Finite Group Databasebeta

← C24×C32×C9×C16 #72198345

Subgroups — C24×C32×C9×C16 #72198345

Order 20736.

64750 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C32×C8×C9C211yesmaximal, characteristic
2C23×C32×C9×C16C2130yesmaximal, direct product factor
3C24×C33×C16C311yesmaximal, characteristic
4C24×C3×C9×C16C3112yesmaximal, direct product factor
5C24×C32×C4×C9C411yescharacteristic
6C23×C32×C8×C9C4115yes—
7C23×C32×C8×C9C22115yes—
8C22×C32×C9×C16C221140yesdirect product factor
9C24×C3×C8×C9C2×C3112yes—
10C24×C33×C8C2×C311yescharacteristic
11C23×C33×C16C2×C3130yes—
12C23×C3×C9×C16C2×C31360yesdirect product factor
13C25×C32×C9C811yescharacteristic
14C23×C32×C4×C9C8115yes—
15C23×C32×C4×C9C2×C4115yes—
16C22×C32×C8×C9C2×C41105yes—
17C22×C32×C8×C9C23135yes—
18C2×C32×C9×C16C231120yesdirect product factor
19C24×C32×C16C919yesdirect product factor
20C24×C32×C16C3214yes—
21C24×C9×C16C3219yesdirect product factor
22C24×C33×C4C3×C411yescharacteristic
23C23×C33×C8C3×C4115yes—
24C23×C33×C8C22×C3115yes—
25C24×C3×C4×C9C3×C4112yes—
26C22×C33×C16C22×C31140yes—
27C23×C3×C8×C9C3×C41180yes—
28C23×C3×C8×C9C22×C31180yes—
29C22×C3×C9×C16C22×C311680yesdirect product factor
30C24×C32×C9C16116yesdirect product factor
31C24×C32×C9C2×C8115yes—
32C22×C32×C4×C9C2×C81105yes—
33C22×C32×C4×C9C22×C4135yes—
34C2×C32×C8×C9C22×C41105yes—
35C2×C32×C8×C9C24115yes—
36C32×C9×C16C24116yesdirect product factor
37C24×C32×C8C2×C919yes—
38C24×C32×C8C2×C3214yes—
39C23×C32×C16C2×C91270yesdirect product factor
40C23×C32×C16C2×C321120yes—
41C24×C8×C9C2×C3219yes—
42C23×C9×C16C2×C321270yesdirect product factor
43C25×C33C3×C811yescharacteristic
44C23×C33×C4C3×C8115yes—
45C23×C33×C4C2×C3×C4115yes—
46C25×C3×C9C3×C8112yes—
47C22×C33×C8C2×C3×C41105yes—
48C22×C33×C8C23×C3135yes—
49C23×C3×C4×C9C3×C81180yes—
50C23×C3×C4×C9C2×C3×C41180yes—
51C2×C33×C16C23×C31120yes—
52C22×C3×C8×C9C2×C3×C411260yes—
53C22×C3×C8×C9C23×C31420yes—
54C2×C3×C9×C16C23×C311440yesdirect product factor
55C24×C3×C16C3×C9112yesdirect product factor
56C24×C3×C16C3311yescharacteristic
57C23×C32×C9C2×C161120yesdirect product factor
58C23×C32×C9C22×C8135yes—
59C2×C32×C4×C9C22×C81105yes—
60C2×C32×C4×C9C23×C4115yes—
61C32×C8×C9C23×C4115yes—
62C32×C8×C9C2511yescharacteristic
63C24×C32×C4C4×C919yes—
64C24×C32×C4C32×C414yes—
65C23×C32×C8C4×C91135yes—
66C23×C32×C8C22×C91135yes—
67C23×C32×C8C32×C4160yes—
68C23×C32×C8C22×C32160yes—
69C24×C4×C9C32×C419yes—
