×#67—21no—
12C2×C9×#67—21no—
13C2×C9×D12C2211yescharacteristic
14C2×C9×D12C2211yescharacteristic
15C2×C4×D3×C9—21no—
16C2×C3×C8×C9C2211yescharacteristic
17C3×C9×C4⋊C4C2211yescharacteristic
18C9×#213—22no—
19C9×(C2×D3)⋊C4—21no—
20C4×C9×Dic3C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22C9×D24—22no—
23C3×#940C2×C311yescharacteristic
24C2×C3×#213C2×C311yescharacteristic
25C2×C3×D24C2×C311yescharacteristic
26C3×#893C2×C311yescharacteristic
27C32×#102C2×C311yescharacteristic
28C3×Dic3⋊C8C2×C311yescharacteristic
29C3×#809C2×C311yescharacteristic
30C2×C9×D8—31no—
31C2×C9×D8—31no—
32C9×#127—31no—
33C4×D4×C9—31no—
34C9×C4⋊C8—31no—
35C9×#102—21no—
36C9×#102—31no—
37C9×#102D311yescharacteristic, complemented
38C23×C3×C9—21no—
39C22×D3×C9—21no—
40C22×D3×C9—21no—
41C3×D4×C9—21no—
42C3×D4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C9×#67—42no—
46C9×#67—42no—
47C2×C9×Dic3D411yescharacteristic
48C2×C9×Dic3D411yescharacteristic
49C9×D12—21no—
50C9×D12—21no—
51C9×D12—21no—
52C9×D12D412yescomplemented
53C4×D3×C9—42no—
54C3×C8×C9—21no—
55C9×C3⋊C8—21no—
56C3×#459—31no—
57#4786C911yescharacteristic, direct product factor
58C2×C32×D4C22×C311yescharacteristic
59C2×C3×#67—21no—
60C2×C3×#67—21no—
61C2×C3×D12C22×C311yescharacteristic
62C2×C3×D12C22×C311yescharacteristic
63C2×C3×C4×D3—21no—
64C2×C32×C8C22×C311yescharacteristic
65C32×C4⋊C4C22×C311yescharacteristic
66C3×#213—22no—
67C3×(C2×D3)⋊C4—21no—
68C3×C4×Dic3C22×C311yescharacteristic
69C2×C3×C3⋊C8C22×C311yescharacteristic
70C3×D24—22no—
71C2×D4×C9—21no—
72C2×D4×C9—31no—
73C2×D4×C9—31no—
74C2×D4×C9—61no—
75C2×D4×C9—61no—
76C2×D4×C9C2×D311yescharacteristic
77C22×C4×C9—61no—
78C9×D8—62no—
79C9×D8—62no—
80C2×C8×C9—21no—
81C2×C8×C9—31no—
82C2×C8×C9C2×D311yescharacteristic
83C9×C4⋊C4—21no—
84C9×C4⋊C4C2×D311yescharacteristic
85C9×C22⋊C4—61no—
86C42×C9—31no—
87C22×C3×C9—41no—
88C22×C3×C9—41no—
89C22×C3×C9—41no—
90C22×C3×C9C2×D411yescharacteristic
91C2×D3×C9—22no—
92C2×D3×C9—41no—
93C2×D3×C9—41no—
94C2×D3×C9—41no—
95C2×D3×C9—41no—
96C2×D3×C9—41no—
97C3×C4×C9—41no—
98C3×C4×C9C2×D411yescharacteristic
99C3×C4×C9#4111yescharacteristic
100C9×Dic3—21no—
101C9×Dic3D812yescomplemented
102C2×C3×D8—31no—
103C2×C3×D8—31no—
104C3×#127—31no—
105C3×C4×D4—31no—
106#940C2×C911yescharacteristic, complemented
107C2×#213C2×C911yescharacteristic, complemented
108C2×D24C2×C911yescharacteristic, complemented
109#893C2×C911yescharacteristic, complemented
110C3×C4⋊C8—31no—
111C3×#102—21no—
112C3×#102—31no—
113C3×#102C2×C911yescharacteristic, complemented
114C3×#102C3×D311yescharacteristic
115Dic3⋊C8C2×C911yescharacteristic, complemented
116#809C2×C911yescharacteristic, complemented
117C23×C32—21no—
118C22×C3×D3—21no—
119C22×C3×D3—21no—
120C32×D4—21no—
121C32×D4—21no—
122C2×C32×C4—21no—
123C2×C32×C4C23×C311yescharacteristic
124C3×#67—42no—
125C3×#67—42no—
126C2×C3×Dic3C3×D411yescharacteristic
127C2×C3×Dic3C3×D411yescharacteristic
128C3×D12—21no—
129C3×D12—21no—
130C3×D12—21no—
131C3×D12C3×D412yes—
132C3×C4×D3—42no—
133C23×C9—21no—
134C23×C9—41no—
135C23×C9—61no—
136C23×C9—61no—
137C32×C8—21no—
138C3×C3⋊C8—21no—
139D4×C9—22no—
140D4×C9—21no—
141D4×C9—21no—
142D4×C9—32no—
143D4×C9—41no—
144D4×C9—61no—
145D4×C9—61no—
146D4×C9—61no—
