Finite Group Databasebeta

← C24×C3×C5×C8×C9 #72195166

Subgroups — C24×C3×C5×C8×C9 #72195166

Order 17280.

19760 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
1C24×C3×C4×C5×C9C211yesmaximal, characteristic
2C23×C3×C5×C8×C9C2130yesmaximal, direct product factor
3C24×C32×C5×C8C311yesmaximal, characteristic
4C24×C5×C8×C9C313yesmaximal, direct product factor
5C25×C3×C5×C9C411yescharacteristic
6C23×C3×C4×C5×C9C4115yes—
7C23×C3×C4×C5×C9C22115yes—
8C22×C3×C5×C8×C9C221140yesdirect product factor
9C24×C3×C8×C9C511yesmaximal, characteristic, direct product factor, Hall
10C24×C32×C4×C5C2×C311yescharacteristic
11C24×C4×C5×C9C2×C313yes—
12C23×C32×C5×C8C2×C3130yes—
13C23×C5×C8×C9C2×C3190yesdirect product factor
14C24×C3×C5×C9C8116yesdirect product factor
15C24×C3×C5×C9C2×C4115yes—
16C22×C3×C4×C5×C9C2×C41105yes—
17C22×C3×C4×C5×C9C23135yes—
18C2×C3×C5×C8×C9C231120yesdirect product factor
19C24×C3×C5×C8C913yesdirect product factor
20C24×C3×C5×C8C3211yescharacteristic
21C24×C3×C4×C9C2×C511yescharacteristic
22C23×C3×C8×C9C2×C5130yesdirect product factor
23C25×C32×C5C3×C411yescharacteristic
24C23×C32×C4×C5C3×C4115yes—
25C23×C32×C4×C5C22×C3115yes—
26C25×C5×C9C3×C413yes—
27C22×C32×C5×C8C22×C31140yes—
28C23×C4×C5×C9C3×C4145yes—
29C23×C4×C5×C9C22×C3145yes—
30C22×C5×C8×C9C22×C31420yesdirect product factor
31C24×C32×C8C3×C511yescharacteristic
32C24×C8×C9C3×C513yesdirect product factor
33C23×C3×C5×C9C2×C81120yesdirect product factor
34C23×C3×C5×C9C22×C4135yes—
35C2×C3×C4×C5×C9C22×C41105yes—
36C2×C3×C4×C5×C9C24115yes—
37C3×C5×C8×C9C24116yesdirect product factor
38C24×C3×C4×C5C2×C913yes—
39C24×C3×C4×C5C2×C3211yescharacteristic
40C23×C3×C5×C8C2×C9190yesdirect product factor
41C23×C3×C5×C8C2×C32130yes—
42C25×C3×C9C4×C511yescharacteristic
43C23×C3×C4×C9C4×C5115yes—
44C23×C3×C4×C9C22×C5115yes—
45C22×C3×C8×C9C22×C51140yesdirect product factor
46C24×C32×C5C3×C8116yes—
47C24×C32×C5C2×C3×C4115yes—
48C22×C32×C4×C5C2×C3×C41105yes—
49C22×C32×C4×C5C23×C3135yes—
50C24×C5×C9C3×C8148yesdirect product factor
51C24×C5×C9C2×C3×C4145yes—
52C2×C32×C5×C8C23×C31120yes—
53C22×C4×C5×C9C2×C3×C41315yes—
54C22×C4×C5×C9C23×C31105yes—
55C2×C5×C8×C9C23×C31360yesdirect product factor
56C24×C5×C8C3×C911yescharacteristic, direct product factor, Hall
57C24×C32×C4C2×C3×C511yescharacteristic
58C23×C32×C8C2×C3×C5130yes—
59C24×C4×C9C2×C3×C513yes—
60C23×C8×C9C2×C3×C5190yesdirect product factor
61C22×C3×C5×C9C22×C81140yesdirect product factor
62C22×C3×C5×C9C23×C4115yes—
63C3×C4×C5×C9C23×C4115yes—
64C3×C4×C5×C9C2511yescharacteristic
65C25×C3×C5C4×C913yes—
66C25×C3×C5C32×C411yescharacteristic
67C23×C3×C4×C5C4×C9145yes—
68C23×C3×C4×C5C22×C9145yes—
69C23×C3×C4×C5C32×C4115yes—
70C23×C3×C4×C5C22×C32115yes—
