Finite Group Databasebeta

← C3×C9×#548 #13154626

Subgroups — C3×C9×#548 #13154626

Order 1728.

1450 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×#41C211yesmaximal, characteristic, complemented
2C22×C3×Q8×C9C211yesmaximal, characteristic, complemented
3C3×C9×#124C213yesmaximal, complemented
4C3×C9×C22⋊Q8C213yesmaximal, complemented
5C3×C9×#121C213yesmaximal, complemented
6C3×C4×Q8×C9C211yesmaximal, characteristic, complemented
7C3×C4×D4×C9C213yesmaximal, complemented
8C32×#548C311yesmaximal, characteristic
9C9×#548C313yesmaximal, direct product factor
10C3×C9×#41—24no—
11C2×C3×Q8×C9—24no—
12C2×C3×Q8×C9C2213yes—
13C2×C3×Q8×C9C2211yescharacteristic
14C2×C3×D4×C9C2213yes—
15C2×C3×D4×C9C2213yes—
16C22×C3×C4×C9C2213yes—
17C22×C3×C4×C9C2213yes—
18C3×C9×C4⋊C4C2213yes—
19C3×C9×C4⋊C4C2213yes—
20C3×C9×C22⋊C4C2211yescharacteristic
21C3×C9×C22⋊C4C2213yes—
22C3×C9×C22⋊C4C2216yes—
23C3×C42×C9C2213yes—
24C2×C32×#41C2×C311yescharacteristic
25C22×C32×Q8C2×C311yescharacteristic
26C32×#124C2×C313yes—
27C32×C22⋊Q8C2×C313yes—
28C32×#121C2×C313yes—
29C32×C4×Q8C2×C311yescharacteristic
30C32×C4×D4C2×C313yes—
31C2×C9×#41C2×C313yescomplemented
32C22×Q8×C9C2×C313yescomplemented
33C9×#124C2×C319yescomplemented
34C9×C22⋊Q8C2×C319yescomplemented
35C9×#121C2×C319yescomplemented
36C4×Q8×C9C2×C313yescomplemented
37C4×D4×C9C2×C319yescomplemented
38C23×C3×C9C2313yes—
39C23×C3×C9C2311yescharacteristic
40C3×Q8×C9—26no—
41C3×Q8×C9D414yescomplemented
42C3×D4×C9—26no—
43C3×D4×C9—26no—
44C2×C3×C4×C9—26no—
45C2×C3×C4×C9—26no—
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3×C4×C9C2313yes—
48C2×C3×C4×C9C2313yes—
49C2×C3×C4×C9C2311yescharacteristic
50C2×C3×C4×C9C2313yes—
51C3×#548C913yesdirect product factor
52C3×#548C3211yescharacteristic
53C32×#41—24no—
54C2×C32×Q8—24no—
55C2×C32×Q8C22×C313yes—
56C2×C32×Q8C22×C311yescharacteristic
57C2×C32×D4C22×C313yes—
58C2×C32×D4C22×C313yes—
59C22×C32×C4C22×C313yes—
60C22×C32×C4C22×C313yes—
61C32×C4⋊C4C22×C313yes—
62C32×C4⋊C4C22×C313yes—
63C32×C22⋊C4C22×C311yescharacteristic
64C32×C22⋊C4C22×C313yes—
65C32×C22⋊C4C22×C316yes—
66C32×C42C22×C313yes—
67C9×#41—212no—
68C2×Q8×C9—212no—
69C2×Q8×C9C22×C319yes—
70C2×Q8×C9C22×C313yes—
71C2×D4×C9C22×C319yes—
72C2×D4×C9C22×C319yes—
73C22×C4×C9C22×C319yes—
74C22×C4×C9C22×C319yes—
75C9×C4⋊C4C22×C319yes—
76C9×C4⋊C4C22×C319yes—
77C9×C22⋊C4C22×C313yes—
78C9×C22⋊C4C22×C319yes—
79C9×C22⋊C4C22×C3118yes—
80C42×C9C22×C319yes—
