Finite Group Databasebeta

← C9×#5134 #13143191

Subgroups — C9×#5134 #13143191

Order 1728.

428 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×M16C211yesmaximal, characteristic, complemented
2C9×#930C211yesmaximal, characteristic
3C9×#924C211yesmaximal, characteristic, complemented
4C2×C9×#204C211yesmaximal, characteristic, complemented
5C9×#821C211yesmaximal, characteristic, complemented
6C9×#818C211yesmaximal, characteristic, complemented
7C9×#816C211yesmaximal, characteristic, complemented
8C3×#5134C311yesmaximal, characteristic
9C9×#468—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×C9×#67—21no—
12C2×C9×D12C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C3×C9×M16—22no—
15C2×C3×C8×C9C2211yescharacteristic
16C2×C3×C8×C9C2211yescharacteristic
17C9×#217—21no—
18C9×(C2×D3)⋊C4—21no—
19C9×C4⋊Dic3C2211yescharacteristic
20C9×C4⋊Dic3C2211yescharacteristic
21C9×Dic3⋊C4—21no—
22C9×#204—22no—
23C2×C32×M16C2×C311yescharacteristic
24C3×#930C2×C311yescharacteristic
25C3×#924C2×C311yescharacteristic
26C2×C3×#204C2×C311yescharacteristic
27C3×#821C2×C311yescharacteristic
28C3×#818C2×C311yescharacteristic
29C3×#816C2×C311yescharacteristic
30C2×C9×SD16—31no—
31C2×C9×M16—21no—
32C2×C9×M16D311yescharacteristic, complemented
33C9×C22⋊Q8—31no—
34C9×#121—31no—
35C9×#107—31no—
36C9×#103—31no—
37C9×#102—31no—
38C23×C3×C9D411yescharacteristic
39C22×D3×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9D411yescharacteristic
43C2×C3×C4×C9C2311yescharacteristic
44C9×#67—42no—
45C2×C9×Dic3—21no—
46C2×C9×Dic3—21no—
47C2×C9×Dic3—21no—
48C9×D12—21no—
49C9×D12—21no—
50C9×Dic6—21no—
51C9×Dic6—21no—
52C3×C8×C9—21no—
53C3×C8×C9D412yescomplemented
54C3×#468—31no—
55#5134C911yescharacteristic, direct product factor
56C22×C32×C4C22×C311yescharacteristic
57C2×C3×#67—21no—
58C2×C3×D12C22×C311yescharacteristic
59C2×C3×Dic6C22×C311yescharacteristic
60C32×M16—22no—
61C2×C32×C8C22×C311yescharacteristic
62C2×C32×C8C22×C311yescharacteristic
63C3×#217—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×C4⋊Dic3C22×C311yescharacteristic
66C3×C4⋊Dic3C22×C311yescharacteristic
67C3×Dic3⋊C4—21no—
68C3×#204—22no—
69C2×Q8×C9—31no—
70C2×D4×C9—31no—
71C2×D4×C9—61no—
72C22×C4×C9—21no—
73C22×C4×C9C2×D311yescharacteristic
74C9×SD16—62no—
75C9×M16—24no—
76C9×M16—22no—
77C2×C8×C9—21no—
78C2×C8×C9—21no—
79C2×C8×C9C2×D311yescharacteristic
80C2×C8×C9C2×D311yescharacteristic
81C9×C4⋊C4—31no—
82C9×C4⋊C4—31no—
83C9×C4⋊C4—61no—
84C9×C22⋊C4—61no—
85C9×C22⋊C4—61no—
86C22×C3×C9—21no—
87C22×C3×C9—21no—
88C22×C3×C9—21no—
89C22×C3×C9C2×D411yescharacteristic
90C2×D3×C9—41no—
91C2×D3×C9—41no—
92C2×D3×C9—41no—
93C3×C4×C9—21no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C9×Dic3—41no—
97C9×Dic3—41no—
98C9×Dic3—41no—
99C2×C3×SD16—31no—
100C2×C3×M16—21no—
101C2×C3×M16C2×C911yescharacteristic, complemented
102C2×C3×M16C3×D311yescharacteristic
103C3×C22⋊Q8—31no—
104C3×#121—31no—
105#930C2×C911yescharacteristic
106#924C2×C911yescharacteristic, complemented
107C2×#204C2×C911yescharacteristic, complemented
108C3×#107—31no—
109C3×#103—31no—
110C3×#102—31no—
111#821C2×C911yescharacteristic, complemented
112#818C2×C911yescharacteristic, complemented
113#816C2×C911yescharacteristic, complemented
