Finite Group Databasebeta

← C9×#5145 #13143255

Subgroups — C9×#5145 #13143255

Order 1728.

374 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#131C211yesmaximal, characteristic, complemented
2C9×#950C211yesmaximal, characteristic
3C9×#901C211yesmaximal, characteristic, complemented
4C9×#837C211yesmaximal, characteristic, complemented
5C9×#825C211yesmaximal, characteristic, complemented
6C9×#822C211yesmaximal, characteristic, complemented
7C9×#815C211yesmaximal, characteristic, complemented
8C3×#5145C311yesmaximal, characteristic
9C9×#444—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C9×#235C2211yescharacteristic
12C3×C9×M16—21no—
13C3×C9×M16C2211yescharacteristic
14C2×C3×C8×C9—21no—
15C2×C3×C8×C9C2211yescharacteristic
16C9×#216C411yescharacteristic, complemented
17C9×#215C411yescharacteristic, complemented
18C9×#214C411yescharacteristic, complemented
19C9×#213C411yescharacteristic, complemented
20C4×C9×Dic3C2211yescharacteristic
21C9×#208C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23C9×ΓL(1,𝔽25)—21no—
24D3×C8×C9—21no—
25C32×#131C2×C311yescharacteristic
26C3×#950C2×C311yescharacteristic
27C3×#901C2×C311yescharacteristic
28C3×#837C2×C311yescharacteristic
29C3×#825C2×C311yescharacteristic
30C3×#822C2×C311yescharacteristic
31C3×#815C2×C311yescharacteristic
32C9×#135—31no—
33C9×#131—21no—
34C9×#131—31no—
35C9×#131D311yescharacteristic, complemented
36C9×#108—31no—
37C9×#104—31no—
38C9×#104—31no—
39C9×#97—31no—
40C3×Q8×C9C2×C411yescharacteristic
41C3×D4×C9—21no—
42C3×D4×C9C2×C411yescharacteristic
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C9×#67—21no—
46C2×C9×Dic3—21no—
47C9×D12C2×C411yescharacteristic
48C4×D3×C9—21no—
49C9×Dic6C2×C411yescharacteristic
50C3×C8×C9—21no—
51C3×C8×C9D411yescharacteristic
52C3×C8×C9D411yescharacteristic
53C9×C3⋊C8—21no—
54C9×C3⋊C8C2×C411yescharacteristic
55C9×C3⋊C8C2×C411yescharacteristic
56C3×#444—31no—
57#5145C911yescharacteristic, direct product factor
58C32×#41C22×C311yescharacteristic
59C3×#235C22×C311yescharacteristic
60C32×M16—21no—
61C32×M16C22×C311yescharacteristic
62C2×C32×C8—21no—
63C2×C32×C8C22×C311yescharacteristic
64C3×#216C3×C411yescharacteristic
65C3×#215C3×C411yescharacteristic
66C3×#214C3×C411yescharacteristic
67C3×#213C3×C411yescharacteristic
68C3×C4×Dic3C22×C311yescharacteristic
69C3×#208C22×C311yescharacteristic
70C2×C3×C3⋊C8C22×C311yescharacteristic
71C3×ΓL(1,𝔽25)—21no—
72C3×D3×C8—21no—
73C9×#41—21no—
74C9×#41—31no—
75C9×#41C2×D311yescharacteristic
76C9×Q16—31no—
77C9×SD16—31no—
78C9×SD16—31no—
79C9×D8—31no—
80C9×M16—22no—
81C9×M16—21no—
82C9×M16—21no—
83C9×M16—31no—
84C9×M16—61no—
85C9×M16C2×D311yescharacteristic
86C2×C8×C9—22no—
87C2×C8×C9—21no—
88C2×C8×C9—21no—
89C2×C8×C9—31no—
90C2×C8×C9—61no—
91C2×C8×C9C2×D311yescharacteristic
92C42×C9—31no—
93C22×C3×C9—21no—
94C22×C3×C9#4111yescharacteristic
95C2×D3×C9—21no—
96C3×C4×C9—21no—
97C3×C4×C9C22×C411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C9×Dic3—21no—
100C9×Dic3—41no—
101C3×#135—31no—
102C3×#131—21no—
103C3×#131—31no—
104C3×#131C2×C911yescharacteristic, complemented
105C3×#131C3×D311yescharacteristic
106#950C2×C911yescharacteristic
107#901C2×C911yescharacteristic, complemented
108C3×#108—31no—
109C3×#104—31no—
110C3×#104—31no—
111C3×#97—31no—
112#837C2×C911yescharacteristic, complemented
