Finite Group Databasebeta

← C2×C32×#1009 #13181721

Subgroups — C2×C32×#1009 #13181721

Order 1728.

6044 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×#41C211yesmaximal, characteristic, complemented
2C32×#1009C2116yesmaximal, direct product factor
3C2×C32×#239C212yesmaximal, complemented
4C2×C32×D3×D4C216yesmaximal, complemented
5C2×C32×#235C213yesmaximal, complemented
6C22×C32×D12C213yesmaximal, complemented
7C2×C32×H(5,𝔽2)—31nomaximal
8C2×C3×#1009C314yesmaximal, direct product factor
9C23×C32×D3C2216yescomplemented
10C33×#41C2218yescomplemented
11C2×C33×Q8C2211yescharacteristic, complemented
12C2×C33×D4C2213yescomplemented
13C22×C33×C4C2213yescomplemented
14C2×C32×#67C2216yescomplemented
15C32×#239C22116yescomplemented
16C32×D3×D4C22148yescomplemented
17C32×#235C22124yescomplemented
18C2×C32×D12C2213yescomplemented
19C2×C32×D12C2216yescomplemented
20C2×C32×D12C22124yescomplemented
21C2×C32×C4×D3C2216yescomplemented
22C2×C32×Dic6C2211yescharacteristic, complemented
23C32×H(5,𝔽2)—316no—
24C2×C32×#41—24no—
25C2×C32×#41—32no—
26C2×C32×#41—33no—
27C2×C32×#41D311yescharacteristic, complemented
28C2×C32×#41C2×C314yescomplemented
29C22×C32×D4—36no—
30C22×C32×D4—33no—
31C3×#1009C2×C3164yesdirect product factor
32C2×C3×#239C2×C318yescomplemented
33C2×C3×D3×D4C2×C3124yescomplemented
34C2×C3×#235C2×C3112yescomplemented
35C22×C3×D12C2×C3112yescomplemented
36C23×C33C2313yes—
37C22×C32×D3—412no—
38C22×C32×D3C2316yes—
39C22×C32×D3C23124yescomplemented
40C33×Q8C2314yescomplemented
41C33×D4C23112yescomplemented
42C2×C33×C4C2311yescharacteristic
43C2×C33×C4C2313yes—
44C2×C33×C4C23112yescomplemented
45C32×#67C23124yescomplemented
46C2×C32×Dic3C2312yes—
47C32×D12C23112yescomplemented
48C32×D12C23124yescomplemented
49C32×C4×D3C23124yescomplemented
50C32×Dic6C2314yescomplemented
51C2×C3×H(5,𝔽2)—34no—
52C2×#1009C3211yescharacteristic, direct product factor
53C24×C32—36no—
54C23×C3×D3C22×C3124yescomplemented
55C32×#41—232no—
56C32×#41—316no—
57C32×#41—324no—
58C32×#41C2×D318yescomplemented
59C32×#41C22×C3132yescomplemented
60C2×C32×Q8—24no—
61C2×C32×Q8—31no—
62C2×C32×Q8C2×D311yescharacteristic, complemented
63C2×C32×Q8C22×C314yescomplemented
64C2×C32×D4—212no—
65C2×C32×D4—36no—
66C2×C32×D4—348no—
67C2×C32×D4—36no—
68C2×C32×D4—33no—
69C2×C32×D4—324no—
70C2×C32×D4C2×D313yescomplemented
71C2×C32×D4C22×C3112yescomplemented
72C22×C32×C4—212no—
73C22×C32×C4—36no—
74C22×C32×C4C2×D313yescomplemented
75C22×C32×C4C22×C3112yescomplemented
76C2×C3×#67C22×C3124yescomplemented
77C3×#239C22×C3164yescomplemented
78C3×D3×D4C22×C31192yescomplemented
79C3×#235C22×C3196yescomplemented
80C2×C3×D12C22×C3112yescomplemented
81C2×C3×D12C22×C3124yescomplemented
82C2×C3×D12C22×C3196yescomplemented
83C2×C3×C4×D3C22×C3124yescomplemented
84C2×C3×Dic6C22×C314yescomplemented
85C22×C33—26no—
86C22×C33C2416yes—
87C22×C33C2411yescharacteristic
88C2×C32×D3—212no—
89C2×C32×D3—424no—
90C2×C32×D3C24112yes—
91C33×C4C2412yes—
92C33×C4C2416yes—
93C32×Dic3C2414yes—
94C3×H(5,𝔽2)—364no—
95C2×C3×#41—24no—
96C2×C3×#41—38no—
97C2×C3×#41—312no—
98C2×C3×#41C3×D314yescomplemented
99C2×C3×#41C2×C3211yescharacteristic, complemented
100C22×C3×D4—324no—
101C22×C3×D4—312no—
102#1009C2×C32116yesdirect product factor
103C2×#239C2×C3212yescomplemented
104C2×D3×D4C2×C3216yescomplemented
105C2×#235C2×C3213yescomplemented
