Finite Group Databasebeta

← C9×#5204 #13143494

Subgroups — C9×#5204 #13143494

Order 1728.

360 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#137C211yesmaximal, characteristic
2C9×#951C211yesmaximal, characteristic
3C9×#948C211yesmaximal, characteristic
4C2×C9×#216C211yesmaximal, characteristic, complemented
5C9×#836C211yesmaximal, characteristic, complemented
6C9×#824C211yesmaximal, characteristic, complemented
7C9×#822C211yesmaximal, characteristic, complemented
8C3×#5204C311yesmaximal, characteristic
9C9×#473—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C2×C3×Q8×C9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C3×C9×Q16—21no—
14C3×C9×SD16—21no—
15C3×C9×M16C2211yescharacteristic
16C9×#216—21no—
17C9×#216—21no—
18C9×#216—21no—
19C9×#214—21no—
20C9×#208—21no—
21C9×#208C2211yescharacteristic
22C9×#208C2211yescharacteristic
23C2×C9×C3⋊C8—21no—
24C2×C9×C3⋊C8C2211yescharacteristic
25C32×#137C2×C311yescharacteristic
26C3×#951C2×C311yescharacteristic
27C3×#948C2×C311yescharacteristic
28C2×C3×#216C2×C311yescharacteristic
29C3×#836C2×C311yescharacteristic
30C3×#824C2×C311yescharacteristic
31C3×#822C2×C311yescharacteristic
32C9×#137—21no—
33C9×#137—31no—
34C9×#137D311yescharacteristic
35C2×C9×Q16—31no—
36C9×#131—31no—
37C9×#108—31no—
38C9×#101—31no—
39C9×#101—31no—
40C3×Q8×C9—21no—
41C3×Q8×C9—21no—
42C3×Q8×C9D411yescharacteristic
43C3×D4×C9—21no—
44C3×D4×C9D411yescharacteristic
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9C2311yescharacteristic
48C2×C9×Dic3—21no—
49C9×Dic6—21no—
50C9×Dic6—21no—
51C3×C8×C9—21no—
52C9×C3⋊C8—21no—
53C9×C3⋊C8—21no—
54C9×C3⋊C8D411yescharacteristic
55C9×C3⋊C8D411yescharacteristic
56C3×#473—31no—
57#5204C911yescharacteristic, direct product factor
58C32×#41C22×C311yescharacteristic
59C2×C32×Q8C22×C311yescharacteristic
60C2×C3×Dic6C22×C311yescharacteristic
61C32×Q16—21no—
62C32×SD16—21no—
63C32×M16C22×C311yescharacteristic
64C3×#216—21no—
65C3×#216—21no—
66C3×#216—21no—
67C3×#214—21no—
68C3×#208—21no—
69C3×#208C22×C311yescharacteristic
70C3×#208C22×C311yescharacteristic
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8C22×C311yescharacteristic
73C9×#41—21no—
74C9×#41C2×D311yescharacteristic
75C2×Q8×C9—21no—
76C2×Q8×C9—31no—
77C2×Q8×C9C2×D311yescharacteristic
78C9×Q16—22no—
79C9×Q16—21no—
80C9×Q16—61no—
81C9×Q16—61no—
82C9×Q16—61no—
83C9×SD16—22no—
84C9×SD16—21no—
85C9×SD16—61no—
86C9×M16—21no—
87C9×M16—31no—
88C9×M16—31no—
89C9×M16—61no—
90C9×M16C2×D311yescharacteristic
91C2×C8×C9—31no—
92C2×C8×C9—61no—
93C22×C3×C9—21no—
94C22×C3×C9#4111yescharacteristic
95C3×C4×C9—21no—
96C3×C4×C9—41no—
97C3×C4×C9C2×D411yescharacteristic
98C3×C4×C9C2×D411yescharacteristic
99C9×Dic3—41no—
100C3×#137—21no—
101C3×#137—31no—
102C3×#137C2×C911yescharacteristic
103C3×#137C3×D311yescharacteristic
104C2×C3×Q16—31no—
105C3×#131—31no—
106#951C2×C911yescharacteristic
107#948C2×C911yescharacteristic