70C22×C32×C16C22×C911260yesdirect product factor
71C22×C32×C16C22×C321560yes—
72C23×C8×C9C32×C41135yes—
73C23×C8×C9C22×C321135yes—
74C22×C9×C16C22×C3211260yesdirect product factor
75C24×C33C3×C16116yes—
76C24×C33C2×C3×C8115yes—
77C22×C33×C4C2×C3×C81105yes—
78C22×C33×C4C22×C3×C4135yes—
79C24×C3×C9C3×C161192yesdirect product factor
80C24×C3×C9C2×C3×C81180yes—
81C2×C33×C8C22×C3×C41105yes—
82C2×C33×C8C24×C3115yes—
83C22×C3×C4×C9C2×C3×C811260yes—
84C22×C3×C4×C9C22×C3×C41420yes—
85C33×C16C24×C3116yes—
86C2×C3×C8×C9C22×C3×C411260yes—
87C2×C3×C8×C9C24×C31180yes—
88C3×C9×C16C24×C31192yesdirect product factor
89C24×C3×C8C2×C3×C9112yes—
90C24×C3×C8C2×C3311yescharacteristic
91C23×C3×C16C2×C3×C91360yesdirect product factor
92C23×C3×C16C2×C33130yes—
93C22×C32×C9C22×C161140yesdirect product factor
94C22×C32×C9C23×C8115yes—
95C32×C4×C9C23×C8115yes—
96C32×C4×C9C24×C411yescharacteristic
97C25×C32C8×C919yes—
98C25×C32C32×C814yes—
99C23×C32×C4C8×C91135yes—
100C23×C32×C4C2×C4×C91135yes—
101C23×C32×C4C32×C8160yes—
102C23×C32×C4C2×C32×C4160yes—
103C25×C9C32×C819yes—
104C22×C32×C8C2×C4×C91945yes—
105C22×C32×C8C23×C91315yes—
106C22×C32×C8C2×C32×C41420yes—
107C22×C32×C8C23×C321140yes—
108C23×C4×C9C32×C81135yes—
109C23×C4×C9C2×C32×C41135yes—
110C2×C32×C16C23×C911080yesdirect product factor
111C2×C32×C16C23×C321480yes—
112C22×C8×C9C2×C32×C41945yes—
113C22×C8×C9C23×C321315yes—
114C2×C9×C16C23×C3211080yesdirect product factor
115C24×C16C32×C911yescharacteristic, direct product factor, Sylow
116C23×C33C2×C3×C161120yes—
117C23×C33C22×C3×C8135yes—
118C2×C33×C4C22×C3×C81105yes—
119C2×C33×C4C23×C3×C4115yes—
120C23×C3×C9C2×C3×C1611440yesdirect product factor
121C23×C3×C9C22×C3×C81420yes—
122C33×C8C23×C3×C4115yes—
123C33×C8C25×C311yescharacteristic
124C2×C3×C4×C9C22×C3×C811260yes—
125C2×C3×C4×C9C23×C3×C41180yes—
126C3×C8×C9C23×C3×C41180yes—
127C3×C8×C9C25×C3112yes—
128C24×C3×C4C3×C4×C9112yes—
129C24×C3×C4C33×C411yescharacteristic
130C23×C3×C8C3×C4×C91180yes—
131C23×C3×C8C22×C3×C91180yes—
132C23×C3×C8C33×C4115yes—
133C23×C3×C8C22×C33115yes—
134C22×C3×C16C22×C3×C911680yesdirect product factor
135C22×C3×C16C22×C331140yes—
136C2×C32×C9C23×C16130yesdirect product factor
137C2×C32×C9C24×C811yescharacteristic
138C24×C32C9×C161144yesdirect product factor
139C24×C32C2×C8×C91135yes—
140C24×C32C32×C16164yes—
141C24×C32C2×C32×C8160yes—
142C22×C32×C4C2×C8×C91945yes—
143C22×C32×C4C22×C4×C91315yes—
144C22×C32×C4C2×C32×C81420yes—