71C22×C3×C5×C8C22×C91420yesdirect product factor
72C22×C3×C5×C8C22×C321140yes—
73C24×C3×C9C5×C8116yesdirect product factor
74C24×C3×C9C2×C4×C5115yes—
75C22×C3×C4×C9C2×C4×C51105yes—
76C22×C3×C4×C9C23×C5135yes—
77C2×C3×C8×C9C23×C51120yesdirect product factor
78C24×C3×C8C5×C913yesdirect product factor
79C24×C3×C8C32×C511yescharacteristic
80C23×C32×C5C2×C3×C81120yes—
81C23×C32×C5C22×C3×C4135yes—
82C2×C32×C4×C5C22×C3×C41105yes—
83C2×C32×C4×C5C24×C3115yes—
84C23×C5×C9C2×C3×C81360yesdirect product factor
85C23×C5×C9C22×C3×C41105yes—
86C32×C5×C8C24×C3116yes—
87C2×C4×C5×C9C22×C3×C41315yes—
88C2×C4×C5×C9C24×C3145yes—
89C5×C8×C9C24×C3148yesdirect product factor
90C24×C4×C5C2×C3×C911yescharacteristic
91C23×C5×C8C2×C3×C9130yesdirect product factor
92C25×C32C3×C4×C511yescharacteristic
93C23×C32×C4C3×C4×C5115yes—
94C23×C32×C4C22×C3×C5115yes—
95C25×C9C3×C4×C513yes—
96C22×C32×C8C22×C3×C51140yes—
97C23×C4×C9C3×C4×C5145yes—
98C23×C4×C9C22×C3×C5145yes—
99C22×C8×C9C22×C3×C51420yesdirect product factor
100C2×C3×C5×C9C23×C8130yesdirect product factor
101C2×C3×C5×C9C24×C411yescharacteristic
102C24×C3×C5C8×C9148yesdirect product factor
103C24×C3×C5C2×C4×C9145yes—
104C24×C3×C5C32×C8116yes—
105C24×C3×C5C2×C32×C4115yes—
106C22×C3×C4×C5C2×C4×C91315yes—
107C22×C3×C4×C5C23×C91105yes—
108C22×C3×C4×C5C2×C32×C41105yes—
109C22×C3×C4×C5C23×C32135yes—
110C2×C3×C5×C8C23×C91360yesdirect product factor
111C2×C3×C5×C8C23×C321120yes—
112C23×C3×C9C2×C5×C81120yesdirect product factor
113C23×C3×C9C22×C4×C5135yes—
114C2×C3×C4×C9C22×C4×C51105yes—
115C2×C3×C4×C9C24×C5115yes—
116C3×C8×C9C24×C5116yesdirect product factor
117C24×C3×C4C2×C5×C913yes—
118C24×C3×C4C2×C32×C511yescharacteristic
119C23×C3×C8C2×C5×C9190yesdirect product factor
120C23×C3×C8C2×C32×C5130yes—
121C22×C32×C5C22×C3×C81140yes—
122C22×C32×C5C23×C3×C4115yes—
123C32×C4×C5C23×C3×C4115yes—
124C32×C4×C5C25×C311yescharacteristic
125C22×C5×C9C22×C3×C81420yesdirect product factor
126C22×C5×C9C23×C3×C4145yes—
127C4×C5×C9C23×C3×C4145yes—
128C4×C5×C9C25×C313yes—
129C25×C5C3×C4×C911yescharacteristic
130C23×C4×C5C3×C4×C9115yes—
131C23×C4×C5C22×C3×C9115yes—
132C22×C5×C8C22×C3×C91140yesdirect product factor
133C24×C32C3×C5×C8116yes—
134C24×C32C2×C3×C4×C5115yes—
135C22×C32×C4C2×C3×C4×C51105yes—
136C22×C32×C4C23×C3×C5135yes—
137C24×C9C3×C5×C8148yesdirect product factor
138C24×C9C2×C3×C4×C5145yes—
139C2×C32×C8C23×C3×C51120yes—
140C22×C4×C9C2×C3×C4×C51315yes—
141C22×C4×C9C23×C3×C51105yes—
142C2×C8×C9C23×C3×C51360yesdirect product factor
143C3×C5×C9C24×C811yescharacteristic, direct product factor, Hall
144C24×C8C3×C5×C911yescharacteristic, direct product factor, Sylow
145C23×C3×C5C2×C8×C91360yesdirect product factor
