Finite Group Databasebeta

← C9×#4996 #13142670

Subgroups — C9×#4996 #13142670

Order 1728.

364 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#117C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic
3C9×#830C211yesmaximal, characteristic
4C9×#810C211yesmaximal, characteristic, complemented
5C9×#810C211yesmaximal, characteristic, complemented
6C9×#808C211yesmaximal, characteristic, complemented
7C9×#807C211yesmaximal, characteristic, complemented
8C3×#4996C311yesmaximal, characteristic
9C9×#485—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C3×C9×C22⋊C4—21no—
15C3×C42×C9—21no—
16C9×#217—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C9×Dic3⋊C4—21no—
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C32×#117C2×C311yescharacteristic
22C3×#924C2×C311yescharacteristic
23C3×#830C2×C311yescharacteristic
24C3×#810C2×C311yescharacteristic
25C3×#810C2×C311yescharacteristic
26C3×#808C2×C311yescharacteristic
27C3×#807C2×C311yescharacteristic
28C9×C22⋊Q8—31no—
29C9×#117—21no—
30C9×#117D311yescharacteristic
31C9×#107—31no—
32C9×#106—31no—
33C9×#103—31no—
34C9×#103—31no—
35C9×C22⋊C8—31no—
36C23×C3×C9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9D411yescharacteristic
42C2×C3×C4×C9C2311yescharacteristic
43C2×C9×Dic3—21no—
44C2×C9×Dic3—21no—
45C9×Dic6—21no—
46C9×Dic6—21no—
47C9×C3⋊C8—21no—
48C9×C3⋊C8—21no—
49C3×#485—31no—
50#4996C911yescharacteristic, direct product factor
51C22×C32×C4C22×C311yescharacteristic
52C2×C3×Dic6C22×C311yescharacteristic
53C32×C4⋊C4C22×C311yescharacteristic
54C32×C4⋊C4C22×C311yescharacteristic
55C32×C22⋊C4—21no—
56C32×C42—21no—
57C3×#217—21no—
58C3×C4⋊Dic3C22×C311yescharacteristic
59C3×Dic3⋊C4—21no—
60C2×C3×C3⋊C8C22×C311yescharacteristic
61C2×C3×C3⋊C8C22×C311yescharacteristic
62C2×Q8×C9—31no—
63C22×C4×C9—21no—
64C22×C4×C9C2×D311yescharacteristic
65C2×C8×C9—31no—
66C2×C8×C9—31no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4—31no—
70C9×C4⋊C4—61no—
71C9×C4⋊C4C2×D311yescharacteristic
72C9×C4⋊C4C2×D311yescharacteristic
73C9×C22⋊C4—22no—
74C9×C22⋊C4—21no—
75C9×C22⋊C4—61no—
76C42×C9—22no—
77C42×C9—21no—
78C22×C3×C9—21no—
79C22×C3×C9—21no—
80C22×C3×C9—21no—
81C22×C3×C9C2×D411yescharacteristic
82C3×C4×C9—21no—
83C3×C4×C9—41no—
84C3×C4×C9—41no—
85C3×C4×C9#4111yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C9×Dic3—41no—
88C9×Dic3—41no—
89C3×C22⋊Q8—31no—
90C3×#117—21no—
91C3×#117C2×C911yescharacteristic
92C3×#117C3×D311yescharacteristic
93#924C2×C911yescharacteristic
94C3×#107—31no—
95C3×#106—31no—
96C3×#103—31no—
97C3×#103—31no—
98C3×C22⋊C8—31no—
99#830C2×C911yescharacteristic
100#810C2×C911yescharacteristic, complemented
101#810C2×C911yescharacteristic, complemented
102#808C2×C911yescharacteristic, complemented
103#807C2×C911yescharacteristic, complemented
104C23×C32C3×D411yescharacteristic
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4C3×D411yescharacteristic
110C2×C32×C4C23×C311yescharacteristic
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C3×Dic6—21no—
114C3×Dic6—21no—
115C23×C9—21no—
116C23×C9#6711yescharacteristic
117C3×C3⋊C8—21no—
118C3×C3⋊C8—21no—
119Q8×C9—61no—
120Q8×C9—61no—
121C2×C4×C9—22no—
122C2×C4×C9—22no—
123C2×C4×C9—21no—
124C2×C4×C9—21no—
125C2×C4×C9—22no—
126C2×C4×C9—22no—
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9—21no—
130C2×C4×C9—21no—
131C2×C4×C9—61no—
132C2×C4×C9—61no—
133C2×C4×C9#6711yescharacteristic
134C2×C4×C9C22×D311yescharacteristic
135C8×C9—61no—
136C8×C9—61no—
137#485—31noSylow
138C2×C3×C9—41no—
139C2×C3×C9#12311yescharacteristic
140C2×C3×C9#13511yescharacteristic
141C2×C3×C9#13711yescharacteristic
142C2×C3×Q8—31no—
143C22×C3×C4—21no—
144C22×C3×C4C22×C911yescharacteristic
145C22×C3×C4C2×C3×D311yescharacteristic
146C2×Dic6C22×C911yescharacteristic
147C2×C3×C8—31no—
148C2×C3×C8—31no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4—61no—
153C3×C4⋊C4C22×C911yescharacteristic
154C3×C4⋊C4C22×C911yescharacteristic
155C3×C4⋊C4C2×C3×D311yescharacteristic
156C3×C4⋊C4C2×C3×D311yescharacteristic
157C3×C22⋊C4—22no—
158C3×C22⋊C4—21no—
159C3×C22⋊C4—21no—
160C3×C22⋊C4—61no—
161C3×C42—22no—
162C3×C42—21no—
163C3×C42—21no—
