Finite Group Databasebeta

← C3×C9×#485 #13145213

Subgroups — C3×C9×#485 #13145213

Order 1728.

480 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic
2C3×C9×#117C211yesmaximal, characteristic
3C3×C9×#107C211yesmaximal, characteristic, complemented
4C3×C9×#106C211yesmaximal, characteristic, complemented
5C3×C9×#103C211yesmaximal, characteristic, complemented
6C3×C9×#103C211yesmaximal, characteristic, complemented
7C3×C9×C22⋊C8C211yesmaximal, characteristic
8C32×#485C311yesmaximal, characteristic
9C9×#485C313yesmaximal, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C2×C3×C8×C9C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4—21no—
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×C9×C4⋊C4C2211yescharacteristic
17C3×C9×C4⋊C4C2211yescharacteristic
18C3×C9×C22⋊C4—21no—
19C3×C9×C22⋊C4—21no—
20C3×C42×C9—21no—
21C32×C22⋊Q8C2×C311yescharacteristic
22C32×#117C2×C311yescharacteristic
23C32×#107C2×C311yescharacteristic
24C32×#106C2×C311yescharacteristic
25C32×#103C2×C311yescharacteristic
26C32×#103C2×C311yescharacteristic
27C32×C22⋊C8C2×C311yescharacteristic
28C9×C22⋊Q8C2×C313yes—
29C9×#117C2×C313yes—
30C9×#107C2×C313yescomplemented
31C9×#106C2×C313yescomplemented
32C9×#103C2×C313yescomplemented
33C9×#103C2×C313yescomplemented
34C9×C22⋊C8C2×C313yes—
35C23×C3×C9D411yescharacteristic
36C3×Q8×C9—21no—
37C3×Q8×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9D411yescharacteristic
45C2×C3×C4×C9C2311yescharacteristic
46C3×C8×C9—21no—
47C3×C8×C9—21no—
48C3×#485C913yesdirect product factor
49C3×#485C3211yescharacteristic
50C2×C32×Q8C22×C311yescharacteristic
51C22×C32×C4C22×C311yescharacteristic
52C2×C32×C8C22×C311yescharacteristic
53C2×C32×C8C22×C311yescharacteristic
54C32×C4⋊C4—21no—
55C32×C4⋊C4C22×C311yescharacteristic
56C32×C4⋊C4C22×C311yescharacteristic
57C32×C4⋊C4C22×C311yescharacteristic
58C32×C22⋊C4—21no—
59C32×C22⋊C4—21no—
60C32×C42—21no—
61C2×Q8×C9C22×C313yes—
62C22×C4×C9C22×C313yes—
63C2×C8×C9C22×C313yes—
64C2×C8×C9C22×C313yes—
65C9×C4⋊C4—23no—
66C9×C4⋊C4C22×C313yes—
67C9×C4⋊C4C22×C313yes—
68C9×C4⋊C4C22×C313yes—
69C9×C22⋊C4—23no—
70C9×C22⋊C4—23no—
71C42×C9—23no—
72C22×C3×C9—21no—
73C22×C3×C9—21no—
74C22×C3×C9—21no—
75C22×C3×C9C2×D411yescharacteristic
76C3×C4×C9—21no—
77C3×C4×C9—41no—
78C3×C4×C9—41no—
79C3×C4×C9—41no—
80C3×C4×C9—41no—
81C3×C4×C9#4111yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3×C22⋊Q8C2×C913yes—
84C3×C22⋊Q8C2×C3211yescharacteristic
85C3×#117C2×C913yes—
86C3×#117C2×C3211yescharacteristic
87C3×#107C2×C913yescomplemented
88C3×#107C2×C3211yescharacteristic
89C3×#106C2×C913yescomplemented
90C3×#106C2×C3211yescharacteristic
91C3×#103C2×C913yescomplemented
92C3×#103C2×C913yescomplemented
93C3×#103C2×C3211yescharacteristic
94C3×#103C2×C3211yescharacteristic
95C3×C22⋊C8C2×C913yes—
96C3×C22⋊C8C2×C3211yescharacteristic
97C23×C32C3×D411yescharacteristic
98C32×Q8—21no—
99C32×Q8—21no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4C3×D411yescharacteristic
107C2×C32×C4C23×C311yescharacteristic
108C23×C9C3×D413yes—
109C32×C8—21no—
110C32×C8—21no—
111Q8×C9—23no—
112Q8×C9—23no—
113C2×C4×C9—23no—
114C2×C4×C9—23no—
115C2×C4×C9—23no—
116C2×C4×C9—23no—
117C2×C4×C9—23no—
118C2×C4×C9—23no—
119C2×C4×C9C3×D413yes—
120C2×C4×C9C23×C313yes—
121C8×C9—23no—
122C8×C9—23no—
123#485C3×C911yescharacteristic, direct product factor, Sylow
124C2×C3×C9—41no—
125C2×C3×C9#12311yescharacteristic
126C2×C3×C9#13511yescharacteristic
127C2×C3×C9#13711yescharacteristic
128C2×C3×Q8C22×C913yes—
129C2×C3×Q8C22×C3211yescharacteristic
130C22×C3×C4C22×C913yes—
131C22×C3×C4C22×C3211yescharacteristic
132C2×C3×C8C22×C913yes—
133C2×C3×C8C22×C913yes—
134C2×C3×C8C22×C3211yescharacteristic
135C2×C3×C8C22×C3211yescharacteristic
136C3×C4⋊C4—23no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4C22×C913yes—
139C3×C4⋊C4C22×C913yes—
140C3×C4⋊C4C22×C913yes—
141C3×C4⋊C4C22×C3211yescharacteristic
142C3×C4⋊C4C22×C3211yescharacteristic
143C3×C4⋊C4C22×C3211yescharacteristic
144C3×C22⋊C4—23no—
145C3×C22⋊C4—23no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—21no—
148C3×C42—23no—
