Finite Group Databasebeta

← C32×#3500 #11770549

Subgroups — C32×#3500 #11770549

Order 1152.

4620 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×2−1+4C211yesmaximal, characteristic, complemented
2C2×C32×H(5,𝔽2)C211yesmaximal, characteristic, complemented
3C32×#548C214yesmaximal, complemented
4C32×#546C2112yesmaximal, complemented
5C32×#536C213yesmaximal, complemented
6C32×#535C213yesmaximal, complemented
7C32×#532C213yesmaximal, complemented
8C2×C32×#124C213yesmaximal, complemented
9C32×C4×#41C211yesmaximal, characteristic, complemented
10C3×#3500C314yesmaximal, direct product factor
11C32×2−1+4—28no—
12C32×H(5,𝔽2)—28no—
13C2×C32×#41C2213yes—
14C2×C32×#41C2212yes—
15C2×C32×#41C2211yescharacteristic
16C2×C32×#41C2213yescomplemented
17C2×C32×#41C2216yescomplemented
18C22×C32×Q8C2213yescomplemented
19C22×C32×Q8C2212yescomplemented
20C22×C32×D4C2213yes—
21C22×C32×D4C2216yes—
22C32×C4⋊Q8C2213yescomplemented
23C32×#127C2213yescomplemented
24C32×#124C2213yescomplemented
25C32×#124C2216yescomplemented
26C32×#124C22112yescomplemented
27C32×#124C2211yescharacteristic, complemented
28C32×#123C22112yescomplemented
29C32×C22⋊Q8C22112yescomplemented
30C32×#121C22112yescomplemented
31C32×#121C22112yescomplemented
32C32×#120C22112yescomplemented
33C32×C4×Q8C2212yescomplemented
34C32×C4×D4C2216yescomplemented
35C32×C4×D4C22112yescomplemented
36C32×#117C2213yescomplemented
37C2×C32×C22⋊C4C22112yescomplemented
38C2×C32×C42C2213yescomplemented
39C2×C3×2−1+4C2×C314yescomplemented
40C2×C3×H(5,𝔽2)C2×C314yescomplemented
41C3×#548C2×C3116yescomplemented
42C3×#546C2×C3148yescomplemented
43C3×#536C2×C3112yescomplemented
44C3×#535C2×C3112yescomplemented
45C3×#532C2×C3112yescomplemented
46C2×C3×#124C2×C3112yescomplemented
47C3×C4×#41C2×C314yescomplemented
48C24×C32C2316yes—
49C32×#41—28no—
50C32×#41—224no—
51C32×#41—212no—
52C32×#41—212no—
53C32×#41D418yescomplemented
54C2×C32×Q8—28no—
55C2×C32×Q8—212no—
56C2×C32×Q8C2316yes—
57C2×C32×Q8C2313yes—
58C2×C32×Q8C2311yescharacteristic
59C2×C32×Q8C2311yescharacteristic
60C2×C32×D4—224no—
61C2×C32×D4—212no—
62C2×C32×D4C2316yes—
63C2×C32×D4C2316yes—
64C2×C32×D4C2313yes—
65C2×C32×D4C2316yes—
66C2×C32×D4C2311yescharacteristic
67C2×C32×D4C23112yes—
68C2×C32×D4C2313yes—
69C22×C32×C4C2316yes—
70C22×C32×C4C2316yes—
71C22×C32×C4C2316yes—
72C22×C32×C4C2316yes—
73C22×C32×C4C2313yes—
74C32×C4⋊C4C2316yes—
75C32×C4⋊C4C23112yes—
76C32×C22⋊C4C23112yes—
77C32×C22⋊C4C23124yes—
78C32×C22⋊C4C2316yes—
79C32×C22⋊C4C2314yes—
80C32×C42C2311yescharacteristic
81C32×C42C2313yes—
82C32×C42C2316yes—
83#3500C3211yescharacteristic, direct product factor, Sylow
84C3×2−1+4—232no—
85C3×H(5,𝔽2)—232no—
86C2×C3×#41C22×C3112yes—
87C2×C3×#41C22×C318yes—
88C2×C3×#41C22×C314yes—
89C2×C3×#41C22×C3112yescomplemented
90C2×C3×#41C22×C3124yescomplemented
91C22×C3×Q8C22×C3112yescomplemented
92C22×C3×Q8C22×C318yescomplemented
93C22×C3×D4C22×C3112yes—
94C22×C3×D4C22×C3124yes—
95C3×C4⋊Q8C22×C3112yescomplemented
96C3×#127C22×C3112yescomplemented
97C3×#124C22×C3112yescomplemented