114C23×C32C3×D411yescharacteristic
115C22×C3×D3—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4C3×D411yescharacteristic
119C2×C32×C4C23×C311yescharacteristic
120C3×#67—42no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—21no—
124C3×D12—21no—
125C3×D12—21no—
126C3×Dic6—21no—
127C3×Dic6—21no—
128C23×C9—21no—
129C23×C9—61no—
130C23×C9D1211yescharacteristic
131C32×C8—21no—
132C32×C8C3×D412yes—
133Q8×C9—61no—
134Q8×C9—61no—
135D4×C9—61no—
136D4×C9—61no—
137D4×C9—122no—
138C2×C4×C9—22no—
139C2×C4×C9—22no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—61no—
145C2×C4×C9—61no—
146C2×C4×C9—61no—
147C2×C4×C9D1211yescharacteristic
148C2×C4×C9C22×D311yescharacteristic
149C8×C9—22no—
150C8×C9—21no—
151C8×C9—22no—
152C8×C9#6712yescomplemented
153#468—31noSylow
154C2×C3×C9—41no—
155C2×C3×C9#12111yescharacteristic
156C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
157C2×C3×C9#13711yescharacteristic
158D3×C9—81no—
159C2×C3×Q8—31no—
160C2×C3×D4—31no—
161C2×C3×D4—61no—
162C22×C3×C4—21no—
163C22×C3×C4C22×C911yescharacteristic
164C22×C3×C4C2×C3×D311yescharacteristic
165C2×#67—21no—
166C2×D12C22×C911yescharacteristic
167C2×Dic6C22×C911yescharacteristic
168C3×SD16—62no—
169C3×M16—24no—
170C3×M16—22no—
171C3×M16—22no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8C22×C911yescharacteristic
175C2×C3×C8C22×C911yescharacteristic
176C2×C3×C8C2×C3×D311yescharacteristic
177C2×C3×C8C2×C3×D311yescharacteristic
178C3×C4⋊C4—31no—
179C3×C4⋊C4—31no—
180C3×C4⋊C4—61no—
181C3×C22⋊C4—61no—
182C3×C22⋊C4—61no—
183#217—21no—
184(C2×D3)⋊C4—21no—
185C4⋊Dic3C22×C911yescharacteristic
186C4⋊Dic3C22×C911yescharacteristic
187Dic3⋊C4—21no—
188#204—22no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32C2×C3×D411yescharacteristic
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C32×C4—21no—
197C32×C4C2×C3×D411yescharacteristic
198C32×C4C3×#4111yescharacteristic
199C3×Dic3—41no—
200C3×Dic3—41no—
201C3×Dic3—41no—
202C22×C9—22no—
203C22×C9—21no—
204C22×C9—21no—
205C22×C9—21no—
206C22×C9—21no—
207C22×C9—41no—
208C22×C9—41no—
209C22×C9—121no—
210C22×C9—121no—
211C22×C9—121no—
212C22×C9C2×D1211yescharacteristic
213C4×C9—22no—
214C4×C9—21no—
215C4×C9—21no—
216C4×C9—21no—
217C4×C9—121no—
218C4×C9—121no—
219C4×C9—121no—
220C4×C9#23511yescharacteristic
221C4×C9C2×#6711yescharacteristic
222C2×SD16—31no—
223C2×M16D3×C911yescharacteristic, complemented
224C22⋊Q8—31no—
225#121—31no—
226#107—31no—
227#103—31no—
228#102—31no—
229C3×C9#46811yescharacteristic, complemented, Sylow
230C23×C3—21no—
231C23×C3—61no—
232C23×C3D4×C911yescharacteristic
233C23×C3C3×D1211yescharacteristic
234C22×D3—21no—
235C3×Q8—61no—
236C3×Q8—61no—
237C3×D4—61no—
238C3×D4—61no—
239C3×D4—122no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4D4×C911yescharacteristic
252C2×C3×C4C23×C911yescharacteristic
253C2×C3×C4C3×D1211yescharacteristic
254C2×C3×C4C22×C3×D311yescharacteristic
255#67—42no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259D12—21no—
260D12—21no—
261Dic6—21no—
262Dic6—21no—
263C3×C8—22no—
264C3×C8—21no—
265C3×C8—21no—
266C3×C8—22no—
267C3×C8D4×C912yescomplemented
268C3×C8C3×#6712yes—
269C2×C32—41no—
270C2×C32C3×#12111yescharacteristic
271C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