149C3×C42—21no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32C2×C3×D411yescharacteristic
154C32×C4—21no—
155C32×C4—41no—
156C32×C4—41no—
157C32×C4—41no—
158C32×C4—41no—
159C32×C4C3×#4111yescharacteristic
160C32×C4C3×#4111yescharacteristic
161C22×C9—23no—
162C22×C9—23no—
163C22×C9—23no—
164C22×C9C2×C3×D413yes—
165C4×C9—23no—
166C4×C9—43no—
167C4×C9—43no—
168C4×C9—43no—
169C4×C9—43no—
170C4×C9C3×#4113yes—
171C4×C9C3×#4113yes—
172C22⋊Q8C2×C3×C911yescharacteristic
173#117C2×C3×C911yescharacteristic
174#107C2×C3×C911yescharacteristic, complemented
175#106C2×C3×C911yescharacteristic, complemented
176#103C2×C3×C911yescharacteristic, complemented
177#103C2×C3×C911yescharacteristic, complemented
178C22⋊C8C2×C3×C911yescharacteristic
179C3×C9#48511yescharacteristic, direct product factor, Sylow
180C23×C3D4×C913yes—
181C23×C3C32×D411yescharacteristic
182C3×Q8—23no—
183C3×Q8—23no—
184C3×Q8—21no—
185C3×Q8—21no—
186C2×C3×C4—23no—
187C2×C3×C4—23no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—23no—
191C2×C3×C4—23no—
192C2×C3×C4—23no—
193C2×C3×C4—23no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4D4×C913yes—
199C2×C3×C4C23×C913yes—
200C2×C3×C4C32×D411yescharacteristic
201C2×C3×C4C23×C3211yescharacteristic
202C3×C8—23no—
203C3×C8—23no—
204C3×C8—21no—
205C3×C8—21no—
206C2×C32—41no—
207C2×C32C3×#12311yescharacteristic
208C2×C32C3×#13511yescharacteristic
209C2×C32C3×#13711yescharacteristic
210C2×C9—43no—
211C2×C9C3×#12313yes—
212C2×C9C3×#13513yes—
213C2×C9C3×#13713yes—
214C2×Q8C22×C3×C911yescharacteristic
215C22×C4C22×C3×C911yescharacteristic
216C2×C8C22×C3×C911yescharacteristic
217C2×C8C22×C3×C911yescharacteristic
218C4⋊C4—21no—
219C4⋊C4C22×C3×C911yescharacteristic
220C4⋊C4C22×C3×C911yescharacteristic
221C4⋊C4C22×C3×C911yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—21no—
224C42—21no—
225C22×C3—23no—
226C22×C3—23no—
227C22×C3—23no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3C2×D4×C913yes—
232C22×C3C2×C32×D411yescharacteristic
233C3×C4—23no—
234C3×C4—21no—
235C3×C4—43no—
236C3×C4—43no—
237C3×C4—43no—
238C3×C4—43no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4C9×#4113yes—
244C3×C4C9×#4113yes—
245C3×C4C32×#4111yescharacteristic
246C3×C4C32×#4111yescharacteristic
247C32C3×#48511yescharacteristic
248C9C3×#48513yesdirect product factor
249C23C3×D4×C911yescharacteristic
250Q8—21no—
251Q8—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4C3×D4×C911yescharacteristic
259C2×C4C23×C3×C911yescharacteristic
260C8—21no—
261C8—21no—
262C2×C3—41no—
263C2×C3—43no—
264C2×C3C9×#12313yes—
265C2×C3C9×#13513yes—
266C2×C3C9×#13713yes—
267C2×C3C32×#12311yescharacteristic
268C2×C3C32×#13511yescharacteristic
269C2×C3C32×#13711yescharacteristic
270C22—21no—
271C22—21no—
272C22—21no—
273C22C2×C3×D4×C911yescharacteristic
274C4—21no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4C3×C9×#4111yescharacteristic
280C4C3×C9×#4111yescharacteristic
281C3C9×#48513yesdirect product factor
282C3C32×#48511yescharacteristic
283C2—41no—
284C2C3×C9×#12311yescharacteristic
285C2C3×C9×#13511yescharacteristic
286C2C3×C9×#13711yescharacteristic
subgroupG/H
centerC22×C3×C9C2×D4
commutatorC2×C4C23×C3×C9
Frattini, cosocleC2×C3×C4C23×C32
socleC22×C32C2×C3×D4
prime radicalC23×C32C3×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C9 ◁ C22×C3×C4×C9 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×C9×#107C2semidirect
4. C3×C9×#106C2semidirect
5. C3×C9×#103C2semidirect
6. C3×C9×#103C2semidirect
9. C9×#485C3direct (N × H)
30. C9×#107C2×C3semidirect
31. C9×#106C2×C3semidirect
32. C9×#103C2×C3semidirect
33. C9×#103C2×C3semidirect
48. C3×#485C9direct (N × H)
87. C3×#107C2×C9semidirect
89. C3×#106C2×C9semidirect
91. C3×#103C2×C9semidirect
92. C3×#103C2×C9semidirect
123. #485C3×C9direct (N × H)
174. #107C2×C3×C9semidirect
175. #106C2×C3×C9semidirect
176. #103C2×C3×C9semidirect
177. #103C2×C3×C9semidirect
179. C3×C9#485direct (N × H)
248. C9C3×#485direct (N × H)
281. C3C9×#485direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C9×C22⋊Q82
C3×C9×#1172
C3×C9×#1072
C3×C9×#1062
C3×C9×#1032
C3×C9×C22⋊C82
C32×#4853
C9×#4853