106C22×D12C2×C3213yescomplemented
107C23×C32—212no—
108C23×C32—36no—
109C23×C32—324no—
110C23×C32—1212no—
111C23×C32C22×D313yes—
112C23×C32C23×C3112yes—
113C22×C3×D3—448no—
114C22×C3×D3C23×C3124yes—
115C22×C3×D3C23×C3196yescomplemented
116C32×Q8—216no—
117C32×Q8—34no—
118C32×Q8C22×D314yescomplemented
119C32×Q8C23×C3116yescomplemented
120C32×D4—248no—
121C32×D4—324no—
122C32×D4—324no—
123C32×D4—312no—
124C32×D4C22×D3112yescomplemented
125C32×D4C23×C3148yescomplemented
126C2×C32×C4—24no—
127C2×C32×C4—212no—
128C2×C32×C4—248no—
129C2×C32×C4—32no—
130C2×C32×C4—324no—
131C2×C32×C4C22×D313yes—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D3112yescomplemented
134C2×C32×C4C23×C314yes—
135C2×C32×C4C23×C3112yes—
136C2×C32×C4C23×C3148yescomplemented
137C3×#67C23×C3196yescomplemented
138C2×C3×Dic3C23×C318yes—
139C3×D12C23×C3148yescomplemented
140C3×D12C23×C3196yescomplemented
141C3×C4×D3C23×C3196yescomplemented
142C3×Dic6C23×C3116yescomplemented
143C2×H(5,𝔽2)—31noSylow
144C2×C33—26no—
145C2×C33H(5,𝔽2)12yescomplemented
146C2×C33C2511yescharacteristic
147C32×D3—212no—
148C24×C3—324no—
149C23×D3C22×C3216yescomplemented
150C3×#41—232no—
151C3×#41—364no—
152C3×#41—396no—
153C3×#41C2×C3×D3132yescomplemented
154C3×#41C22×C3218yescomplemented
155C2×C3×Q8—24no—
156C2×C3×Q8—34no—
157C2×C3×Q8C2×C3×D314yescomplemented
158C2×C3×Q8C22×C3211yescharacteristic, complemented
159C2×C3×D4—212no—
160C2×C3×D4—324no—
161C2×C3×D4—3192no—
162C2×C3×D4—324no—
163C2×C3×D4—312no—
164C2×C3×D4—396no—
165C2×C3×D4C2×C3×D3112yescomplemented
166C2×C3×D4C22×C3213yescomplemented
167C22×C3×C4—212no—
168C22×C3×C4—324no—
169C22×C3×C4C2×C3×D3112yescomplemented
170C22×C3×C4C22×C3213yescomplemented
171C2×#67C22×C3216yescomplemented
172#239C22×C32116yescomplemented
173D3×D4C22×C32148yescomplemented
174#235C22×C32124yescomplemented
175C2×D12C22×C3213yescomplemented
176C2×D12C22×C3216yescomplemented
177C2×D12C22×C32124yescomplemented
178C2×C4×D3C22×C3216yescomplemented
179C2×Dic6C22×C3211yescharacteristic, complemented
180C22×C32—26no—
181C22×C32—24no—
182C22×C32—224no—
183C22×C32—224no—
184C22×C32—312no—
185C22×C32—424no—
186C22×C32—612no—
187C22×C32—1224no—
188C22×C32C23×D316yes—
189C22×C32C23×D311yescharacteristic
190C22×C32C24×C3124yes—
191C22×C32C24×C314yes—
192C2×C3×D3—248no—
193C2×C3×D3—496no—
194C2×C3×D3C24×C3148yes—
195C32×C4—28no—
196C32×C4—224no—
197C32×C4—34no—
198C32×C4C23×D316yes—
199C32×C4C23×D312yes—
200C32×C4C24×C318yes—
201C32×C4C24×C3124yes—
202C3×Dic3C24×C3116yes—
203H(5,𝔽2)—316no—
204C2×#41—32no—
205C2×#41—33no—
206C2×#41C32×D311yescharacteristic, complemented
207C22×D4—36no—
208C22×D4—33no—
209C33C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
210C23×C3—212no—
211C23×C3—324no—
212C23×C3—396no—
213C23×C3—1248no—
214C23×C3C22×C3×D3112yes—
215C23×C3C23×C3213yes—
216C22×D3—412no—
217C22×D3C23×C3216yes—
218C22×D3C23×C32124yescomplemented
219C3×Q8—216no—
220C3×Q8—316no—
221C3×Q8C22×C3×D3116yescomplemented
222C3×Q8C23×C3214yescomplemented
223C3×D4—248no—
224C3×D4—396no—
225C3×D4—396no—
226C3×D4—348no—
227C3×D4C22×C3×D3148yescomplemented
228C3×D4C23×C32112yescomplemented
229C2×C3×C4—24no—
230C2×C3×C4—212no—
231C2×C3×C4—248no—
232C2×C3×C4—38no—
233C2×C3×C4—396no—
234C2×C3×C4C22×C3×D3112yes—
235C2×C3×C4C22×C3×D314yes—
236C2×C3×C4C22×C3×D3148yescomplemented
237C2×C3×C4C23×C3211yescharacteristic