98C3×#124C22×C3124yescomplemented
99C3×#124C22×C3148yescomplemented
100C3×#124C22×C314yescomplemented
101C3×#123C22×C3148yescomplemented
102C3×C22⋊Q8C22×C3148yescomplemented
103C3×#121C22×C3148yescomplemented
104C3×#121C22×C3148yescomplemented
105C3×#120C22×C3148yescomplemented
106C3×C4×Q8C22×C318yescomplemented
107C3×C4×D4C22×C3124yescomplemented
108C3×C4×D4C22×C3148yescomplemented
109C3×#117C22×C3112yescomplemented
110C2×C3×C22⋊C4C22×C3148yescomplemented
111C2×C3×C42C22×C3112yescomplemented
112C23×C32—212no—
113C23×C32—412no—
114C23×C32C2416yes—
115C23×C32C2412yes—
116C23×C32C2413yes—
117C32×Q8—212no—
118C32×Q8—26no—
119C32×Q8—22no—
120C32×Q8C2×D414yes—
121C32×D4—212no—
122C32×D4—212no—
123C32×D4—212no—
124C32×D4—26no—
125C32×D4—22no—
126C32×D4—412no—
127C32×D4C2×D4112yes—
128C2×C32×C4—212no—
129C2×C32×C4—212no—
130C2×C32×C4—212no—
131C2×C32×C4—46no—
132C2×C32×C4C2×D4112yes—
133C2×C32×C4C2412yes—
134C2×C32×C4C2416yes—
135C2×C32×C4C2416yes—
136C2×C32×C4C2412yes—
137C2×C32×C4C2411yescharacteristic
138C2×C32×C4C2413yes—
139C2×2−1+4C2×C3211yescharacteristic, complemented
140C2×H(5,𝔽2)C2×C3211yescharacteristic, complemented
141#548C2×C3214yescomplemented
142#546C2×C32112yescomplemented
143#536C2×C3213yescomplemented
144#535C2×C3213yescomplemented
145#532C2×C3213yescomplemented
146C2×#124C2×C3213yescomplemented
147C4×#41C2×C3211yescharacteristic, complemented
148C24×C3C23×C3124yes—
149C3×#41—232no—
150C3×#41—296no—
151C3×#41—248no—
152C3×#41—248no—
153C3×#41C3×D4132yescomplemented
154C2×C3×Q8—232no—
155C2×C3×Q8—248no—
156C2×C3×Q8C23×C3124yes—
157C2×C3×Q8C23×C3112yes—
158C2×C3×Q8C23×C314yes—
159C2×C3×Q8C23×C314yes—
160C2×C3×D4—296no—
161C2×C3×D4—248no—
162C2×C3×D4C23×C3124yes—
163C2×C3×D4C23×C3124yes—
164C2×C3×D4C23×C3124yes—
165C2×C3×D4C23×C3112yes—
166C2×C3×D4C23×C314yes—
167C2×C3×D4C23×C3148yes—
168C2×C3×D4C23×C3112yes—
169C22×C3×C4C23×C3124yes—
170C22×C3×C4C23×C3124yes—
171C22×C3×C4C23×C3124yes—
172C22×C3×C4C23×C3124yes—
173C22×C3×C4C23×C3112yes—
174C3×C4⋊C4C23×C3124yes—
175C3×C4⋊C4C23×C3148yes—
176C3×C22⋊C4C23×C3148yes—
177C3×C22⋊C4C23×C3196yes—
178C3×C22⋊C4C23×C3124yes—
179C3×C22⋊C4C23×C3116yes—
180C3×C42C23×C314yes—
181C3×C42C23×C3112yes—
182C3×C42C23×C3124yes—
183C22×C32—26no—
184C22×C32—22no—
185C22×C32—26no—
186C22×C32—24no—
187C22×C32—212no—
188C22×C32—812no—
189C22×C32C22×D416yes—
190C22×C32C2511yescharacteristic
191C32×C4—22no—
192C32×C4—26no—
193C32×C4—22no—
194C32×C4—26no—
195C32×C4C22×D416yes—
196C32×C4C22×D412yes—
1972−1+4—28no—
198H(5,𝔽2)—28no—
199C2×#41C22×C3213yes—
200C2×#41C22×C3212yes—
201C2×#41C22×C3211yescharacteristic
202C2×#41C22×C3213yescomplemented
203C2×#41C22×C3216yescomplemented
204C22×Q8C22×C3213yescomplemented
205C22×Q8C22×C3212yescomplemented
206C22×D4C22×C3213yes—
207C22×D4C22×C3216yes—
208C4⋊Q8C22×C3213yescomplemented
209#127C22×C3213yescomplemented
210#124C22×C3213yescomplemented
211#124C22×C3216yescomplemented
212#124C22×C32112yescomplemented
213#124C22×C3211yescharacteristic, complemented
214#123C22×C32112yescomplemented