113#825C2×C911yescharacteristic, complemented
114#822C2×C911yescharacteristic, complemented
115#815C2×C911yescharacteristic, complemented
116C32×Q8C2×C3×C411yescharacteristic
117C32×D4—21no—
118C32×D4C2×C3×C411yescharacteristic
119C2×C32×C4—21no—
120C2×C32×C4C23×C311yescharacteristic
121C3×#67—21no—
122C2×C3×Dic3—21no—
123C3×D12C2×C3×C411yescharacteristic
124C3×C4×D3—21no—
125C3×Dic6C2×C3×C411yescharacteristic
126C32×C8—21no—
127C32×C8C3×D411yescharacteristic
128C32×C8C3×D411yescharacteristic
129C3×C3⋊C8—21no—
130C3×C3⋊C8C2×C3×C411yescharacteristic
131C3×C3⋊C8C2×C3×C411yescharacteristic
132Q8×C9—21no—
133Q8×C9—31no—
134Q8×C9C4×D311yescharacteristic
135D4×C9—22no—
136D4×C9—21no—
137D4×C9—21no—
138D4×C9—31no—
139D4×C9—61no—
140D4×C9C4×D311yescharacteristic
141C2×C4×C9—22no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—61no—
145C2×C4×C9—61no—
146C2×C4×C9C22×D311yescharacteristic
147C8×C9—22no—
148C8×C9—21no—
149C8×C9—21no—
150C8×C9—21no—
151C8×C9—31no—
152C8×C9—31no—
153C8×C9—61no—
154C8×C9#6711yescharacteristic
155C8×C9#6711yescharacteristic
156#444—31noSylow
157C2×C3×C9—21no—
158C2×C3×C9—41no—
159C2×C3×C9C4×D411yescharacteristic
160D3×C9—41no—
161C3×#41—21no—
162C3×#41—31no—
163C3×#41C22×C911yescharacteristic
164C3×#41C2×C3×D311yescharacteristic
165#235C22×C911yescharacteristic
166C3×Q16—31no—
167C3×SD16—31no—
168C3×SD16—31no—
169C3×D8—31no—
170C3×M16—22no—
171C3×M16—21no—
172C3×M16—21no—
173C3×M16—21no—
174C3×M16—31no—
175C3×M16—61no—
176C3×M16C22×C911yescharacteristic
177C3×M16C2×C3×D311yescharacteristic
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—31no—
183C2×C3×C8—61no—
184C2×C3×C8C22×C911yescharacteristic
185C2×C3×C8C2×C3×D311yescharacteristic
186C3×C42—31no—
187#216C4×C911yescharacteristic, complemented
188#215C4×C911yescharacteristic, complemented
189#214C4×C911yescharacteristic, complemented
190#213C4×C911yescharacteristic, complemented
191C4×Dic3C22×C911yescharacteristic
192#208C22×C911yescharacteristic
193C2×C3⋊C8C22×C911yescharacteristic
194ΓL(1,𝔽25)—21no—
195D3×C8—21no—
196C22×C32—21no—
197C22×C32C3×#4111yescharacteristic
198C2×C3×D3—21no—
199C32×C4—21no—
200C32×C4C22×C3×C411yescharacteristic
201C32×C4C2×C3×D411yescharacteristic
202C3×Dic3—21no—
203C3×Dic3—41no—
204C22×C9—22no—
205C22×C9—21no—
206C22×C9—21no—
207C22×C9—61no—
208C22×C9#23511yescharacteristic
209C4×C9—22no—
210C4×C9—21no—
211C4×C9—21no—
212C4×C9—21no—
213C4×C9—61no—
214C4×C9—121no—
215C4×C9C2×C4×D311yescharacteristic
216C4×C9C2×#6711yescharacteristic
217#135—31no—
218#131—31no—
219#131D3×C911yescharacteristic, complemented
220#108—31no—
221#104—31no—
222#104—31no—
223#97—31no—
224C3×C9#44411yescharacteristic, complemented, Sylow
225C3×Q8—21no—
226C3×Q8—31no—
227C3×Q8C2×C4×C911yescharacteristic
228C3×Q8C3×C4×D311yescharacteristic
229C3×D4—22no—
230C3×D4—21no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—31no—
234C3×D4—61no—
235C3×D4C2×C4×C911yescharacteristic
236C3×D4C3×C4×D311yescharacteristic
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4C23×C911yescharacteristic
244C2×C3×C4C22×C3×D311yescharacteristic
245#67—21no—
246C2×Dic3—21no—
247D12C2×C4×C911yescharacteristic
248C4×D3—21no—
249Dic6C2×C4×C911yescharacteristic
250C3×C8—22no—
251C3×C8—21no—
252C3×C8—21no—
253C3×C8—21no—
254C3×C8—21no—