108C2×#216C2×C911yescharacteristic, complemented
109C3×#108—31no—
110C3×#101—31no—
111C3×#101—31no—
112#836C2×C911yescharacteristic, complemented
113#824C2×C911yescharacteristic, complemented
114#822C2×C911yescharacteristic, complemented
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8C3×D411yescharacteristic
118C32×D4—21no—
119C32×D4C3×D411yescharacteristic
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4C23×C311yescharacteristic
123C2×C3×Dic3—21no—
124C3×Dic6—21no—
125C3×Dic6—21no—
126C32×C8—21no—
127C3×C3⋊C8—21no—
128C3×C3⋊C8—21no—
129C3×C3⋊C8C3×D411yescharacteristic
130C3×C3⋊C8C3×D411yescharacteristic
131Q8×C9—22no—
132Q8×C9—21no—
133Q8×C9—21no—
134Q8×C9—21no—
135Q8×C9—41no—
136Q8×C9—61no—
137Q8×C9—61no—
138Q8×C9#6711yescharacteristic
139D4×C9—21no—
140D4×C9—21no—
141D4×C9—41no—
142D4×C9#6711yescharacteristic
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9—21no—
146C2×C4×C9—41no—
147C2×C4×C9—41no—
148C2×C4×C9—61no—
149C2×C4×C9C22×D311yescharacteristic
150C8×C9—22no—
151C8×C9—21no—
152C8×C9—31no—
153C8×C9—31no—
154C8×C9—61no—
155C8×C9—61no—
156#473—31noSylow
157C2×C3×C9—21no—
158C2×C3×C9—41no—
159C2×C3×C9#12111yescharacteristic
160C3×#41—21no—
161C3×#41C22×C911yescharacteristic
162C3×#41C2×C3×D311yescharacteristic
163C2×C3×Q8—21no—
164C2×C3×Q8—31no—
165C2×C3×Q8C22×C911yescharacteristic
166C2×C3×Q8C2×C3×D311yescharacteristic
167C2×Dic6C22×C911yescharacteristic
168C3×Q16—22no—
169C3×Q16—21no—
170C3×Q16—21no—
171C3×Q16—61no—
172C3×Q16—61no—
173C3×Q16—61no—
174C3×SD16—22no—
175C3×SD16—21no—
176C3×SD16—21no—
177C3×SD16—61no—
178C3×M16—21no—
179C3×M16—31no—
180C3×M16—31no—
181C3×M16—61no—
182C3×M16C22×C911yescharacteristic
183C3×M16C2×C3×D311yescharacteristic
184C2×C3×C8—31no—
185C2×C3×C8—61no—
186#216—21no—
187#216—21no—
188#216—21no—
189#214—21no—
190#208—21no—
191#208C22×C911yescharacteristic
192#208C22×C911yescharacteristic
193C2×C3⋊C8—21no—
194C2×C3⋊C8C22×C911yescharacteristic
195C22×C32—21no—
196C22×C32C3×#4111yescharacteristic
197C32×C4—21no—
198C32×C4—41no—
199C32×C4C2×C3×D411yescharacteristic
200C32×C4C2×C3×D411yescharacteristic
201C3×Dic3—41no—
202C22×C9—21no—
203C22×C9—21no—
204C22×C9—41no—
205C22×C9#23711yescharacteristic
206C4×C9—21no—
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—41no—
210C4×C9—42no—
211C4×C9—41no—
212C4×C9—121no—
213C4×C9D3×D411yescharacteristic
214C4×C9C2×#6711yescharacteristic
215#137—31no—
216#137D3×C911yescharacteristic
217C2×Q16—31no—
218#131—31no—
219#108—31no—
220#101—31no—
221#101—31no—
222C3×C9#47311yescharacteristic, complemented, Sylow
223C3×Q8—22no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—21no—
227C3×Q8—21no—
228C3×Q8—21no—
229C3×Q8—41no—
230C3×Q8—61no—
231C3×Q8—61no—
232C3×Q8D4×C911yescharacteristic
233C3×Q8C3×#6711yescharacteristic
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—21no—
237C3×D4—41no—
238C3×D4D4×C911yescharacteristic
