Finite Group Databasebeta

← C4×C9×#239 #13152694

Subgroups — C4×C9×#239 #13152694

Order 1728.

1000 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#239C211yesmaximal, characteristic
2C3×C4×Q8×C9C211yesmaximal, characteristic, complemented
3Q8×C9×Dic3C211yesmaximal, characteristic, complemented
4C9×#893C213yesmaximal, complemented
5C9×#892C213yesmaximal, complemented
6C4×C9×D12C213yesmaximal, complemented
7C42×D3×C9C213yesmaximal, complemented
8C3×C4×#239C311yesmaximal, characteristic
9C4×C9×#41—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C9×#239C418yesdirect product factor
12C2×C9×D12C2213yes—
13C2×C4×D3×C9C2213yes—
14C2×C4×D3×C9C2213yes—
15C2×C4×D3×C9C2213yes—
16C3×C9×C4⋊C4C2213yes—
17C3×C42×C9C2213yes—
18C9×(C2×D3)⋊C4C2216yes—
19C9×C4⋊Dic3C2213yes—
20C4×C9×Dic3C2211yescharacteristic
21C4×C9×Dic3C2213yes—
22C4×C9×Dic3C2213yes—
23C2×C3×#239C2×C311yescharacteristic
24C32×C4×Q8C2×C311yescharacteristic
25C3×Q8×Dic3C2×C311yescharacteristic
26C3×#893C2×C313yes—
27C3×#892C2×C313yes—
28C3×C4×D12C2×C313yes—
29C3×C42×D3C2×C313yes—
30C2×C9×#41—31no—
31C4×Q8×C9—21no—
32C4×Q8×C9—31no—
33C4×Q8×C9D311yescharacteristic, complemented
34C4×D4×C9—33no—
35C4×D4×C9—33no—
36C9×#117—33no—
37C2×C42×C9—33no—
38C22×D3×C9C2313yes—
39C3×Q8×C9C2×C414yescomplemented
40C2×C3×C4×C9C2313yes—
41C2×C3×C4×C9C2313yes—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C9×Dic3C2313yes—
44C2×C9×Dic3C2311yescharacteristic
45C2×C9×Dic3C2311yescharacteristic
46C9×D12C2×C4112yescomplemented
47C4×D3×C9—26no—
48C4×D3×C9—26no—
49C4×D3×C9C2×C4112yescomplemented
50C3×C4×#41—31no—
51C4×#239C911yescharacteristic, direct product factor
52C2×C32×Q8C22×C311yescharacteristic
53C3×#239C3×C418yes—
54C2×C3×D12C22×C313yes—
55C2×C3×C4×D3C22×C313yes—
56C2×C3×C4×D3C22×C313yes—
57C2×C3×C4×D3C22×C313yes—
58C32×C4⋊C4C22×C313yes—
59C32×C42C22×C313yes—
60C3×(C2×D3)⋊C4C22×C316yes—
61C3×C4⋊Dic3C22×C313yes—
62C3×C4×Dic3C22×C313yes—
63C3×C4×Dic3C22×C313yes—
64C3×C4×Dic3C22×C311yescharacteristic
65C9×#41—38no—
66C2×Q8×C9—21no—
67C2×Q8×C9C2×D311yescharacteristic
68C2×D4×C9—33no—
69C22×C4×C9—33no—
70C22×C4×C9—33no—
71C22×C4×C9—33no—
72C9×C4⋊C4—23no—
73C9×C4⋊C4—33no—
74C9×C4⋊C4C2×D313yes—
75C9×C22⋊C4—36no—
76C42×C9—23no—
77C42×C9—33no—
78C42×C9—33no—
79C42×C9—31no—
80C42×C9C2×D313yes—
81C22×C3×C9C2411yescharacteristic
82C2×D3×C9—23no—
83C2×D3×C9—23no—
84C2×D3×C9C22×C416yes—
85C3×C4×C9—23no—
86C3×C4×C9C22×C416yes—
87C3×C4×C9#4112yescomplemented
88C9×Dic3—23no—
89C9×Dic3C22×C412yes—
90C9×Dic3#4112yescomplemented
91C2×C3×#41—31no—
92C2×#239C2×C911yescharacteristic
93C3×C4×Q8—21no—
94C3×C4×Q8—31no—
95C3×C4×Q8C2×C911yescharacteristic, complemented
96C3×C4×Q8C3×D311yescharacteristic
97C3×C4×D4—33no—
98C3×C4×D4—33no—
99C3×#117—33no—
100C2×C3×C42—33no—
101Q8×Dic3C2×C911yescharacteristic, complemented
102#893C2×C913yescomplemented
103#892C2×C913yescomplemented
104C4×D12C2×C913yescomplemented
105C42×D3C2×C913yescomplemented
106C22×C3×D3C23×C313yes—
107C32×Q8C2×C3×C414yes—
108C2×C32×C4C23×C313yes—
109C2×C32×C4C23×C313yes—
110C2×C32×C4C23×C311yescharacteristic
111C2×C3×Dic3C23×C313yes—
112C2×C3×Dic3C23×C311yescharacteristic
113C2×C3×Dic3C23×C311yescharacteristic
114C3×D12C2×C3×C4112yes—
115C3×C4×D3—26no—
116C3×C4×D3—26no—
117C3×C4×D3C2×C3×C4112yes—
118C23×C9—33no—
119Q8×C9—24no—
120Q8×C9C4×D314yescomplemented
121D4×C9—312no—
122C2×C4×C9—21no—
123C2×C4×C9—23no—
124C2×C4×C9—23no—
125C2×C4×C9—31no—
126C2×C4×C9—31no—
127C2×C4×C9—312no—
128C2×C4×C9—33no—
129C2×C4×C9—66no—
130C2×C4×C9—66no—
131C2×C4×C9C22×D313yes—
132C2×C4×C9C22×D313yes—
133C2×C4×C9C22×D311yescharacteristic
134C4×#41—31noSylow
135C2×C3×C9C23×C411yescharacteristic
136C2×C3×C9C2×#4111yescharacteristic
137C2×C3×C9C2×#4111yescharacteristic
138D3×C9—26no—
139C3×#41—38no—
140C2×C3×Q8—21no—
141C2×C3×Q8C22×C911yescharacteristic
142C2×C3×Q8C2×C3×D311yescharacteristic
143C2×C3×D4—33no—
144C22×C3×C4—33no—
145C22×C3×C4—33no—
146C22×C3×C4—33no—
147#239C4×C918yesdirect product factor
148C2×D12C22×C913yes—
149C2×C4×D3C22×C913yes—
150C2×C4×D3C22×C913yes—
151C2×C4×D3C22×C913yes—
