Finite Group Databasebeta

← C26×C4×C8×C9 #72196243

Subgroups — C26×C4×C8×C9 #72196243

Order 18432.

9082680 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C26×C42×C9C211yesmaximal, characteristic
2C27×C8×C9C212yesmaximal
3C25×C4×C8×C9C21252yesmaximal, direct product factor
4C26×C3×C4×C8C311yesmaximal, characteristic
5C27×C4×C9C412yes—
6C27×C4×C9C2211yescharacteristic
7C25×C42×C9C41126yes—
8C25×C42×C9C221126yes—
9C26×C8×C9C41256yesdirect product factor
10C26×C8×C9C221252yes—
11C24×C4×C8×C9C22110416yesdirect product factor
12C26×C3×C42C2×C311yescharacteristic
13C27×C3×C8C2×C312yes—
14C25×C3×C4×C8C2×C31252yes—
15C28×C9C2×C411yescharacteristic
16C26×C4×C9C81256yesdirect product factor
17C26×C4×C9C2×C4163yes—
18C26×C4×C9C2×C41128yes—
19C26×C4×C9C2×C41252yes—
20C26×C4×C9C23163yes—
21C24×C42×C9C2×C417812yes—
22C24×C42×C9C2312604yes—
23C25×C8×C9C2×C4116128yesdirect product factor
24C25×C8×C9C2315208yes—
25C23×C4×C8×C9C23189280yesdirect product factor
26C26×C4×C8C911yescharacteristic, direct product factor, Sylow
27C27×C3×C4C3×C412yes—
28C27×C3×C4C22×C311yescharacteristic
29C25×C3×C42C3×C41126yes—
30C25×C3×C42C22×C31126yes—
31C26×C3×C8C3×C41256yes—
32C26×C3×C8C22×C31252yes—
33C24×C3×C4×C8C22×C3110416yes—
34C26×C42C2×C911yescharacteristic
35C25×C4×C8C2×C91252yesdirect product factor
36C27×C8C2×C912yes—
37C27×C9C42164yes—
38C27×C9C2×C81128yes—
39C27×C9C22×C4163yes—
40C25×C4×C9C4214032yes—
41C25×C4×C9C2×C8116128yesdirect product factor
42C25×C4×C9C22×C415208yes—
43C25×C4×C9C22×C411953yes—
44C25×C4×C9C22×C414032yes—
45C25×C4×C9C241651yes—
46C24×C8×C9C22×C41166656yesdirect product factor
47C24×C8×C9C24122320yes—
48C23×C42×C9C22×C4178120yes—
49C23×C42×C9C24111160yes—
50C22×C4×C8×C9C241166656yesdirect product factor
51C28×C3C2×C3×C411yescharacteristic
52C26×C3×C4C3×C81256yes—
53C26×C3×C4C2×C3×C4163yes—
54C26×C3×C4C2×C3×C41128yes—
55C26×C3×C4C2×C3×C41252yes—
56C26×C3×C4C23×C3163yes—
57C25×C3×C8C2×C3×C4116128yes—
58C25×C3×C8C23×C315208yes—
59C24×C3×C42C2×C3×C417812yes—
60C24×C3×C42C23×C312604yes—
61C23×C3×C4×C8C23×C3189280yes—
62C26×C9C4×C814096yesdirect product factor
63C26×C9C2×C4212016yes—
64C26×C9C22×C814032yes—
65C26×C9C23×C41651yes—
66C24×C4×C9C2×C42162496yes—
67C24×C4×C9C22×C81166656yesdirect product factor
68C24×C4×C9C23×C419765yes—
69C24×C4×C9C23×C4122320yes—
70C24×C4×C9C23×C4120832yes—
71C24×C4×C9C2511395yes—
72C23×C8×C9C23×C41357120yesdirect product factor
73C23×C8×C9C25120832yes—
74C22×C42×C9C23×C41156240yes—
75C22×C42×C9C25110416yes—
76C2×C4×C8×C9C25164512yesdirect product factor
77C27×C4C4×C912yes—
78C27×C4C22×C911yescharacteristic
79C26×C8C4×C91256yesdirect product factor
80C26×C8C22×C91252yes—
81C25×C42C4×C91126yes—
82C25×C42C22×C91126yes—
83C24×C4×C8C22×C9110416yesdirect product factor
84C27×C3C3×C42164yes—
85C27×C3C2×C3×C81128yes—
86C27×C3C22×C3×C4163yes—
87C25×C3×C4C3×C4214032yes—
88C25×C3×C4C2×C3×C8116128yes—
89C25×C3×C4C22×C3×C415208yes—
90C25×C3×C4C22×C3×C411953yes—