215C22⋊Q8C22×C32112yescomplemented
216#121C22×C32112yescomplemented
217#121C22×C32112yescomplemented
218#120C22×C32112yescomplemented
219C4×Q8C22×C3212yescomplemented
220C4×D4C22×C3216yescomplemented
221C4×D4C22×C32112yescomplemented
222#117C22×C3213yescomplemented
223C2×C22⋊C4C22×C32112yescomplemented
224C2×C42C22×C3213yescomplemented
225C23×C3—248no—
226C23×C3—448no—
227C23×C3C24×C3124yes—
228C23×C3C24×C318yes—
229C23×C3C24×C3112yes—
230C3×Q8—248no—
231C3×Q8—224no—
232C3×Q8—28no—
233C3×Q8C2×C3×D4116yes—
234C3×D4—248no—
235C3×D4—248no—
236C3×D4—248no—
237C3×D4—224no—
238C3×D4—28no—
239C3×D4—448no—
240C3×D4C2×C3×D4148yes—
241C2×C3×C4—248no—
242C2×C3×C4—248no—
243C2×C3×C4—248no—
244C2×C3×C4—424no—
245C2×C3×C4C2×C3×D4148yes—
246C2×C3×C4C24×C318yes—
247C2×C3×C4C24×C3124yes—
248C2×C3×C4C24×C3124yes—
249C2×C3×C4C24×C318yes—
250C2×C3×C4C24×C314yes—
251C2×C3×C4C24×C3112yes—
252C2×C32—26no—
253C2×C32—42no—
254C2×C32—46no—
255C2×C32C23×D411yescharacteristic
256C2×C32#58512yes—
257C24C23×C3216yes—
258#41—28no—
259#41—224no—
260#41—212no—
261#41—212no—
262#41C32×D418yescomplemented
263C2×Q8—28no—
264C2×Q8—212no—
265C2×Q8C23×C3216yes—
266C2×Q8C23×C3213yes—
267C2×Q8C23×C3211yescharacteristic
268C2×Q8C23×C3211yescharacteristic
269C2×D4—224no—
270C2×D4—212no—
271C2×D4C23×C3216yes—
272C2×D4C23×C3216yes—
273C2×D4C23×C3216yes—
274C2×D4C23×C3213yes—
275C2×D4C23×C3211yescharacteristic
276C2×D4C23×C32112yes—
277C2×D4C23×C3213yes—
278C22×C4C23×C3216yes—
279C22×C4C23×C3216yes—
280C22×C4C23×C3216yes—
281C22×C4C23×C3216yes—
282C22×C4C23×C3213yes—
283C4⋊C4C23×C3216yes—
284C4⋊C4C23×C32112yes—
285C22⋊C4C23×C32112yes—
286C22⋊C4C23×C32124yes—
287C22⋊C4C23×C3216yes—
288C22⋊C4C23×C3214yes—
289C42C23×C3211yescharacteristic
290C42C23×C3213yes—
291C42C23×C3216yes—
292C22×C3—224no—
293C22×C3—28no—
294C22×C3—224no—
295C22×C3—216no—
296C22×C3—248no—
297C22×C3—848no—
298C22×C3C22×C3×D4124yes—
299C22×C3C25×C314yes—
300C3×C4—28no—
301C3×C4—224no—
302C3×C4—28no—
303C3×C4—224no—
304C3×C4C22×C3×D4124yes—
305C3×C4C22×C3×D418yes—
306C32#350011yescharacteristic, direct product factor, Sylow
307C23—212no—
308C23—412no—
309C23C24×C3216yes—
310C23C24×C3212yes—
311C23C24×C3213yes—
312Q8—212no—
313Q8—26no—
314Q8—22no—
315Q8C2×C32×D414yes—
316D4—212no—
317D4—212no—
318D4—212no—
319D4—26no—
320D4—22no—
321D4—412no—
322D4C2×C32×D4112yes—
323C2×C4—212no—
324C2×C4—212no—
325C2×C4—212no—
326C2×C4—46no—
327C2×C4C2×C32×D4112yes—
328C2×C4C24×C3212yes—
329C2×C4C24×C3216yes—
330C2×C4C24×C3216yes—
331C2×C4C24×C3212yes—
332C2×C4C24×C3211yescharacteristic
333C2×C4C24×C3213yes—
334C2×C3—224no—
335C2×C3—424no—
336C2×C3—48no—
337C2×C3C23×C3×D414yes—
338C2×C3C3×#58518yes—
339C22—26no—
340C22—22no—
341C22—26no—
342C22—24no—
343C22—212no—
344C22—812no—
345C22C22×C32×D416yes—
346C22C25×C3211yescharacteristic
347C4—22no—
348C4—26no—
349C4—22no—
350C4—26no—
351C4C22×C32×D416yes—
352C4C22×C32×D412yes—
353C3C3×#350014yesdirect product factor
354C2—26no—
355C2—46no—
356C2—42no—
357C2C23×C32×D411yescharacteristic
358C2C32×#58512yes—
subgroupG/H