145C22×C32×C4C22×C32×C41140yes—
146C24×C9C32×C161144yesdirect product factor
147C24×C9C2×C32×C81135yes—
148C2×C32×C8C22×C4×C91945yes—
149C2×C32×C8C24×C91135yes—
150C2×C32×C8C22×C32×C41420yes—
151C2×C32×C8C24×C32160yes—
152C22×C4×C9C2×C32×C81945yes—
153C22×C4×C9C22×C32×C41315yes—
154C32×C16C24×C91144yesdirect product factor
155C32×C16C24×C32164yes—
156C2×C8×C9C22×C32×C41945yes—
157C2×C8×C9C24×C321135yes—
158C9×C16C24×C321144yesdirect product factor
159C24×C8C2×C32×C911yescharacteristic
160C23×C16C2×C32×C9130yesdirect product factor
161C22×C33C22×C3×C161140yes—
162C22×C33C23×C3×C8115yes—
163C33×C4C23×C3×C8115yes—
164C33×C4C24×C3×C411yescharacteristic
165C22×C3×C9C22×C3×C1611680yesdirect product factor
166C22×C3×C9C23×C3×C81180yes—
167C3×C4×C9C23×C3×C81180yes—
168C3×C4×C9C24×C3×C4112yes—
169C25×C3C3×C8×C9112yes—
170C25×C3C33×C811yescharacteristic
171C23×C3×C4C3×C8×C91180yes—
172C23×C3×C4C2×C3×C4×C91180yes—
173C23×C3×C4C33×C8115yes—
174C23×C3×C4C2×C33×C4115yes—
175C22×C3×C8C2×C3×C4×C911260yes—
176C22×C3×C8C23×C3×C91420yes—
177C22×C3×C8C2×C33×C41105yes—
178C22×C3×C8C23×C33135yes—
179C2×C3×C16C23×C3×C911440yesdirect product factor
180C2×C3×C16C23×C331120yes—
181C32×C9C24×C1611yescharacteristic, direct product factor, Sylow
182C23×C32C2×C9×C1611080yesdirect product factor
183C23×C32C22×C8×C91315yes—
184C23×C32C2×C32×C161480yes—
185C23×C32C22×C32×C81140yes—
186C2×C32×C4C22×C8×C91945yes—
187C2×C32×C4C23×C4×C91135yes—
188C2×C32×C4C22×C32×C81420yes—
189C2×C32×C4C23×C32×C4160yes—
190C23×C9C2×C32×C1611080yesdirect product factor
191C23×C9C22×C32×C81315yes—
192C32×C8C23×C4×C91135yes—
193C32×C8C25×C919yes—
194C32×C8C23×C32×C4160yes—
195C32×C8C25×C3214yes—
196C2×C4×C9C22×C32×C81945yes—
197C2×C4×C9C23×C32×C41135yes—
198C8×C9C23×C32×C41135yes—
199C8×C9C25×C3219yes—
200C24×C4C32×C4×C911yescharacteristic
201C23×C8C32×C4×C9115yes—
202C23×C8C22×C32×C9115yes—
203C22×C16C22×C32×C91140yesdirect product factor
204C2×C33C23×C3×C16130yes—
205C2×C33C24×C3×C811yescharacteristic
206C2×C3×C9C23×C3×C161360yesdirect product factor
207C2×C3×C9C24×C3×C8112yes—
208C24×C3C3×C9×C161192yesdirect product factor
209C24×C3C2×C3×C8×C91180yes—
210C24×C3C33×C16116yes—
211C24×C3C2×C33×C8115yes—
212C22×C3×C4C2×C3×C8×C911260yes—
213C22×C3×C4C22×C3×C4×C91420yes—
214C22×C3×C4C2×C33×C81105yes—
215C22×C3×C4C22×C33×C4135yes—
216C2×C3×C8C22×C3×C4×C911260yes—
217C2×C3×C8C24×C3×C91180yes—
218C2×C3×C8C22×C33×C41105yes—