255C3×C8—31no—
256C3×C8—31no—
257C3×C8—61no—
258C3×C8D4×C911yescharacteristic
259C3×C8D4×C911yescharacteristic
260C3×C8C3×#6711yescharacteristic
261C3×C8C3×#6711yescharacteristic
262C3⋊C8—21no—
263C3⋊C8C2×C4×C911yescharacteristic
264C3⋊C8C2×C4×C911yescharacteristic
265C2×C32—21no—
266C2×C32—41no—
267C2×C32C3×C4×D411yescharacteristic
268C3×D3—41no—
269C2×C9—21no—
270C2×C9—21no—
271C2×C9—41no—
272C2×C9—42no—
273C2×C9—41no—
274C2×C9—121no—
275C2×C9C4×#6711yescharacteristic
276#41—31no—
277#41C2×D3×C911yescharacteristic
278Q16—31no—
279SD16—31no—
280SD16—31no—
281D8—31no—
282M16—21no—
283M16—31no—
284M16—61no—
285M16C2×D3×C911yescharacteristic
286C2×C8—21no—
287C2×C8—31no—
288C2×C8—61no—
289C2×C8C2×D3×C911yescharacteristic
290C42—31no—
291C22×C3—22no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—61no—
296C22×C3C9×#4111yescharacteristic
297C22×C3C3×#23511yescharacteristic
298C2×D3—21no—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—61no—
305C3×C4—121no—
306C3×C4C22×C4×C911yescharacteristic
307C3×C4C2×D4×C911yescharacteristic
308C3×C4C2×C3×C4×D311yescharacteristic
309C3×C4C2×C3×#6711yescharacteristic
310Dic3—21no—
311Dic3—41no—
312C32C3×#44411yescharacteristic
313C9—21no—
314C9#514511yescharacteristic, direct product factor
315Q8—31no—
316Q8C4×D3×C911yescharacteristic
317D4—21no—
318D4—31no—
319D4—61no—
320D4C4×D3×C911yescharacteristic
321C2×C4—21no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4C22×D3×C911yescharacteristic
325C8—21no—
326C8—31no—
327C8—31no—
328C8—61no—
329C8C9×#6711yescharacteristic
330C8C9×#6711yescharacteristic
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—41no—
335C2×C3—42no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—121no—
339C2×C3C4×D4×C911yescharacteristic
340C2×C3C3×C4×#6711yescharacteristic
341D3—41no—
342C22—21no—
343C22—61no—
344C22C9×#23511yescharacteristic
345C4—21no—
346C4—61no—
347C4—121no—
348C4C2×C4×D3×C911yescharacteristic
349C4C2×C9×#6711yescharacteristic
350C3—21no—
351C3C9×#44411yescharacteristic, complemented
352C3C3×#514511yescharacteristic
353C2—21no—
354C2—41no—
355C2—121no—
356C2C4×C9×#6711yescharacteristic
subgroupG/H
centerC4×C9C2×#67
commutatorC3×C4C22×C4×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#131C2
socleC2×C32C3×C4×D4
prime radicalC3×#950C2×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#131D3
hypocenterC3C9×#444
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C9 ◁ C2×C8×C9 ◁ C9×#131

Complements

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

NH ≅ G/Ntype
1. C3×C9×#131C2semidirect
3. C9×#901C2semidirect
4. C9×#837C2semidirect
5. C9×#825C2semidirect
6. C9×#822C2semidirect
7. C9×#815C2semidirect
16. C9×#216C4semidirect
17. C9×#215C4semidirect
18. C9×#214C4semidirect
19. C9×#213C4semidirect
35. C9×#131D3semidirect
57. #5145C9direct (N × H)
104. C3×#131C2×C9semidirect
107. #901C2×C9semidirect
112. #837C2×C9semidirect
113. #825C2×C9semidirect
114. #822C2×C9semidirect
115. #815C2×C9semidirect
187. #216C4×C9semidirect
188. #215C4×C9semidirect
189. #214C4×C9semidirect
190. #213C4×C9semidirect
219. #131D3×C9semidirect
224. C3×C9#444semidirect
314. C9#5145direct (N × H)
351. C3C9×#444semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1312
C9×#9502
C9×#9012
C9×#8372
C9×#8252
C9×#8222
C9×#8152
C3×#51453
C9×#4443