147D4×C9—122no—
148D4×C9—122no—
149C2×C4×C9—21no—
150C2×C4×C9—21no—
151C2×C4×C9—31no—
152C2×C4×C9—31no—
153C2×C4×C9—41no—
154C2×C4×C9—122no—
155C2×C4×C9C22×D311yescharacteristic
156C8×C9—21no—
157C8×C9—41no—
158C8×C9—61no—
159#459—31noSylow
160C2×C3×C9—81no—
161C2×C3×C9#12111yescharacteristic
162C2×C3×C9C2×D811yescharacteristic
163C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
164D3×C9—42no—
165D3×C9—81no—
166C2×C3×D4—21no—
167C2×C3×D4—31no—
168C2×C3×D4—31no—
169C2×C3×D4—61no—
170C2×C3×D4—61no—
171C2×C3×D4C22×C911yescharacteristic
172C2×C3×D4C2×C3×D311yescharacteristic
173C22×C3×C4—61no—
174C2×#67—21no—
175C2×#67—21no—
176C2×D12C22×C911yescharacteristic
177C2×D12C22×C911yescharacteristic
178C2×C4×D3—21no—
179C3×D8—62no—
180C3×D8—62no—
181C2×C3×C8—21no—
182C2×C3×C8—31no—
183C2×C3×C8C22×C911yescharacteristic
184C2×C3×C8C2×C3×D311yescharacteristic
185C3×C4⋊C4—21no—
186C3×C4⋊C4C22×C911yescharacteristic
187C3×C4⋊C4C2×C3×D311yescharacteristic
188C3×C22⋊C4—61no—
189C3×C42—31no—
190#213—22no—
191(C2×D3)⋊C4—21no—
192C4×Dic3C22×C911yescharacteristic
193C2×C3⋊C8C22×C911yescharacteristic
194D24—22no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32C2×C3×D411yescharacteristic
199C2×C3×D3—22no—
200C2×C3×D3—41no—
201C2×C3×D3—41no—
202C2×C3×D3—41no—
203C2×C3×D3—41no—
204C2×C3×D3—41no—
205C32×C4—41no—
206C32×C4C2×C3×D411yescharacteristic
207C32×C4C3×#4111yescharacteristic
208C3×Dic3—21no—
209C3×Dic3C3×D812yes—
210C22×C9—21no—
211C22×C9—42no—
212C22×C9—41no—
213C22×C9—41no—
214C22×C9—41no—
215C22×C9—62no—
216C22×C9—81no—
217C22×C9—81no—
218C22×C9—121no—
219C22×C9—121no—
220C22×C9—121no—
221C22×C9—121no—
222C22×C9—121no—
223C22×C9D3×D411yescharacteristic
224C4×C9—21no—
225C4×C9—21no—
226C4×C9—32no—
227C4×C9—41no—
228C4×C9—61no—
229C4×C9—81no—
230C4×C9#23511yescharacteristic
231C4×C9D3×D411yescharacteristic
232C2×D8—31no—
233C2×D8—31no—
234#127—31no—
235C4×D4—31no—
236C4⋊C8—31no—
237#102—31no—
238#102D3×C911yescharacteristic, complemented
239C3×C9#45911yescharacteristic, complemented, Sylow
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—41no—
243C23×C3—61no—
244C23×C3—61no—
245C22×D3—21no—
246C22×D3—21no—
247C3×D4—22no—
248C3×D4—21no—
249C3×D4—21no—
250C3×D4—21no—
251C3×D4—21no—
252C3×D4—32no—
253C3×D4—41no—
254C3×D4—61no—
255C3×D4—61no—
256C3×D4—61no—
257C3×D4—122no—
258C3×D4—122no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—41no—
265C2×C3×C4—122no—
266C2×C3×C4C23×C911yescharacteristic
267C2×C3×C4C22×C3×D311yescharacteristic
268#67—42no—
269#67—42no—
270C2×Dic3D4×C911yescharacteristic
271C2×Dic3D4×C911yescharacteristic
272D12—21no—
273D12—21no—
274D12—21no—
275D12D4×C912yescomplemented
276C4×D3—42no—
277C3×C8—21no—
278C3×C8—21no—
279C3×C8—41no—
280C3×C8—61no—
281C3⋊C8—21no—
282C2×C32—81no—
283C2×C32C3×#12111yescharacteristic
284C2×C32C2×C3×D811yescharacteristic
285C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
286C3×D3—42no—
287C3×D3—81no—
288C2×C9—21no—
289C2×C9—21no—
290C2×C9—21no—
291C2×C9—82no—
292C2×C9—81no—
293C2×C9—122no—
294C2×C9—241no—
295C2×C9#88411yescharacteristic
296C2×C9D3×D811yescharacteristic
297C2×C9#91411yescharacteristic
298C2×D4—31no—
299C2×D4—31no—
300C2×D4—61no—
301C2×D4—61no—