219C2×C3×C8C24×C33115yes—
220C3×C16C24×C3×C91192yesdirect product factor
221C3×C16C24×C33116yes—
222C22×C32C22×C9×C1611260yesdirect product factor
223C22×C32C23×C8×C91135yes—
224C22×C32C22×C32×C161560yes—
225C22×C32C23×C32×C8160yes—
226C32×C4C23×C8×C91135yes—
227C32×C4C24×C4×C919yes—
228C32×C4C23×C32×C8160yes—
229C32×C4C24×C32×C414yes—
230C22×C9C22×C32×C1611260yesdirect product factor
231C22×C9C23×C32×C81135yes—
232C4×C9C23×C32×C81135yes—
233C4×C9C24×C32×C419yes—
234C25C32×C8×C911yescharacteristic
235C23×C4C32×C8×C9115yes—
236C23×C4C2×C32×C4×C9115yes—
237C22×C8C2×C32×C4×C91105yes—
238C22×C8C23×C32×C9135yes—
239C2×C16C23×C32×C91120yesdirect product factor
240C33C24×C3×C1611yescharacteristic
241C3×C9C24×C3×C16112yesdirect product factor
242C23×C3C2×C3×C9×C1611440yesdirect product factor
243C23×C3C22×C3×C8×C91420yes—
244C23×C3C2×C33×C161120yes—
245C23×C3C22×C33×C8135yes—
246C2×C3×C4C22×C3×C8×C911260yes—
247C2×C3×C4C23×C3×C4×C91180yes—
248C2×C3×C4C22×C33×C81105yes—
249C2×C3×C4C23×C33×C4115yes—
250C3×C8C23×C3×C4×C91180yes—
251C3×C8C25×C3×C9112yes—
252C3×C8C23×C33×C4115yes—
253C3×C8C25×C3311yescharacteristic
254C2×C32C23×C9×C161270yesdirect product factor
255C2×C32C24×C8×C919yes—
256C2×C32C23×C32×C161120yes—
257C2×C32C24×C32×C814yes—
258C2×C9C23×C32×C161270yesdirect product factor
259C2×C9C24×C32×C819yes—
260C24C32×C9×C16116yesdirect product factor
261C24C2×C32×C8×C9115yes—
262C22×C4C2×C32×C8×C91105yes—
263C22×C4C22×C32×C4×C9135yes—
264C2×C8C22×C32×C4×C91105yes—
265C2×C8C24×C32×C9115yes—
266C16C24×C32×C9116yesdirect product factor
267C22×C3C22×C3×C9×C1611680yesdirect product factor
268C22×C3C23×C3×C8×C91180yes—
269C22×C3C22×C33×C161140yes—
270C22×C3C23×C33×C8115yes—
271C3×C4C23×C3×C8×C91180yes—
272C3×C4C24×C3×C4×C9112yes—
273C3×C4C23×C33×C8115yes—
274C3×C4C24×C33×C411yescharacteristic
275C32C24×C9×C1619yesdirect product factor
276C32C24×C32×C1614yes—
277C9C24×C32×C1619yesdirect product factor
278C23C2×C32×C9×C161120yesdirect product factor
279C23C22×C32×C8×C9135yes—
280C2×C4C22×C32×C8×C91105yes—
281C2×C4C23×C32×C4×C9115yes—
282C8C23×C32×C4×C9115yes—
283C8C25×C32×C911yescharacteristic
284C2×C3C23×C3×C9×C161360yesdirect product factor
285C2×C3C23×C33×C16130yes—
286C2×C3C24×C33×C811yescharacteristic
287C2×C3C24×C3×C8×C9112yes—
288C22C22×C32×C9×C161140yesdirect product factor
289C22C23×C32×C8×C9115yes—
290C4C23×C32×C8×C9115yes—
291C4C24×C32×C4×C911yescharacteristic
292C3C24×C3×C9×C16112yesdirect product factor