146C23×C3×C5C22×C4×C91105yes—
147C23×C3×C5C2×C32×C81120yes—
148C23×C3×C5C22×C32×C4135yes—
149C2×C3×C4×C5C22×C4×C91315yes—
150C2×C3×C4×C5C24×C9145yes—
151C2×C3×C4×C5C22×C32×C41105yes—
152C2×C3×C4×C5C24×C32115yes—
153C3×C5×C8C24×C9148yesdirect product factor
154C3×C5×C8C24×C32116yes—
155C22×C3×C9C22×C5×C81140yesdirect product factor
156C22×C3×C9C23×C4×C5115yes—
157C3×C4×C9C23×C4×C5115yes—
158C3×C4×C9C25×C511yescharacteristic
159C25×C3C4×C5×C913yes—
160C25×C3C32×C4×C511yescharacteristic
161C23×C3×C4C4×C5×C9145yes—
162C23×C3×C4C22×C5×C9145yes—
163C23×C3×C4C32×C4×C5115yes—
164C23×C3×C4C22×C32×C5115yes—
165C22×C3×C8C22×C5×C91420yesdirect product factor
166C22×C3×C8C22×C32×C51140yes—
167C2×C32×C5C23×C3×C8130yes—
168C2×C32×C5C24×C3×C411yescharacteristic
169C2×C5×C9C23×C3×C8190yesdirect product factor
170C2×C5×C9C24×C3×C413yes—
171C24×C5C3×C8×C9116yesdirect product factor
172C24×C5C2×C3×C4×C9115yes—
173C22×C4×C5C2×C3×C4×C91105yes—
174C22×C4×C5C23×C3×C9135yes—
175C2×C5×C8C23×C3×C91120yesdirect product factor
176C23×C32C2×C3×C5×C81120yes—
177C23×C32C22×C3×C4×C5135yes—
178C2×C32×C4C22×C3×C4×C51105yes—
179C2×C32×C4C24×C3×C5115yes—
180C23×C9C2×C3×C5×C81360yesdirect product factor
181C23×C9C22×C3×C4×C51105yes—
182C32×C8C24×C3×C5116yes—
183C2×C4×C9C22×C3×C4×C51315yes—
184C2×C4×C9C24×C3×C5145yes—
185C8×C9C24×C3×C5148yesdirect product factor
186C24×C4C2×C3×C5×C911yescharacteristic
187C23×C8C2×C3×C5×C9130yesdirect product factor
188C22×C3×C5C22×C8×C91420yesdirect product factor
189C22×C3×C5C23×C4×C9145yes—
190C22×C3×C5C22×C32×C81140yes—
191C22×C3×C5C23×C32×C4115yes—
192C3×C4×C5C23×C4×C9145yes—
193C3×C4×C5C25×C913yes—
194C3×C4×C5C23×C32×C4115yes—
195C3×C4×C5C25×C3211yescharacteristic
196C2×C3×C9C23×C5×C8130yesdirect product factor
197C2×C3×C9C24×C4×C511yescharacteristic
198C24×C3C5×C8×C9148yesdirect product factor
199C24×C3C2×C4×C5×C9145yes—
200C24×C3C32×C5×C8116yes—
201C24×C3C2×C32×C4×C5115yes—
202C22×C3×C4C2×C4×C5×C91315yes—
203C22×C3×C4C23×C5×C91105yes—
204C22×C3×C4C2×C32×C4×C51105yes—
205C22×C3×C4C23×C32×C5135yes—
206C2×C3×C8C23×C5×C91360yesdirect product factor
207C2×C3×C8C23×C32×C51120yes—
208C32×C5C24×C3×C811yescharacteristic
209C5×C9C24×C3×C813yesdirect product factor
210C23×C5C2×C3×C8×C91120yesdirect product factor
211C23×C5C22×C3×C4×C9135yes—
212C2×C4×C5C22×C3×C4×C91105yes—
213C2×C4×C5C24×C3×C9115yes—
214C5×C8C24×C3×C9116yesdirect product factor
215C22×C32C22×C3×C5×C81140yes—
216C22×C32C23×C3×C4×C5115yes—
217C32×C4C23×C3×C4×C5115yes—
218C32×C4C25×C3×C511yescharacteristic
219C22×C9C22×C3×C5×C81420yesdirect product factor
220C22×C9C23×C3×C4×C5145yes—
221C4×C9C23×C3×C4×C5145yes—
222C4×C9C25×C3×C513yes—
223C25C3×C4×C5×C911yescharacteristic