152C3×C4⋊C4—23no—
153C3×C4⋊C4—33no—
154C3×C4⋊C4C22×C913yes—
155C3×C4⋊C4C2×C3×D313yes—
156C3×C22⋊C4—36no—
157C3×C42—23no—
158C3×C42—31no—
159C3×C42—33no—
160C3×C42—33no—
161C3×C42C22×C913yes—
162C3×C42C2×C3×D313yes—
163(C2×D3)⋊C4C22×C916yes—
164C4⋊Dic3C22×C913yes—
165C4×Dic3C22×C911yescharacteristic
166C4×Dic3C22×C913yes—
167C4×Dic3C22×C913yes—
168C22×C32C24×C311yescharacteristic
169C2×C3×D3—23no—
170C2×C3×D3—23no—
171C2×C3×D3C22×C3×C416yes—
172C32×C4—23no—
173C32×C4C22×C3×C416yes—
174C32×C4C3×#4112yes—
175C3×Dic3—23no—
176C3×Dic3C22×C3×C412yes—
177C3×Dic3C3×#4112yes—
178C22×C9—21no—
179C22×C9—36no—
180C22×C9—63no—
181C22×C9—63no—
182C22×C9C23×D311yescharacteristic
183C4×C9—23no—
184C4×C9—26no—
185C4×C9—22no—
186C4×C9—32no—
187C4×C9—32no—
188C4×C9—43no—
189C4×C9—63no—
190C4×C9C2×C4×D316yes—
191C4×C9#23912yesdirect product factor
192C2×#41—31no—
193C4×Q8—31no—
194C4×Q8D3×C911yescharacteristic, complemented
195C4×D4—33no—
196C4×D4—33no—
197#117—33no—
198C2×C42—33no—
199C3×C9C4×#4111yescharacteristic, complemented, Sylow
200C23×C3—33no—
201C22×D3C23×C913yes—
202C3×Q8—24no—
203C3×Q8C2×C4×C914yescomplemented
204C3×Q8C3×C4×D314yes—
205C3×D4—312no—
206C2×C3×C4—21no—
207C2×C3×C4—23no—
208C2×C3×C4—23no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—312no—
212C2×C3×C4—33no—
213C2×C3×C4—66no—
214C2×C3×C4—66no—
215C2×C3×C4C23×C913yes—
216C2×C3×C4C23×C913yes—
217C2×C3×C4C23×C911yescharacteristic
218C2×C3×C4C22×C3×D313yes—
219C2×C3×C4C22×C3×D313yes—
220C2×C3×C4C22×C3×D311yescharacteristic
221C2×Dic3C23×C913yes—
222C2×Dic3C23×C911yescharacteristic
223C2×Dic3C23×C911yescharacteristic
224D12C2×C4×C9112yescomplemented
225C4×D3—26no—
226C4×D3—26no—
227C4×D3C2×C4×C9112yescomplemented
228C2×C32C23×C3×C411yescharacteristic
229C2×C32C2×C3×#4111yescharacteristic
230C2×C32C2×C3×#4111yescharacteristic
231C3×D3—26no—
232C2×C9—21no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9—66no—
236C2×C9C22×C4×D311yescharacteristic
237C2×C9C2×#23911yescharacteristic
238C2×C9D3×#4111yescharacteristic
239#41—38no—
240C2×Q8C2×D3×C911yescharacteristic
241C2×D4—33no—
242C22×C4—33no—
243C22×C4—33no—
244C22×C4—33no—
245C4⋊C4—33no—
246C4⋊C4C2×D3×C913yes—
247C22⋊C4—36no—
248C42—31no—
249C42—33no—
250C42—33no—
251C42C2×D3×C913yes—
252C22×C3—21no—
253C22×C3—36no—
254C22×C3—63no—
255C22×C3—63no—
256C22×C3C24×C911yescharacteristic
257C22×C3C23×C3×D311yescharacteristic
258C2×D3—23no—
259C2×D3—23no—
260C2×D3C22×C4×C916yes—
261C3×C4—23no—
262C3×C4—23no—
263C3×C4—26no—
264C3×C4—22no—
265C3×C4—32no—
266C3×C4—32no—
267C3×C4—43no—
268C3×C4—63no—
269C3×C4C22×C4×C916yes—
270C3×C4C9×#4112yescomplemented
271C3×C4C2×C3×C4×D316yes—
272C3×C4C3×#23912yes—
273Dic3—23no—
274Dic3C22×C4×C912yes—
275Dic3C9×#4112yescomplemented
276C32C3×C4×#4111yescharacteristic
277C9—21no—
278C9C4×#23911yescharacteristic, direct product factor
279C23—33no—
280Q8C4×D3×C914yescomplemented
281D4—312no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—312no—
285C2×C4—33no—
286C2×C4—66no—
287C2×C4—66no—
288C2×C4C22×D3×C913yes—
289C2×C4C22×D3×C913yes—
290C2×C4C22×D3×C911yescharacteristic
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—66no—
295C2×C3C23×C4×C911yescharacteristic
296C2×C3C2×C9×#4111yescharacteristic
297C2×C3C2×C9×#4111yescharacteristic
298C2×C3C22×C3×C4×D311yescharacteristic
299C2×C3C2×C3×#23911yescharacteristic
300C2×C3C3×D3×#4111yescharacteristic
301D3—26no—
302C22—36no—
303C22—63no—
304C22—63no—
305C22C23×D3×C911yescharacteristic
306C4—23no—
307C4—32no—
308C4—32no—
309C4—63no—
310C4C2×C4×D3×C916yes—
311C4C9×#23912yesdirect product factor
312C3—21no—
313C3C4×C9×#4111yescharacteristic, complemented
314C3C3×C4×#23911yescharacteristic
315C2—66no—
316C2C22×C4×D3×C911yescharacteristic
317C2C2×C9×#23911yescharacteristic
318C2D3×C9×#4111yescharacteristic
subgroupG/H
centerC2×C4×C9C22×D3
commutatorC2×C3C23×C4×C9
FrattiniC22×C3C23×C3×D3
FittingC3×C4×Q8×C9C2
socle, cosocleC22×C32C24×C3
prime radicalC2×C3×#239C2×C3
hypercenterC4×Q8×C9D3
hypocenterC3C4×C9×#41
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4×C9 ◁ C4×Q8×C9