91C25×C3×C4C22×C3×C414032yes—
92C25×C3×C4C24×C31651yes—
93C24×C3×C8C22×C3×C41166656yes—
94C24×C3×C8C24×C3122320yes—
95C23×C3×C42C22×C3×C4178120yes—
96C23×C3×C42C24×C3111160yes—
97C22×C3×C4×C8C24×C31166656yes—
98C25×C9C2×C4×C8164512yesdirect product factor
99C25×C9C22×C42110416yes—
100C25×C9C23×C8120832yes—
101C25×C9C24×C411395yes—
102C23×C4×C9C22×C421156240yes—
103C23×C4×C9C23×C81357120yesdirect product factor
104C23×C4×C9C24×C419765yes—
105C23×C4×C9C24×C4122320yes—
106C23×C4×C9C24×C4120832yes—
107C23×C4×C9C261651yes—
108C22×C8×C9C24×C41166656yesdirect product factor
109C22×C8×C9C2614032yes—
110C2×C42×C9C24×C4162496yes—
111C2×C42×C9C2612016yes—
112C4×C8×C9C2614096yesdirect product factor
113C28C2×C4×C911yescharacteristic
114C26×C4C8×C91256yesdirect product factor
115C26×C4C2×C4×C9163yes—
116C26×C4C2×C4×C91128yes—
117C26×C4C2×C4×C91252yes—
118C26×C4C23×C9163yes—
119C25×C8C2×C4×C9116128yesdirect product factor
120C25×C8C23×C915208yes—
121C24×C42C2×C4×C917812yes—
122C24×C42C23×C912604yes—
123C23×C4×C8C23×C9189280yesdirect product factor
124C26×C3C3×C4×C814096yes—
125C26×C3C2×C3×C4212016yes—
126C26×C3C22×C3×C814032yes—
127C26×C3C23×C3×C41651yes—
128C24×C3×C4C2×C3×C42162496yes—
129C24×C3×C4C22×C3×C81166656yes—
130C24×C3×C4C23×C3×C419765yes—
131C24×C3×C4C23×C3×C4122320yes—
132C24×C3×C4C23×C3×C4120832yes—
133C24×C3×C4C25×C311395yes—
134C23×C3×C8C23×C3×C41357120yes—
135C23×C3×C8C25×C3120832yes—
136C22×C3×C42C23×C3×C41156240yes—
137C22×C3×C42C25×C3110416yes—
138C2×C3×C4×C8C25×C3164512yes—
139C24×C9C22×C4×C81166656yesdirect product factor
140C24×C9C23×C42111160yes—
141C24×C9C24×C8122320yes—
142C24×C9C25×C41651yes—
143C22×C4×C9C23×C42178120yes—
144C22×C4×C9C24×C81166656yesdirect product factor
145C22×C4×C9C25×C415208yes—
146C22×C4×C9C25×C411953yes—
147C22×C4×C9C25×C414032yes—
148C22×C4×C9C27163yes—
149C2×C8×C9C25×C4116128yesdirect product factor
150C2×C8×C9C271128yes—
151C42×C9C25×C414032yes—
152C42×C9C27164yes—
153C27C42×C9164yes—
154C27C2×C8×C91128yes—
155C27C22×C4×C9163yes—
156C25×C4C42×C914032yes—
157C25×C4C2×C8×C9116128yesdirect product factor
158C25×C4C22×C4×C915208yes—
159C25×C4C22×C4×C911953yes—
160C25×C4C22×C4×C914032yes—
161C25×C4C24×C91651yes—
162C24×C8C22×C4×C91166656yesdirect product factor
163C24×C8C24×C9122320yes—
164C23×C42C22×C4×C9178120yes—
165C23×C42C24×C9111160yes—
166C22×C4×C8C24×C91166656yesdirect product factor
167C25×C3C2×C3×C4×C8164512yes—
168C25×C3C22×C3×C42110416yes—
169C25×C3C23×C3×C8120832yes—
170C25×C3C24×C3×C411395yes—
171C23×C3×C4C22×C3×C421156240yes—
172C23×C3×C4C23×C3×C81357120yes—
173C23×C3×C4C24×C3×C419765yes—
174C23×C3×C4C24×C3×C4122320yes—
175C23×C3×C4C24×C3×C4120832yes—
176C23×C3×C4C26×C31651yes—
177C22×C3×C8C24×C3×C41166656yes—
178C22×C3×C8C26×C314032yes—
179C2×C3×C42C24×C3×C4162496yes—
180C2×C3×C42C26×C312016yes—
181C3×C4×C8C26×C314096yes—
182C23×C9C23×C4×C8189280yesdirect product factor
183C23×C9C24×C4212604yes—
184C23×C9C25×C815208yes—