293C3C24×C33×C1611yescharacteristic
294C2C23×C32×C9×C16130yesdirect product factor
295C2C24×C32×C8×C911yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC3×C8C25×C33
socle, prime radicalC25×C33C3×C8
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C32×C9×C16C2direct (N × H)
4. C24×C3×C9×C16C3direct (N × H)
8. C22×C32×C9×C16C22direct (N × H)
12. C23×C3×C9×C16C2×C3direct (N × H)
18. C2×C32×C9×C16C23direct (N × H)
19. C24×C32×C16C9direct (N × H)
21. C24×C9×C16C32direct (N × H)
29. C22×C3×C9×C16C22×C3direct (N × H)
30. C24×C32×C9C16direct (N × H)
36. C32×C9×C16C24direct (N × H)
39. C23×C32×C16C2×C9direct (N × H)
42. C23×C9×C16C2×C32direct (N × H)
54. C2×C3×C9×C16C23×C3direct (N × H)
55. C24×C3×C16C3×C9direct (N × H)
57. C23×C32×C9C2×C16direct (N × H)
70. C22×C32×C16C22×C9direct (N × H)
74. C22×C9×C16C22×C32direct (N × H)
79. C24×C3×C9C3×C16direct (N × H)
88. C3×C9×C16C24×C3direct (N × H)
91. C23×C3×C16C2×C3×C9direct (N × H)
93. C22×C32×C9C22×C16direct (N × H)
110. C2×C32×C16C23×C9direct (N × H)
114. C2×C9×C16C23×C32direct (N × H)
115. C24×C16C32×C9direct (N × H)
120. C23×C3×C9C2×C3×C16direct (N × H)
134. C22×C3×C16C22×C3×C9direct (N × H)
136. C2×C32×C9C23×C16direct (N × H)
138. C24×C32C9×C16direct (N × H)
146. C24×C9C32×C16direct (N × H)
154. C32×C16C24×C9direct (N × H)
158. C9×C16C24×C32direct (N × H)
160. C23×C16C2×C32×C9direct (N × H)
165. C22×C3×C9C22×C3×C16direct (N × H)
179. C2×C3×C16C23×C3×C9direct (N × H)
181. C32×C9C24×C16direct (N × H)
182. C23×C32C2×C9×C16direct (N × H)
190. C23×C9C2×C32×C16direct (N × H)
203. C22×C16C22×C32×C9direct (N × H)
206. C2×C3×C9C23×C3×C16direct (N × H)
208. C24×C3C3×C9×C16direct (N × H)
220. C3×C16C24×C3×C9direct (N × H)
222. C22×C32C22×C9×C16direct (N × H)
230. C22×C9C22×C32×C16direct (N × H)
239. C2×C16C23×C32×C9direct (N × H)
241. C3×C9C24×C3×C16direct (N × H)
242. C23×C3C2×C3×C9×C16direct (N × H)
254. C2×C32C23×C9×C16direct (N × H)
258. C2×C9C23×C32×C16direct (N × H)
260. C24C32×C9×C16direct (N × H)
266. C16C24×C32×C9direct (N × H)
267. C22×C3C22×C3×C9×C16direct (N × H)
275. C32C24×C9×C16direct (N × H)
277. C9C24×C32×C16direct (N × H)
278. C23C2×C32×C9×C16direct (N × H)
284. C2×C3C23×C3×C9×C16direct (N × H)
288. C22C22×C32×C9×C16direct (N × H)
292. C3C24×C3×C9×C16direct (N × H)
294. C2C23×C32×C9×C16direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C32×C8×C92
C23×C32×C9×C162
C24×C33×C163
C24×C3×C9×C163