302C2×D4C2×D3×C911yescharacteristic
303C22×C4—61no—
304D8—62no—
305D8—62no—
306C2×C8—31no—
307C2×C8C2×D3×C911yescharacteristic
308C4⋊C4C2×D3×C911yescharacteristic
309C22⋊C4—61no—
310C42—31no—
311C22×C3—21no—
312C22×C3—42no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—62no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—121no—
323C22×C3—121no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3C2×D4×C911yescharacteristic
328C22×C3C3×D3×D411yescharacteristic
329C2×D3—22no—
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—41no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—32no—
338C3×C4—41no—
339C3×C4—41no—
340C3×C4—61no—
341C3×C4—81no—
342C3×C4C2×D4×C911yescharacteristic
343C3×C4C9×#4111yescharacteristic
344C3×C4C3×#23511yescharacteristic
345C3×C4C3×D3×D411yescharacteristic
346Dic3—21no—
347Dic3C9×D812yescomplemented
348C32C3×#45911yescharacteristic
349C9—21no—
350C9#478611yescharacteristic, direct product factor
351C23—21no—
352C23—61no—
353C23—61no—
354D4—21no—
355D4—21no—
356D4—32no—
357D4—61no—
358D4—61no—
359D4—61no—
360D4—122no—
361D4—122no—
362C2×C4—21no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—122no—
366C2×C4C22×D3×C911yescharacteristic
367C8—21no—
368C8—61no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—82no—
373C2×C3—81no—
374C2×C3—81no—
375C2×C3—122no—
376C2×C3—241no—
377C2×C3C9×#12111yescharacteristic
378C2×C3C2×C9×D811yescharacteristic
379C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
380C2×C3C3×#88411yescharacteristic
381C2×C3C3×D3×D811yescharacteristic
382C2×C3C3×#91411yescharacteristic
383D3—42no—
384D3—81no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—62no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22D3×D4×C911yescharacteristic
395C4—32no—
396C4—41no—
397C4—61no—
398C4C9×#23511yescharacteristic
399C4D3×D4×C911yescharacteristic
400C3—21no—
401C3C9×#45911yescharacteristic, complemented
402C3C3×#478611yescharacteristic
403C2—81no—
404C2—122no—
405C2—241no—
406C2C9×#88411yescharacteristic
407C2D3×C9×D811yescharacteristic
408C2C9×#91411yescharacteristic
subgroupG/H
centerC22×C9D3×D4
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#102C2
socleC22×C32C2×C3×D4
prime radicalC3×#4786C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#102D3
hypocenterC3C9×#459
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C9×#102

Complements

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

NH ≅ G/Ntype
1. C9×#940C2semidirect
2. C2×C9×#213C2semidirect
3. C2×C9×D24C2semidirect
4. C9×#893C2semidirect
5. C3×C9×#102C2semidirect
6. C9×Dic3⋊C8C2semidirect
7. C9×#809C2semidirect
37. C9×#102D3semidirect
52. C9×D12D4semidirect
57. #4786C9direct (N × H)
101. C9×Dic3D8semidirect
106. #940C2×C9semidirect
107. C2×#213C2×C9semidirect
108. C2×D24C2×C9semidirect
109. #893C2×C9semidirect
113. C3×#102C2×C9semidirect
115. Dic3⋊C8C2×C9semidirect
116. #809C2×C9semidirect
238. #102D3×C9semidirect
239. C3×C9#459semidirect
275. D12D4×C9semidirect
347. Dic3C9×D8semidirect
350. C9#4786direct (N × H)
401. C3C9×#459semidirect

Maximal subgroups

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

subgroupindex
C9×#9402
C2×C9×#2132
C2×C9×D242
C9×#8932
C3×C9×#1022
C9×Dic3⋊C82
C9×#8092
C3×#47863
C9×#4593