Finite Group Databasebeta

← C9×#4812 #13141889

Subgroups — C9×#4812 #13141889

Order 1728.

402 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#946C211yesmaximal, characteristic, complemented
2C2×C9×#215C211yesmaximal, characteristic, complemented
3C2×C9×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
4C9×#895C211yesmaximal, characteristic
5C3×C9×#103C211yesmaximal, characteristic, complemented
6C9×#821C211yesmaximal, characteristic, complemented
7C9×#807C211yesmaximal, characteristic, complemented
8C3×#4812C311yesmaximal, characteristic
9C9×#468—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C9×D12—21no—
12C2×C9×D12C2211yescharacteristic
13C2×C4×D3×C9C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C9×#215—22no—
17C9×(C2×D3)⋊C4—21no—
18C9×(C2×D3)⋊C4—21no—
19C9×C4⋊Dic3C2211yescharacteristic
20C9×Dic3⋊C4—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C9×ΓL(1,𝔽25)—22no—
23C3×#946C2×C311yescharacteristic
24C2×C3×#215C2×C311yescharacteristic
25C2×C3×ΓL(1,𝔽25)C2×C311yescharacteristic
26C3×#895C2×C311yescharacteristic
27C32×#103C2×C311yescharacteristic
28C3×#821C2×C311yescharacteristic
29C3×#807C2×C311yescharacteristic
30C2×C9×SD16—31no—
31C2×C9×M16—31no—
32C9×C22⋊Q8—31no—
33C9×#121—31no—
34C9×#107—31no—
35C9×#103—21no—
36C9×#103D311yescharacteristic, complemented
37C9×#102—31no—
38C22×D3×C9—21no—
39C22×D3×C9D411yescharacteristic
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C9×Dic3—21no—
46C2×C9×Dic3D411yescharacteristic
47C9×D12—21no—
48C9×D12—21no—
49C9×D12—42no—
50C4×D3×C9—21no—
51C4×D3×C9—21no—
52C3×C8×C9—21no—
53C9×C3⋊C8D412yescomplemented
54C3×#468—31no—
55#4812C911yescharacteristic, direct product factor
56C2×C32×Q8C22×C311yescharacteristic
57C2×C3×D12—21no—
58C2×C3×D12C22×C311yescharacteristic
59C2×C3×C4×D3C22×C311yescharacteristic
60C2×C32×C8C22×C311yescharacteristic
61C32×C4⋊C4C22×C311yescharacteristic
62C3×#215—22no—
63C3×(C2×D3)⋊C4—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×C4⋊Dic3C22×C311yescharacteristic
66C3×Dic3⋊C4—21no—
67C2×C3×C3⋊C8C22×C311yescharacteristic
68C3×ΓL(1,𝔽25)—22no—
69C2×Q8×C9—21no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×D4×C9—31no—
72C2×D4×C9—61no—
73C22×C4×C9—31no—
74C9×SD16—62no—
75C9×M16—62no—
76C2×C8×C9—21no—
77C2×C8×C9—31no—
78C2×C8×C9C2×D311yescharacteristic
79C9×C4⋊C4—21no—
80C9×C4⋊C4—31no—
81C9×C4⋊C4—61no—
82C9×C4⋊C4C2×D311yescharacteristic
83C9×C22⋊C4—61no—
84C9×C22⋊C4—61no—
85C22×C3×C9C2×D411yescharacteristic
86C2×D3×C9—21no—
87C2×D3×C9—21no—
88C2×D3×C9—21no—
89C2×D3×C9—41no—
90C2×D3×C9—41no—
91C2×D3×C9—41no—
92C3×C4×C9—41no—
93C3×C4×C9—41no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C9×Dic3—21no—
97C9×Dic3—41no—
98C2×C3×SD16—31no—
99C2×C3×M16—31no—
100C3×C22⋊Q8—31no—
101C3×#121—31no—
102#946C2×C911yescharacteristic, complemented
103C2×#215C2×C911yescharacteristic, complemented
104C2×ΓL(1,𝔽25)C2×C911yescharacteristic, complemented
105#895C2×C911yescharacteristic
106C3×#107—31no—
107C3×#103—21no—
108C3×#103C2×C911yescharacteristic, complemented
109C3×#103C3×D311yescharacteristic
110C3×#102—31no—
111#821C2×C911yescharacteristic, complemented
112#807C2×C911yescharacteristic, complemented
113C22×C3×D3—21no—
114C22×C3×D3C3×D411yescharacteristic