81C22×C3×C9—21no—
82C22×C3×C9—21no—
83C22×C3×C9—23no—
84C22×C3×C9—23no—
85C22×C3×C9—23no—
86C22×C3×C9#4112yescomplemented
87C22×C3×C9C2411yescharacteristic
88C3×C4×C9—21no—
89C3×C4×C9—23no—
90C3×C4×C9—23no—
91C3×C4×C9—21no—
92C3×C4×C9C2×D416yes—
93C2×C3×#41C2×C913yescomplemented
94C2×C3×#41C2×C3211yescharacteristic
95C22×C3×Q8C2×C913yescomplemented
96C22×C3×Q8C2×C3211yescharacteristic
97C3×#124C2×C919yescomplemented
98C3×#124C2×C3213yes—
99C3×C22⋊Q8C2×C919yescomplemented
100C3×C22⋊Q8C2×C3213yes—
101C3×#121C2×C919yescomplemented
102C3×#121C2×C3213yes—
103C3×C4×Q8C2×C913yescomplemented
104C3×C4×Q8C2×C3211yescharacteristic
105C3×C4×D4C2×C919yescomplemented
106C3×C4×D4C2×C3213yes—
107C23×C32C23×C313yes—
108C23×C32C23×C311yescharacteristic
109C32×Q8—26no—
110C32×Q8C3×D414yes—
111C32×D4—26no—
112C32×D4—26no—
113C2×C32×C4—26no—
114C2×C32×C4—26no—
115C2×C32×C4C23×C311yescharacteristic
116C2×C32×C4C23×C313yes—
117C2×C32×C4C23×C313yes—
118C2×C32×C4C23×C311yescharacteristic
119C2×C32×C4C23×C313yes—
120C23×C9C23×C319yes—
121C23×C9C23×C313yes—
122Q8×C9—218no—
123Q8×C9C3×D4112yescomplemented
124D4×C9—218no—
125D4×C9—218no—
126C2×C4×C9—218no—
127C2×C4×C9—218no—
128C2×C4×C9C23×C313yes—
129C2×C4×C9C23×C319yes—
130C2×C4×C9C23×C319yes—
131C2×C4×C9C23×C313yes—
132C2×C4×C9C23×C319yes—
133#548C3×C911yescharacteristic, direct product factor, Sylow
134C2×C3×C9—22no—
135C2×C3×C9—43no—
136C2×C3×C9C22×D411yescharacteristic
137C2×C3×C9C2×#4111yescharacteristic
138C2×C3×C92−1+411yescharacteristic
139C3×#41—212no—
140C3×#41—24no—
141C2×C3×Q8—212no—
142C2×C3×Q8—24no—
143C2×C3×Q8C22×C919yes—
144C2×C3×Q8C22×C913yes—
145C2×C3×Q8C22×C3213yes—
146C2×C3×Q8C22×C3211yescharacteristic
147C2×C3×D4C22×C919yes—
148C2×C3×D4C22×C919yes—
149C2×C3×D4C22×C3213yes—
150C2×C3×D4C22×C3213yes—
151C22×C3×C4C22×C919yes—
152C22×C3×C4C22×C919yes—
153C22×C3×C4C22×C3213yes—
154C22×C3×C4C22×C3213yes—
155C3×C4⋊C4C22×C919yes—
156C3×C4⋊C4C22×C919yes—
157C3×C4⋊C4C22×C3213yes—
158C3×C4⋊C4C22×C3213yes—
159C3×C22⋊C4C22×C913yes—
160C3×C22⋊C4C22×C919yes—
161C3×C22⋊C4C22×C9118yes—
162C3×C22⋊C4C22×C3211yescharacteristic
163C3×C22⋊C4C22×C3213yes—
164C3×C22⋊C4C22×C3216yes—
165C3×C42C22×C919yes—
166C3×C42C22×C3213yes—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—23no—
170C22×C32—23no—
171C22×C32—23no—
172C22×C32C3×#4112yes—
173C22×C32C24×C311yescharacteristic
174C32×C4—21no—
175C32×C4—23no—
176C32×C4—23no—
177C32×C4—21no—