Complements

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

NH ≅ G/Ntype
2. C3×C4×Q8×C9C2semidirect
3. Q8×C9×Dic3C2semidirect
4. C9×#893C2semidirect
5. C9×#892C2semidirect
6. C4×C9×D12C2semidirect
7. C42×D3×C9C2semidirect
11. C9×#239C4direct (N × H)
33. C4×Q8×C9D3semidirect
39. C3×Q8×C9C2×C4semidirect
46. C9×D12C2×C4semidirect
49. C4×D3×C9C2×C4semidirect
51. C4×#239C9direct (N × H)
87. C3×C4×C9#41semidirect
90. C9×Dic3#41semidirect
95. C3×C4×Q8C2×C9semidirect
101. Q8×Dic3C2×C9semidirect
102. #893C2×C9semidirect
103. #892C2×C9semidirect
104. C4×D12C2×C9semidirect
105. C42×D3C2×C9semidirect
120. Q8×C9C4×D3semidirect
147. #239C4×C9direct (N × H)
191. C4×C9#239direct (N × H)
194. C4×Q8D3×C9semidirect
199. C3×C9C4×#41semidirect
203. C3×Q8C2×C4×C9semidirect
224. D12C2×C4×C9semidirect
227. C4×D3C2×C4×C9semidirect
270. C3×C4C9×#41semidirect
275. Dic3C9×#41semidirect
278. C9C4×#239direct (N × H)
280. Q8C4×D3×C9semidirect
311. C4C9×#239direct (N × H)
313. C3C4×C9×#41semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2392
C3×C4×Q8×C92
Q8×C9×Dic32
C9×#8932
C9×#8922
C4×C9×D122
C42×D3×C92
C3×C4×#2393
C4×C9×#413