115C32×Q8—21no—
116C32×Q8—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4C23×C311yescharacteristic
120C2×C3×Dic3—21no—
121C2×C3×Dic3C3×D411yescharacteristic
122C3×D12—21no—
123C3×D12—21no—
124C3×D12—42no—
125C3×C4×D3—21no—
126C3×C4×D3—21no—
127C23×C9—31no—
128C23×C9—61no—
129C32×C8—21no—
130C3×C3⋊C8C3×D412yes—
131Q8×C9—22no—
132Q8×C9—21no—
133Q8×C9—21no—
134Q8×C9—41no—
135D4×C9—61no—
136D4×C9—61no—
137D4×C9—122no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—21no—
141C2×C4×C9—31no—
142C2×C4×C9—41no—
143C2×C4×C9—41no—
144C2×C4×C9—61no—
145C2×C4×C9—61no—
146C2×C4×C9—61no—
147C2×C4×C9C22×D311yescharacteristic
148C8×C9—21no—
149C8×C9—32no—
150C8×C9—41no—
151#468—31noSylow
152C2×C3×C9#12111yescharacteristic
153C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
154C2×C3×C9#13711yescharacteristic
155D3×C9—41no—
156D3×C9—81no—
157C2×C3×Q8—21no—
158C2×C3×Q8C22×C911yescharacteristic
159C2×C3×Q8C2×C3×D311yescharacteristic
160C2×C3×D4—31no—
161C2×C3×D4—61no—
162C22×C3×C4—31no—
163C2×D12—21no—
164C2×D12C22×C911yescharacteristic
165C2×C4×D3C22×C911yescharacteristic
166C3×SD16—62no—
167C3×M16—62no—
168C2×C3×C8—21no—
169C2×C3×C8—31no—
170C2×C3×C8C22×C911yescharacteristic
171C2×C3×C8C2×C3×D311yescharacteristic
172C3×C4⋊C4—21no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—61no—
175C3×C4⋊C4C22×C911yescharacteristic
176C3×C4⋊C4C2×C3×D311yescharacteristic
177C3×C22⋊C4—61no—
178C3×C22⋊C4—61no—
179#215—22no—
180(C2×D3)⋊C4—21no—
181(C2×D3)⋊C4—21no—
182C4⋊Dic3C22×C911yescharacteristic
183Dic3⋊C4—21no—
184C2×C3⋊C8C22×C911yescharacteristic
185ΓL(1,𝔽25)—22no—
186C22×C32C2×C3×D411yescharacteristic
187C2×C3×D3—21no—
188C2×C3×D3—21no—
189C2×C3×D3—21no—
190C2×C3×D3—41no—
191C2×C3×D3—41no—
192C2×C3×D3—41no—
193C32×C4—41no—
194C32×C4—41no—
195C32×C4C2×C3×D411yescharacteristic
196C32×C4C3×#4111yescharacteristic
197C3×Dic3—21no—
198C3×Dic3—41no—
199C22×C9—21no—
200C22×C9—61no—
201C22×C9—61no—
202C22×C9—61no—
203C22×C9—121no—
204C22×C9—121no—
205C22×C9—121no—
206C22×C9D3×D411yescharacteristic
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—42no—
210C4×C9—41no—
211C4×C9—41no—
212C4×C9—61no—
213C4×C9—81no—
214C4×C9—121no—
215C4×C9#23511yescharacteristic
216C4×C9D3×D411yescharacteristic
217C2×SD16—31no—
218C2×M16—31no—
219C22⋊Q8—31no—
220#121—31no—
221#107—31no—
222#103D3×C911yescharacteristic, complemented
223#102—31no—
224C3×C9#46811yescharacteristic, complemented, Sylow
225C23×C3—31no—
226C23×C3—61no—
227C22×D3—21no—
228C22×D3D4×C911yescharacteristic
229C3×Q8—22no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—21no—
234C3×Q8—41no—
235C3×D4—61no—
236C3×D4—61no—
237C3×D4—122no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—31no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4C23×C911yescharacteristic
250C2×C3×C4C22×C3×D311yescharacteristic
251C2×Dic3—21no—
252C2×Dic3D4×C911yescharacteristic
253D12—21no—
254D12—21no—
255D12—42no—
256C4×D3—21no—
257C4×D3—21no—
258C3×C8—21no—
259C3×C8—21no—
260C3×C8—32no—
261C3×C8—41no—
262C3⋊C8D4×C912yescomplemented
263C2×C32C3×#12111yescharacteristic
264C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
265C2×C32C3×#13711yescharacteristic
266C3×D3—41no—
267C3×D3—81no—