178C32×C4C2×C3×D416yes—
179C22×C9—23no—
180C22×C9—23no—
181C22×C9—29no—
182C22×C9—29no—
183C22×C9—29no—
184C22×C9C3×#4116yescomplemented
185C22×C9C24×C313yes—
186C4×C9—23no—
187C4×C9—29no—
188C4×C9—29no—
189C4×C9—23no—
190C4×C9C2×C3×D4118yes—
191C2×#41C2×C3×C911yescharacteristic, complemented
192C22×Q8C2×C3×C911yescharacteristic, complemented
193#124C2×C3×C913yescomplemented
194C22⋊Q8C2×C3×C913yescomplemented
195#121C2×C3×C913yescomplemented
196C4×Q8C2×C3×C911yescharacteristic, complemented
197C4×D4C2×C3×C913yescomplemented
198C3×C9#54811yescharacteristic, direct product factor, Sylow
199C23×C3C23×C919yes—
200C23×C3C23×C913yes—
201C23×C3C23×C3213yes—
202C23×C3C23×C3211yescharacteristic
203C3×Q8—218no—
204C3×Q8—26no—
205C3×Q8D4×C9112yescomplemented
206C3×Q8C32×D414yes—
207C3×D4—218no—
208C3×D4—26no—
209C3×D4—218no—
210C3×D4—26no—
211C2×C3×C4—218no—
212C2×C3×C4—218no—
213C2×C3×C4—26no—
214C2×C3×C4—26no—
215C2×C3×C4C23×C913yes—
216C2×C3×C4C23×C919yes—
217C2×C3×C4C23×C919yes—
218C2×C3×C4C23×C913yes—
219C2×C3×C4C23×C919yes—
220C2×C3×C4C23×C3211yescharacteristic
221C2×C3×C4C23×C3213yes—
222C2×C3×C4C23×C3213yes—
223C2×C3×C4C23×C3211yescharacteristic
224C2×C3×C4C23×C3213yes—
225C2×C32—22no—
226C2×C32—43no—
227C2×C32C22×C3×D411yescharacteristic
228C2×C32C2×C3×#4111yescharacteristic
229C2×C32C3×2−1+411yescharacteristic
230C2×C9—26no—
231C2×C9—49no—
232C2×C9C22×C3×D413yes—
233C2×C9C2×C3×#4113yes—
234C2×C9C3×2−1+413yes—
235#41—24no—
236C2×Q8—24no—
237C2×Q8C22×C3×C913yes—
238C2×Q8C22×C3×C911yescharacteristic
239C2×D4C22×C3×C913yes—
240C2×D4C22×C3×C913yes—
241C22×C4C22×C3×C913yes—
242C22×C4C22×C3×C913yes—
243C4⋊C4C22×C3×C913yes—
244C4⋊C4C22×C3×C913yes—
245C22⋊C4C22×C3×C911yescharacteristic
246C22⋊C4C22×C3×C913yes—
247C22⋊C4C22×C3×C916yes—
248C42C22×C3×C913yes—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—23no—
252C22×C3—23no—
253C22×C3—23no—
254C22×C3—23no—
255C22×C3—23no—
256C22×C3—29no—
257C22×C3—29no—
258C22×C3—29no—
259C22×C3C9×#4116yescomplemented
260C22×C3C24×C913yes—
261C22×C3C32×#4112yes—
262C22×C3C24×C3211yescharacteristic
263C3×C4—21no—
264C3×C4—23no—
265C3×C4—23no—
266C3×C4—21no—
267C3×C4—23no—
268C3×C4—29no—
269C3×C4—29no—
270C3×C4—23no—
271C3×C4C2×D4×C9118yes—
272C3×C4C2×C32×D416yes—
273C32C3×#54811yescharacteristic
274C9C3×#54813yesdirect product factor
275C23C23×C3×C913yes—
276C23C23×C3×C911yescharacteristic
277Q8—26no—
278Q8C3×D4×C914yescomplemented