164#217—21no—
165C4⋊Dic3C22×C911yescharacteristic
166Dic3⋊C4—21no—
167C2×C3⋊C8C22×C911yescharacteristic
168C2×C3⋊C8C22×C911yescharacteristic
169C22×C32—21no—
170C22×C32—21no—
171C22×C32—21no—
172C22×C32C2×C3×D411yescharacteristic
173C32×C4—21no—
174C32×C4—41no—
175C32×C4—41no—
176C32×C4C3×#4111yescharacteristic
177C32×C4C3×#4111yescharacteristic
178C3×Dic3—41no—
179C3×Dic3—41no—
180C22×C9—22no—
181C22×C9—21no—
182C22×C9—21no—
183C22×C9—21no—
184C22×C9—21no—
185C22×C9—41no—
186C22×C9—41no—
187C22×C9C2×#6711yescharacteristic
188C4×C9—22no—
189C4×C9—21no—
190C4×C9—21no—
191C4×C9—21no—
192C4×C9—42no—
193C4×C9—42no—
194C4×C9—41no—
195C4×C9—41no—
196C4×C9—121no—
197C4×C9—121no—
198C4×C9#23511yescharacteristic
199C4×C9#23511yescharacteristic
200C22⋊Q8—31no—
201#117D3×C911yescharacteristic
202#107—31no—
203#106—31no—
204#103—31no—
205#103—31no—
206C22⋊C8—31no—
207C3×C9#48511yescharacteristic, complemented, Sylow
208C23×C3—21no—
209C23×C3D4×C911yescharacteristic
210C23×C3C3×#6711yescharacteristic
211C3×Q8—61no—
212C3×Q8—61no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4D4×C911yescharacteristic
230C2×C3×C4C23×C911yescharacteristic
231C2×C3×C4C3×#6711yescharacteristic
232C2×C3×C4C22×C3×D311yescharacteristic
233C2×Dic3—21no—
234C2×Dic3—21no—
235Dic6—21no—
236Dic6—21no—
237C3×C8—61no—
238C3×C8—61no—
239C3⋊C8—21no—
240C3⋊C8—21no—
241C2×C32—41no—
242C2×C32C3×#12311yescharacteristic
243C2×C32C3×#13511yescharacteristic
244C2×C32C3×#13711yescharacteristic
245C2×C9—21no—
246C2×C9—21no—
247C2×C9—21no—
248C2×C9—42no—
249C2×C9—41no—
250C2×C9#92911yescharacteristic
251C2×C9#95011yescharacteristic
252C2×C9#95111yescharacteristic
253C2×Q8—31no—
254C22×C4C2×D3×C911yescharacteristic
255C2×C8—31no—
256C2×C8—31no—
257C4⋊C4—31no—
258C4⋊C4—61no—
259C4⋊C4C2×D3×C911yescharacteristic
260C4⋊C4C2×D3×C911yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—61no—
263C42—21no—
264C22×C3—22no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3C2×D4×C911yescharacteristic
275C22×C3C2×C3×#6711yescharacteristic
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—42no—
282C3×C4—42no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4—121no—
288C3×C4—121no—
289C3×C4C9×#4111yescharacteristic
290C3×C4C9×#4111yescharacteristic
291C3×C4C3×#23511yescharacteristic
292C3×C4C3×#23511yescharacteristic
293Dic3—41no—
294Dic3—41no—
295C32C3×#48511yescharacteristic
296C9—21no—
297C9#499611yescharacteristic, direct product factor
298C23C9×#6711yescharacteristic
299Q8—61no—
300Q8—61no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4C9×#6711yescharacteristic
308C2×C4C22×D3×C911yescharacteristic
309C8—61no—
310C8—61no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—42no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3C9×#12311yescharacteristic
318C2×C3C9×#13511yescharacteristic
319C2×C3C9×#13711yescharacteristic
320C2×C3C3×#92911yescharacteristic
321C2×C3C3×#95011yescharacteristic
322C2×C3C3×#95111yescharacteristic
323C22—21no—
324C22—21no—
325C22—21no—
326C22C2×C9×#6711yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—121no—
331C4—121no—
332C4C9×#23511yescharacteristic
333C4C9×#23511yescharacteristic
334C3—21no—
335C3C9×#48511yescharacteristic, complemented
336C3C3×#499611yescharacteristic
337C2—41no—
338C2C9×#92911yescharacteristic
339C2C9×#95011yescharacteristic
340C2C9×#95111yescharacteristic
subgroupG/H
centerC22×C9C2×#67
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#117C2
socleC22×C32C2×C3×D4
prime radicalC23×C32C3×D4
cosocleC2×C32×C4C23×C3
hypercenterC9×#117D3
hypocenterC3C9×#485
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C22×C4×C9 ◁ C9×#117

Complements

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

NH ≅ G/Ntype
4. C9×#810C2semidirect
5. C9×#810C2semidirect
6. C9×#808C2semidirect
7. C9×#807C2semidirect
50. #4996C9direct (N × H)
100. #810C2×C9semidirect
101. #810C2×C9semidirect
102. #808C2×C9semidirect
103. #807C2×C9semidirect
207. C3×C9#485semidirect
297. C9#4996direct (N × H)
335. C3C9×#485semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1172
C9×#9242
C9×#8302
C9×#8102
C9×#8082
C9×#8072
C3×#49963
C9×#4853