268C2×C9—21no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9—121no—
272C2×C9—241no—
273C2×C9#88411yescharacteristic
274C2×C9#91411yescharacteristic
275C2×C9#91811yescharacteristic
276C2×Q8C2×D3×C911yescharacteristic
277C2×D4—31no—
278C2×D4—61no—
279C22×C4—31no—
280SD16—62no—
281M16—62no—
282C2×C8—31no—
283C2×C8C2×D3×C911yescharacteristic
284C4⋊C4—31no—
285C4⋊C4—61no—
286C4⋊C4C2×D3×C911yescharacteristic
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22×C3—21no—
290C22×C3—61no—
291C22×C3—61no—
292C22×C3—61no—
293C22×C3—121no—
294C22×C3—121no—
295C22×C3—121no—
296C22×C3C2×D4×C911yescharacteristic
297C22×C3C3×D3×D411yescharacteristic
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—41no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—42no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4—61no—
312C3×C4—81no—
313C3×C4—121no—
314C3×C4C2×D4×C911yescharacteristic
315C3×C4C9×#4111yescharacteristic
316C3×C4C3×#23511yescharacteristic
317C3×C4C3×D3×D411yescharacteristic
318Dic3—21no—
319Dic3—41no—
320C32C3×#46811yescharacteristic
321C9—21no—
322C9#481211yescharacteristic, direct product factor
323C23—31no—
324C23—61no—
325Q8—21no—
326Q8—21no—
327D4—61no—
328D4—61no—
329D4—122no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—31no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4C22×D3×C911yescharacteristic
337C8—21no—
338C8—32no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—121no—
343C2×C3—241no—
344C2×C3C9×#12111yescharacteristic
345C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
346C2×C3C9×#13711yescharacteristic
347C2×C3C3×#88411yescharacteristic
348C2×C3C3×#91411yescharacteristic
349C2×C3C3×#91811yescharacteristic
350D3—41no—
351D3—81no—
352C22—61no—
353C22—61no—
354C22—61no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22D3×D4×C911yescharacteristic
359C4—41no—
360C4—41no—
361C4—61no—
362C4—121no—
363C4C9×#23511yescharacteristic
364C4D3×D4×C911yescharacteristic
365C3—21no—
366C3C9×#46811yescharacteristic, complemented
367C3C3×#481211yescharacteristic
368C2—121no—
369C2—241no—
370C2C9×#88411yescharacteristic
371C2C9×#91411yescharacteristic
372C2C9×#91811yescharacteristic
subgroupG/H
centerC22×C9D3×D4
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#103C2
socleC22×C32C2×C3×D4
prime radicalC3×#895C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#103D3
hypocenterC3C9×#468
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C9×#103

Complements

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

NH ≅ G/Ntype
1. C9×#946C2semidirect
2. C2×C9×#215C2semidirect
3. C2×C9×ΓL(1,𝔽25)C2semidirect
5. C3×C9×#103C2semidirect
6. C9×#821C2semidirect
7. C9×#807C2semidirect
36. C9×#103D3semidirect
53. C9×C3⋊C8D4semidirect
55. #4812C9direct (N × H)
102. #946C2×C9semidirect
103. C2×#215C2×C9semidirect
104. C2×ΓL(1,𝔽25)C2×C9semidirect
108. C3×#103C2×C9semidirect
111. #821C2×C9semidirect
112. #807C2×C9semidirect
222. #103D3×C9semidirect
224. C3×C9#468semidirect
262. C3⋊C8D4×C9semidirect
322. C9#4812direct (N × H)
366. C3C9×#468semidirect

Maximal subgroups

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

subgroupindex
C9×#9462
C2×C9×#2152
C2×C9×ΓL(1,𝔽25)2
C9×#8952
C3×C9×#1032
C9×#8212
C9×#8072
C3×#48123
C9×#4683