185C23×C9C26×C4163yes—
186C2×C4×C9C24×C4217812yes—
187C2×C4×C9C25×C8116128yesdirect product factor
188C2×C4×C9C26×C4163yes—
189C2×C4×C9C26×C41128yes—
190C2×C4×C9C26×C41252yes—
191C2×C4×C9C2811yescharacteristic
192C8×C9C26×C41256yesdirect product factor
193C26C4×C8×C914096yesdirect product factor
194C26C2×C42×C912016yes—
195C26C22×C8×C914032yes—
196C26C23×C4×C91651yes—
197C24×C4C2×C42×C9162496yes—
198C24×C4C22×C8×C91166656yesdirect product factor
199C24×C4C23×C4×C919765yes—
200C24×C4C23×C4×C9122320yes—
201C24×C4C23×C4×C9120832yes—
202C24×C4C25×C911395yes—
203C23×C8C23×C4×C91357120yesdirect product factor
204C23×C8C25×C9120832yes—
205C22×C42C23×C4×C91156240yes—
206C22×C42C25×C9110416yes—
207C2×C4×C8C25×C9164512yesdirect product factor
208C24×C3C22×C3×C4×C81166656yes—
209C24×C3C23×C3×C42111160yes—
210C24×C3C24×C3×C8122320yes—
211C24×C3C25×C3×C41651yes—
212C22×C3×C4C23×C3×C42178120yes—
213C22×C3×C4C24×C3×C81166656yes—
214C22×C3×C4C25×C3×C415208yes—
215C22×C3×C4C25×C3×C411953yes—
216C22×C3×C4C25×C3×C414032yes—
217C22×C3×C4C27×C3163yes—
218C2×C3×C8C25×C3×C4116128yes—
219C2×C3×C8C27×C31128yes—
220C3×C42C25×C3×C414032yes—
221C3×C42C27×C3164yes—
222C22×C9C24×C4×C8110416yesdirect product factor
223C22×C9C25×C421126yes—
224C22×C9C26×C81252yes—
225C22×C9C27×C411yescharacteristic
226C4×C9C25×C421126yes—
227C4×C9C26×C81256yesdirect product factor
228C4×C9C27×C412yes—
229C25C2×C4×C8×C9164512yesdirect product factor
230C25C22×C42×C9110416yes—
231C25C23×C8×C9120832yes—
232C25C24×C4×C911395yes—
233C23×C4C22×C42×C91156240yes—
234C23×C4C23×C8×C91357120yesdirect product factor
235C23×C4C24×C4×C919765yes—
236C23×C4C24×C4×C9122320yes—
237C23×C4C24×C4×C9120832yes—
238C23×C4C26×C91651yes—
239C22×C8C24×C4×C91166656yesdirect product factor
240C22×C8C26×C914032yes—
241C2×C42C24×C4×C9162496yes—
242C2×C42C26×C912016yes—
243C4×C8C26×C914096yesdirect product factor
244C23×C3C23×C3×C4×C8189280yes—
245C23×C3C24×C3×C4212604yes—
246C23×C3C25×C3×C815208yes—
247C23×C3C26×C3×C4163yes—
248C2×C3×C4C24×C3×C4217812yes—
249C2×C3×C4C25×C3×C8116128yes—
250C2×C3×C4C26×C3×C4163yes—
251C2×C3×C4C26×C3×C41128yes—
252C2×C3×C4C26×C3×C41252yes—
253C2×C3×C4C28×C311yescharacteristic
254C3×C8C26×C3×C41256yes—
255C2×C9C27×C812yes—
256C2×C9C25×C4×C81252yesdirect product factor
257C2×C9C26×C4211yescharacteristic
258C24C22×C4×C8×C91166656yesdirect product factor
259C24C23×C42×C9111160yes—
260C24C24×C8×C9122320yes—
261C24C25×C4×C91651yes—
262C22×C4C23×C42×C9178120yes—
263C22×C4C24×C8×C91166656yesdirect product factor
264C22×C4C25×C4×C915208yes—
265C22×C4C25×C4×C911953yes—
266C22×C4C25×C4×C914032yes—
267C22×C4C27×C9163yes—
268C2×C8C25×C4×C9116128yesdirect product factor
269C2×C8C27×C91128yes—
270C42C25×C4×C914032yes—
271C42C27×C9164yes—
272C22×C3C24×C3×C4×C8110416yes—
273C22×C3C26×C3×C81252yes—
274C22×C3C25×C3×C421126yes—
275C22×C3C27×C3×C411yescharacteristic
276C3×C4C26×C3×C81256yes—
277C3×C4C25×C3×C421126yes—
278C3×C4C27×C3×C412yes—