239C3×D4C3×#6711yescharacteristic
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—61no—
248C2×C3×C4C23×C911yescharacteristic
249C2×C3×C4C22×C3×D311yescharacteristic
250C2×Dic3—21no—
251Dic6—21no—
252Dic6—21no—
253C3×C8—22no—
254C3×C8—21no—
255C3×C8—21no—
256C3×C8—31no—
257C3×C8—31no—
258C3×C8—61no—
259C3×C8—61no—
260C3⋊C8—21no—
261C3⋊C8—21no—
262C3⋊C8D4×C911yescharacteristic
263C3⋊C8D4×C911yescharacteristic
264C2×C32—21no—
265C2×C32—41no—
266C2×C32C3×#12111yescharacteristic
267C2×C9—21no—
268C2×C9—21no—
269C2×C9—41no—
270C2×C9—41no—
271C2×C9—81no—
272C2×C9#93911yescharacteristic
273#41C2×D3×C911yescharacteristic
274C2×Q8—31no—
275C2×Q8C2×D3×C911yescharacteristic
276Q16—21no—
277Q16—61no—
278Q16—61no—
279Q16—61no—
280SD16—21no—
281SD16—61no—
282M16—31no—
283M16—31no—
284M16—61no—
285M16C2×D3×C911yescharacteristic
286C2×C8—31no—
287C2×C8—61no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—41no—
292C22×C3C9×#4111yescharacteristic
293C22×C3C3×#23711yescharacteristic
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—41no—
299C3×C4—42no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4—121no—
303C3×C4C2×D4×C911yescharacteristic
304C3×C4C2×D4×C911yescharacteristic
305C3×C4C3×D3×D411yescharacteristic
306C3×C4C2×C3×#6711yescharacteristic
307Dic3—41no—
308C32C3×#47311yescharacteristic
309C9—21no—
310C9#520411yescharacteristic, direct product factor
311Q8—21no—
312Q8—21no—
313Q8—61no—
314Q8—61no—
315Q8C9×#6711yescharacteristic
316D4—21no—
317D4C9×#6711yescharacteristic
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—61no—
321C2×C4C22×D3×C911yescharacteristic
322C8—21no—
323C8—31no—
324C8—31no—
325C8—61no—
326C8—61no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—81no—
334C2×C3C9×#12111yescharacteristic
335C2×C3C3×#93911yescharacteristic
336C22—21no—
337C22C9×#23711yescharacteristic
338C4—21no—
339C4—41no—
340C4—121no—
341C4D3×D4×C911yescharacteristic
342C4C2×C9×#6711yescharacteristic
343C3—21no—
344C3C9×#47311yescharacteristic, complemented
345C3C3×#520411yescharacteristic
346C2—21no—
347C2—41no—
348C2C9×#93911yescharacteristic
subgroupG/H
centerC2×C9#939
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#137C2
socleC2×C32C3×#121
prime radicalC32×#41C22×C3
cosocleC2×C32×C4C23×C3
hypercenterC9×#137D3
hypocenterC3C9×#473
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C9×#137

Complements

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

NH ≅ G/Ntype
4. C2×C9×#216C2semidirect
5. C9×#836C2semidirect
6. C9×#824C2semidirect
7. C9×#822C2semidirect
57. #5204C9direct (N × H)
108. C2×#216C2×C9semidirect
112. #836C2×C9semidirect
113. #824C2×C9semidirect
114. #822C2×C9semidirect
222. C3×C9#473semidirect
310. C9#5204direct (N × H)
344. C3C9×#473semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1372
C9×#9512
C9×#9482
C2×C9×#2162
C9×#8362
C9×#8242
C9×#8222
C3×#52043
C9×#4733