272C2×C32C3×#13711yescharacteristic
273C3×D3—81no—
274C2×C9—21no—
275C2×C9—21no—
276C2×C9—21no—
277C2×C9—42no—
278C2×C9—41no—
279C2×C9—241no—
280C2×C9#90811yescharacteristic
281C2×C9#90911yescharacteristic
282C2×C9#93011yescharacteristic
283C2×Q8—31no—
284C2×D4—31no—
285C2×D4—61no—
286C22×C4C2×D3×C911yescharacteristic
287SD16—62no—
288M16—22no—
289C2×C8C2×D3×C911yescharacteristic
290C2×C8C2×D3×C911yescharacteristic
291C4⋊C4—31no—
292C4⋊C4—31no—
293C4⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C22×C3—22no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—121no—
307C22×C3—121no—
308C22×C3—121no—
309C22×C3C2×D4×C911yescharacteristic
310C22×C3C2×C3×D1211yescharacteristic
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C3×C4—22no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—121no—
320C3×C4—121no—
321C3×C4—121no—
322C3×C4C2×D4×C911yescharacteristic
323C3×C4C9×#4111yescharacteristic
324C3×C4C3×#23511yescharacteristic
325C3×C4C2×C3×#6711yescharacteristic
326Dic3—41no—
327Dic3—41no—
328Dic3—41no—
329C32C3×#46811yescharacteristic
330C9—21no—
331C9#513411yescharacteristic, direct product factor
332C23—61no—
333C23C9×D1211yescharacteristic
334Q8—61no—
335Q8—61no—
336D4—61no—
337D4—61no—
338D4—122no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4C9×D1211yescharacteristic
345C2×C4C22×D3×C911yescharacteristic
346C8—21no—
347C8C9×#6712yescomplemented
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—42no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—241no—
355C2×C3C9×#12111yescharacteristic
356C2×C3C9×AGL(1,ℤ/8)11yescharacteristic
357C2×C3C9×#13711yescharacteristic
358C2×C3C3×#90811yescharacteristic
359C2×C3C3×#90911yescharacteristic
360C2×C3C3×#93011yescharacteristic
361D3—81no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22C2×C9×D1211yescharacteristic
369C4—21no—
370C4—121no—
371C4—121no—
372C4—121no—
373C4C9×#23511yescharacteristic
374C4C2×C9×#6711yescharacteristic
375C3—21no—
376C3C9×#46811yescharacteristic, complemented
377C3C3×#513411yescharacteristic
378C2—41no—
379C2—241no—
380C2C9×#90811yescharacteristic
381C2C9×#90911yescharacteristic
382C2C9×#93011yescharacteristic
subgroupG/H
centerC22×C9C2×D12
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C3×C9×M16C2
socleC22×C32C2×C3×D4
prime radicalC3×#930C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC2×C9×M16D3
hypocenterC3C9×#468
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C22×C4×C9 ◁ C2×C9×M16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×M16C2semidirect
3. C9×#924C2semidirect
4. C2×C9×#204C2semidirect
5. C9×#821C2semidirect
6. C9×#818C2semidirect
7. C9×#816C2semidirect
32. C2×C9×M16D3semidirect
53. C3×C8×C9D4semidirect
55. #5134C9direct (N × H)
101. C2×C3×M16C2×C9semidirect
106. #924C2×C9semidirect
107. C2×#204C2×C9semidirect
111. #821C2×C9semidirect
112. #818C2×C9semidirect
113. #816C2×C9semidirect
152. C8×C9#67semidirect
223. C2×M16D3×C9semidirect
229. C3×C9#468semidirect
267. C3×C8D4×C9semidirect
331. C9#5134direct (N × H)
347. C8C9×#67semidirect
376. C3C9×#468semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×M162
C9×#9302
C9×#9242
C2×C9×#2042
C9×#8212
C9×#8182
C9×#8162
C3×#51343
C9×#4683