center, socleC22×C32C25
commutator, Frattini, cosocleC22C25×C32
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C32 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C32×2−1+4C2semidirect
2. C2×C32×H(5,𝔽2)C2semidirect
3. C32×#548C2semidirect
4. C32×#546C2semidirect
5. C32×#536C2semidirect
6. C32×#535C2semidirect
7. C32×#532C2semidirect
8. C2×C32×#124C2semidirect
9. C32×C4×#41C2semidirect
10. C3×#3500C3direct (N × H)
16. C2×C32×#41C22semidirect
17. C2×C32×#41C22semidirect
18. C22×C32×Q8C22semidirect
19. C22×C32×Q8C22semidirect
22. C32×C4⋊Q8C22semidirect
23. C32×#127C22semidirect
24. C32×#124C22semidirect
25. C32×#124C22semidirect
26. C32×#124C22semidirect
27. C32×#124C22semidirect
28. C32×#123C22semidirect
29. C32×C22⋊Q8C22semidirect
30. C32×#121C22semidirect
31. C32×#121C22semidirect
32. C32×#120C22semidirect
33. C32×C4×Q8C22semidirect
34. C32×C4×D4C22semidirect
35. C32×C4×D4C22semidirect
36. C32×#117C22semidirect
37. C2×C32×C22⋊C4C22semidirect
38. C2×C32×C42C22semidirect
39. C2×C3×2−1+4C2×C3semidirect
40. C2×C3×H(5,𝔽2)C2×C3semidirect
41. C3×#548C2×C3semidirect
42. C3×#546C2×C3semidirect
43. C3×#536C2×C3semidirect
44. C3×#535C2×C3semidirect
45. C3×#532C2×C3semidirect
46. C2×C3×#124C2×C3semidirect
47. C3×C4×#41C2×C3semidirect
53. C32×#41D4semidirect
83. #3500C32direct (N × H)
89. C2×C3×#41C22×C3semidirect
90. C2×C3×#41C22×C3semidirect
91. C22×C3×Q8C22×C3semidirect
92. C22×C3×Q8C22×C3semidirect
95. C3×C4⋊Q8C22×C3semidirect
96. C3×#127C22×C3semidirect
97. C3×#124C22×C3semidirect
98. C3×#124C22×C3semidirect
99. C3×#124C22×C3semidirect
100. C3×#124C22×C3semidirect
101. C3×#123C22×C3semidirect
102. C3×C22⋊Q8C22×C3semidirect
103. C3×#121C22×C3semidirect
104. C3×#121C22×C3semidirect
105. C3×#120C22×C3semidirect
106. C3×C4×Q8C22×C3semidirect
107. C3×C4×D4C22×C3semidirect
108. C3×C4×D4C22×C3semidirect
109. C3×#117C22×C3semidirect
110. C2×C3×C22⋊C4C22×C3semidirect
111. C2×C3×C42C22×C3semidirect
139. C2×2−1+4C2×C32semidirect
140. C2×H(5,𝔽2)C2×C32semidirect
141. #548C2×C32semidirect
142. #546C2×C32semidirect
143. #536C2×C32semidirect
144. #535C2×C32semidirect
145. #532C2×C32semidirect
146. C2×#124C2×C32semidirect
147. C4×#41C2×C32semidirect
153. C3×#41C3×D4semidirect
202. C2×#41C22×C32semidirect
203. C2×#41C22×C32semidirect
204. C22×Q8C22×C32semidirect
205. C22×Q8C22×C32semidirect
208. C4⋊Q8C22×C32semidirect
209. #127C22×C32semidirect
210. #124C22×C32semidirect
211. #124C22×C32semidirect
212. #124C22×C32semidirect
213. #124C22×C32semidirect
214. #123C22×C32semidirect
215. C22⋊Q8C22×C32semidirect
216. #121C22×C32semidirect
217. #121C22×C32semidirect
218. #120C22×C32semidirect
219. C4×Q8C22×C32semidirect
220. C4×D4C22×C32semidirect
221. C4×D4C22×C32semidirect
222. #117C22×C32semidirect
223. C2×C22⋊C4C22×C32semidirect
224. C2×C42C22×C32semidirect
262. #41C32×D4semidirect
306. C32#3500direct (N × H)
353. C3C3×#3500direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×2−1+42
C2×C32×H(5,𝔽2)2
C32×#5482
C32×#5462
C32×#5362
C32×#5352
C32×#5322
C2×C32×#1242
C32×C4×#412
C3×#35003