224C23×C4C3×C4×C5×C9115yes—
225C23×C4C22×C3×C5×C9115yes—
226C22×C8C22×C3×C5×C91140yesdirect product factor
227C2×C3×C5C23×C8×C9190yesdirect product factor
228C2×C3×C5C24×C4×C913yes—
229C2×C3×C5C23×C32×C8130yes—
230C2×C3×C5C24×C32×C411yescharacteristic
231C3×C9C24×C5×C811yescharacteristic, direct product factor, Sylow
232C23×C3C2×C5×C8×C91360yesdirect product factor
233C23×C3C22×C4×C5×C91105yes—
234C23×C3C2×C32×C5×C81120yes—
235C23×C3C22×C32×C4×C5135yes—
236C2×C3×C4C22×C4×C5×C91315yes—
237C2×C3×C4C24×C5×C9145yes—
238C2×C3×C4C22×C32×C4×C51105yes—
239C2×C3×C4C24×C32×C5115yes—
240C3×C8C24×C5×C9148yesdirect product factor
241C3×C8C24×C32×C5116yes—
242C22×C5C22×C3×C8×C91140yesdirect product factor
243C22×C5C23×C3×C4×C9115yes—
244C4×C5C23×C3×C4×C9115yes—
245C4×C5C25×C3×C911yescharacteristic
246C2×C32C23×C3×C5×C8130yes—
247C2×C32C24×C3×C4×C511yescharacteristic
248C2×C9C23×C3×C5×C8190yesdirect product factor
249C2×C9C24×C3×C4×C513yes—
250C24C3×C5×C8×C9116yesdirect product factor
251C24C2×C3×C4×C5×C9115yes—
252C22×C4C2×C3×C4×C5×C91105yes—
253C22×C4C23×C3×C5×C9135yes—
254C2×C8C23×C3×C5×C91120yesdirect product factor
255C3×C5C24×C8×C913yesdirect product factor
256C3×C5C24×C32×C811yescharacteristic
257C22×C3C22×C5×C8×C91420yesdirect product factor
258C22×C3C23×C4×C5×C9145yes—
259C22×C3C22×C32×C5×C81140yes—
260C22×C3C23×C32×C4×C5115yes—
261C3×C4C23×C4×C5×C9145yes—
262C3×C4C25×C5×C913yes—
263C3×C4C23×C32×C4×C5115yes—
264C3×C4C25×C32×C511yescharacteristic
265C2×C5C23×C3×C8×C9130yesdirect product factor
266C2×C5C24×C3×C4×C911yescharacteristic
267C32C24×C3×C5×C811yescharacteristic
268C9C24×C3×C5×C813yesdirect product factor
269C23C2×C3×C5×C8×C91120yesdirect product factor
270C23C22×C3×C4×C5×C9135yes—
271C2×C4C22×C3×C4×C5×C91105yes—
272C2×C4C24×C3×C5×C9115yes—
273C8C24×C3×C5×C9116yesdirect product factor
274C2×C3C23×C5×C8×C9190yesdirect product factor
275C2×C3C23×C32×C5×C8130yes—
276C2×C3C24×C4×C5×C913yes—
277C2×C3C24×C32×C4×C511yescharacteristic
278C5C24×C3×C8×C911yescharacteristic, direct product factor, Sylow
279C22C22×C3×C5×C8×C91140yesdirect product factor
280C22C23×C3×C4×C5×C9115yes—
281C4C23×C3×C4×C5×C9115yes—
282C4C25×C3×C5×C911yescharacteristic
283C3C24×C5×C8×C913yesdirect product factor
284C3C24×C32×C5×C811yescharacteristic
285C2C23×C3×C5×C8×C9130yesdirect product factor
286C2C24×C3×C4×C5×C911yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC3×C4C25×C32×C5
socle, prime radicalC25×C32×C5C3×C4
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C3×C5×C8×C9C2direct (N × H)
4. C24×C5×C8×C9C3direct (N × H)
8. C22×C3×C5×C8×C9C22direct (N × H)
9. C24×C3×C8×C9C5direct (N × H)
13. C23×C5×C8×C9C2×C3direct (N × H)
14. C24×C3×C5×C9C8direct (N × H)