279C9C26×C4×C811yescharacteristic, direct product factor, Sylow
280C23C23×C4×C8×C9189280yesdirect product factor
281C23C25×C8×C915208yes—
282C23C24×C42×C912604yes—
283C23C26×C4×C9163yes—
284C2×C4C25×C8×C9116128yesdirect product factor
285C2×C4C24×C42×C917812yes—
286C2×C4C26×C4×C9163yes—
287C2×C4C26×C4×C91128yes—
288C2×C4C26×C4×C91252yes—
289C2×C4C28×C911yescharacteristic
290C8C26×C4×C91256yesdirect product factor
291C2×C3C25×C3×C4×C81252yes—
292C2×C3C27×C3×C812yes—
293C2×C3C26×C3×C4211yescharacteristic
294C22C24×C4×C8×C9110416yesdirect product factor
295C22C26×C8×C91252yes—
296C22C25×C42×C91126yes—
297C22C27×C4×C911yescharacteristic
298C4C26×C8×C91256yesdirect product factor
299C4C25×C42×C91126yes—
300C4C27×C4×C912yes—
301C3C26×C3×C4×C811yescharacteristic
302C2C25×C4×C8×C91252yesdirect product factor
303C2C27×C8×C912yes—
304C2C26×C42×C911yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC2×C3×C4C28×C3
socle, prime radicalC28×C3C2×C3×C4
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
3. C25×C4×C8×C9C2direct (N × H)
9. C26×C8×C9C4direct (N × H)
11. C24×C4×C8×C9C22direct (N × H)
16. C26×C4×C9C8direct (N × H)
23. C25×C8×C9C2×C4direct (N × H)
25. C23×C4×C8×C9C23direct (N × H)
26. C26×C4×C8C9direct (N × H)
35. C25×C4×C8C2×C9direct (N × H)
41. C25×C4×C9C2×C8direct (N × H)
46. C24×C8×C9C22×C4direct (N × H)
50. C22×C4×C8×C9C24direct (N × H)
62. C26×C9C4×C8direct (N × H)
67. C24×C4×C9C22×C8direct (N × H)
72. C23×C8×C9C23×C4direct (N × H)
76. C2×C4×C8×C9C25direct (N × H)
79. C26×C8C4×C9direct (N × H)
83. C24×C4×C8C22×C9direct (N × H)
98. C25×C9C2×C4×C8direct (N × H)
103. C23×C4×C9C23×C8direct (N × H)
108. C22×C8×C9C24×C4direct (N × H)
112. C4×C8×C9C26direct (N × H)
114. C26×C4C8×C9direct (N × H)
119. C25×C8C2×C4×C9direct (N × H)
123. C23×C4×C8C23×C9direct (N × H)
139. C24×C9C22×C4×C8direct (N × H)
144. C22×C4×C9C24×C8direct (N × H)
149. C2×C8×C9C25×C4direct (N × H)
157. C25×C4C2×C8×C9direct (N × H)
162. C24×C8C22×C4×C9direct (N × H)
166. C22×C4×C8C24×C9direct (N × H)
182. C23×C9C23×C4×C8direct (N × H)
187. C2×C4×C9C25×C8direct (N × H)
192. C8×C9C26×C4direct (N × H)
193. C26C4×C8×C9direct (N × H)
198. C24×C4C22×C8×C9direct (N × H)
203. C23×C8C23×C4×C9direct (N × H)
207. C2×C4×C8C25×C9direct (N × H)
222. C22×C9C24×C4×C8direct (N × H)
227. C4×C9C26×C8direct (N × H)
229. C25C2×C4×C8×C9direct (N × H)
234. C23×C4C23×C8×C9direct (N × H)
239. C22×C8C24×C4×C9direct (N × H)
243. C4×C8C26×C9direct (N × H)
256. C2×C9C25×C4×C8direct (N × H)
258. C24C22×C4×C8×C9direct (N × H)
263. C22×C4C24×C8×C9direct (N × H)
268. C2×C8C25×C4×C9direct (N × H)
279. C9C26×C4×C8direct (N × H)
280. C23C23×C4×C8×C9direct (N × H)
284. C2×C4C25×C8×C9direct (N × H)
290. C8C26×C4×C9direct (N × H)
294. C22C24×C4×C8×C9direct (N × H)
298. C4C26×C8×C9direct (N × H)
302. C2C25×C4×C8×C9direct (N × H)

Maximal subgroups

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

subgroupindex
C26×C42×C92
C27×C8×C92
C25×C4×C8×C92
C26×C3×C4×C83