279D4—26no—
280D4—26no—
281C2×C4—26no—
282C2×C4—26no—
283C2×C4C23×C3×C911yescharacteristic
284C2×C4C23×C3×C913yes—
285C2×C4C23×C3×C913yes—
286C2×C4C23×C3×C911yescharacteristic
287C2×C4C23×C3×C913yes—
288C2×C3—26no—
289C2×C3—22no—
290C2×C3—49no—
291C2×C3—43no—
292C2×C3C22×D4×C913yes—
293C2×C3C2×C9×#4113yes—
294C2×C3C9×2−1+413yes—
295C2×C3C22×C32×D411yescharacteristic
296C2×C3C2×C32×#4111yescharacteristic
297C2×C3C32×2−1+411yescharacteristic
298C22—21no—
299C22—21no—
300C22—23no—
301C22—23no—
302C22—23no—
303C22C3×C9×#4112yescomplemented
304C22C24×C3×C911yescharacteristic
305C4—21no—
306C4—23no—
307C4—23no—
308C4—21no—
309C4C2×C3×D4×C916yes—
310C3C9×#54813yesdirect product factor
311C3C32×#54811yescharacteristic
312C2—22no—
313C2—43no—
314C2C22×C3×D4×C911yescharacteristic
315C2C2×C3×C9×#4111yescharacteristic
316C2C3×C9×2−1+411yescharacteristic
subgroupG/H
centerC22×C3×C9C24
commutatorC22C24×C3×C9
Frattini, cosocleC22×C3C24×C32
socleC22×C32C24×C3
prime radicalC32×#548C3
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C9 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×#41C2semidirect
2. C22×C3×Q8×C9C2semidirect
3. C3×C9×#124C2semidirect
4. C3×C9×C22⋊Q8C2semidirect
5. C3×C9×#121C2semidirect
6. C3×C4×Q8×C9C2semidirect
7. C3×C4×D4×C9C2semidirect
9. C9×#548C3direct (N × H)
31. C2×C9×#41C2×C3semidirect
32. C22×Q8×C9C2×C3semidirect
33. C9×#124C2×C3semidirect
34. C9×C22⋊Q8C2×C3semidirect
35. C9×#121C2×C3semidirect
36. C4×Q8×C9C2×C3semidirect
37. C4×D4×C9C2×C3semidirect
41. C3×Q8×C9D4semidirect
51. C3×#548C9direct (N × H)
86. C22×C3×C9#41semidirect
93. C2×C3×#41C2×C9semidirect
95. C22×C3×Q8C2×C9semidirect
97. C3×#124C2×C9semidirect
99. C3×C22⋊Q8C2×C9semidirect
101. C3×#121C2×C9semidirect
103. C3×C4×Q8C2×C9semidirect
105. C3×C4×D4C2×C9semidirect
123. Q8×C9C3×D4semidirect
133. #548C3×C9direct (N × H)
184. C22×C9C3×#41semidirect
191. C2×#41C2×C3×C9semidirect
192. C22×Q8C2×C3×C9semidirect
193. #124C2×C3×C9semidirect
194. C22⋊Q8C2×C3×C9semidirect
195. #121C2×C3×C9semidirect
196. C4×Q8C2×C3×C9semidirect
197. C4×D4C2×C3×C9semidirect
198. C3×C9#548direct (N × H)
205. C3×Q8D4×C9semidirect
259. C22×C3C9×#41semidirect
274. C9C3×#548direct (N × H)
278. Q8C3×D4×C9semidirect
303. C22C3×C9×#41semidirect
310. C3C9×#548direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C9×#412
C22×C3×Q8×C92
C3×C9×#1242
C3×C9×C22⋊Q82
C3×C9×#1212
C3×C4×Q8×C92
C3×C4×D4×C92
C32×#5483
C9×#5483