18. C2×C3×C5×C8×C9C23direct (N × H)
19. C24×C3×C5×C8C9direct (N × H)
22. C23×C3×C8×C9C2×C5direct (N × H)
30. C22×C5×C8×C9C22×C3direct (N × H)
32. C24×C8×C9C3×C5direct (N × H)
33. C23×C3×C5×C9C2×C8direct (N × H)
37. C3×C5×C8×C9C24direct (N × H)
40. C23×C3×C5×C8C2×C9direct (N × H)
45. C22×C3×C8×C9C22×C5direct (N × H)
50. C24×C5×C9C3×C8direct (N × H)
55. C2×C5×C8×C9C23×C3direct (N × H)
56. C24×C5×C8C3×C9direct (N × H)
60. C23×C8×C9C2×C3×C5direct (N × H)
61. C22×C3×C5×C9C22×C8direct (N × H)
71. C22×C3×C5×C8C22×C9direct (N × H)
73. C24×C3×C9C5×C8direct (N × H)
77. C2×C3×C8×C9C23×C5direct (N × H)
78. C24×C3×C8C5×C9direct (N × H)
84. C23×C5×C9C2×C3×C8direct (N × H)
89. C5×C8×C9C24×C3direct (N × H)
91. C23×C5×C8C2×C3×C9direct (N × H)
99. C22×C8×C9C22×C3×C5direct (N × H)
100. C2×C3×C5×C9C23×C8direct (N × H)
102. C24×C3×C5C8×C9direct (N × H)
110. C2×C3×C5×C8C23×C9direct (N × H)
112. C23×C3×C9C2×C5×C8direct (N × H)
116. C3×C8×C9C24×C5direct (N × H)
119. C23×C3×C8C2×C5×C9direct (N × H)
125. C22×C5×C9C22×C3×C8direct (N × H)
132. C22×C5×C8C22×C3×C9direct (N × H)
137. C24×C9C3×C5×C8direct (N × H)
142. C2×C8×C9C23×C3×C5direct (N × H)
143. C3×C5×C9C24×C8direct (N × H)
144. C24×C8C3×C5×C9direct (N × H)
145. C23×C3×C5C2×C8×C9direct (N × H)
153. C3×C5×C8C24×C9direct (N × H)
155. C22×C3×C9C22×C5×C8direct (N × H)
165. C22×C3×C8C22×C5×C9direct (N × H)
169. C2×C5×C9C23×C3×C8direct (N × H)
171. C24×C5C3×C8×C9direct (N × H)
175. C2×C5×C8C23×C3×C9direct (N × H)
180. C23×C9C2×C3×C5×C8direct (N × H)
185. C8×C9C24×C3×C5direct (N × H)
187. C23×C8C2×C3×C5×C9direct (N × H)
188. C22×C3×C5C22×C8×C9direct (N × H)
196. C2×C3×C9C23×C5×C8direct (N × H)
198. C24×C3C5×C8×C9direct (N × H)
206. C2×C3×C8C23×C5×C9direct (N × H)
209. C5×C9C24×C3×C8direct (N × H)
210. C23×C5C2×C3×C8×C9direct (N × H)
214. C5×C8C24×C3×C9direct (N × H)
219. C22×C9C22×C3×C5×C8direct (N × H)
226. C22×C8C22×C3×C5×C9direct (N × H)
227. C2×C3×C5C23×C8×C9direct (N × H)
231. C3×C9C24×C5×C8direct (N × H)
232. C23×C3C2×C5×C8×C9direct (N × H)
240. C3×C8C24×C5×C9direct (N × H)
242. C22×C5C22×C3×C8×C9direct (N × H)
248. C2×C9C23×C3×C5×C8direct (N × H)
250. C24C3×C5×C8×C9direct (N × H)
254. C2×C8C23×C3×C5×C9direct (N × H)
255. C3×C5C24×C8×C9direct (N × H)
257. C22×C3C22×C5×C8×C9direct (N × H)
265. C2×C5C23×C3×C8×C9direct (N × H)
268. C9C24×C3×C5×C8direct (N × H)
269. C23C2×C3×C5×C8×C9direct (N × H)
273. C8C24×C3×C5×C9direct (N × H)
274. C2×C3C23×C5×C8×C9direct (N × H)
278. C5C24×C3×C8×C9direct (N × H)
279. C22C22×C3×C5×C8×C9direct (N × H)
283. C3C24×C5×C8×C9direct (N × H)
285. C2C23×C3×C5×C8×C9direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C4×C5×C92
C23×C3×C5×C8×C92
C24×C32×C5×C83
C24×C5×C8×C93
C24×C3×C8×C95