238C2×C3×C4C23×C3213yes—
239C2×C3×C4C23×C32112yescomplemented
240#67C23×C32124yescomplemented
241C2×Dic3C23×C3212yes—
242D12C23×C32112yescomplemented
243D12C23×C32124yescomplemented
244C4×D3C23×C32124yescomplemented
245Dic6C23×C3214yescomplemented
246C2×C32—26no—
247C2×C32—28no—
248C2×C32—24no—
249C2×C32—224no—
250C2×C32—424no—
251C2×C32—612no—
252C2×C32#100912yesdirect product factor
253C2×C32C3×H(5,𝔽2)18yescomplemented
254C2×C32C24×D311yescharacteristic
255C2×C32C25×C314yes—
256C3×D3—248no—
257C24—36no—
258#41—316no—
259#41—324no—
260#41C2×C32×D318yescomplemented
261C2×Q8—31no—
262C2×Q8C2×C32×D311yescharacteristic, complemented
263C2×D4—36no—
264C2×D4—348no—
265C2×D4—36no—
266C2×D4—33no—
267C2×D4—324no—
268C2×D4C2×C32×D313yescomplemented
269C22×C4—36no—
270C22×C4C2×C32×D313yescomplemented
271C22×C3—224no—
272C22×C3—26no—
273C22×C3—24no—
274C22×C3—224no—
275C22×C3—348no—
276C22×C3—424no—
277C22×C3—648no—
278C22×C3—1296no—
279C22×C3C23×C3×D3124yes—
280C22×C3C23×C3×D314yes—
281C22×C3C24×C3216yes—
282C22×C3C24×C3211yescharacteristic
283C2×D3—212no—
284C2×D3—424no—
285C2×D3C24×C32112yes—
286C3×C4—28no—
287C3×C4—224no—
288C3×C4—316no—
289C3×C4C23×C3×D3124yes—
290C3×C4C23×C3×D318yes—
291C3×C4C24×C3212yes—
292C3×C4C24×C3216yes—
293Dic3C24×C3214yes—
294C32—24no—
295C32C2×#100911yescharacteristic, direct product factor
296C32C2×C3×H(5,𝔽2)14yescomplemented
297C23—36no—
298C23—324no—
299C23—1212no—
300C23C22×C32×D313yes—
301Q8—34no—
302Q8C22×C32×D314yescomplemented
303D4—324no—
304D4—324no—
305D4—312no—
306D4C22×C32×D3112yescomplemented
307C2×C4—32no—
308C2×C4—324no—
309C2×C4C22×C32×D313yes—
310C2×C4C22×C32×D311yescharacteristic
311C2×C4C22×C32×D3112yescomplemented
312C2×C3—224no—
313C2×C3—26no—
314C2×C3—28no—
315C2×C3—24no—
316C2×C3—424no—
317C2×C3—648no—
318C2×C3C3×#100918yesdirect product factor
319C2×C3C32×H(5,𝔽2)12yescomplemented
320C2×C3C24×C3×D314yes—
321C2×C3C25×C3211yescharacteristic
322D3—212no—
323C22—26no—
324C22—312no—
325C22—612no—
326C22—1224no—
327C22C23×C32×D316yes—
328C22C23×C32×D311yescharacteristic
329C4—34no—
330C4C23×C32×D316yes—
331C4C23×C32×D312yes—
332C3—24no—
333C3C2×C3×#100914yesdirect product factor
334C3C2×C32×H(5,𝔽2)11yescharacteristic, complemented
335C2—26no—
336C2—612no—
337C2C32×#100912yesdirect product factor
338C2C24×C32×D311yescharacteristic
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC2×C3C25×C32
FrattiniC2C24×C32×D3
FittingC2×C33×#41C2
socleC22×C33C24
hypercenterC2×C32×#41D3
hypocenterC3C2×C32×H(5,𝔽2)
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C2×C32×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. C32×#1009C2direct (N × H)
3. C2×C32×#239C2semidirect
4. C2×C32×D3×D4C2semidirect
5. C2×C32×#235C2semidirect
6. C22×C32×D12C2semidirect
8. C2×C3×#1009C3direct (N × H)
9. C23×C32×D3C22semidirect
10. C33×#41C22semidirect
11. C2×C33×Q8C22semidirect
12. C2×C33×D4C22semidirect
13. C22×C33×C4C22semidirect
14. C2×C32×#67C22semidirect
15. C32×#239C22semidirect
16. C32×D3×D4C22semidirect
17. C32×#235C22semidirect
18. C2×C32×D12C22semidirect
19. C2×C32×D12C22semidirect
20. C2×C32×D12C22semidirect
21. C2×C32×C4×D3C22semidirect
22. C2×C32×Dic6C22semidirect
27. C2×C32×#41D3semidirect
28. C2×C32×#41C2×C3semidirect
31. C3×#1009C2×C3direct (N × H)
32. C2×C3×#239C2×C3semidirect
33. C2×C3×D3×D4C2×C3semidirect
34. C2×C3×#235C2×C3semidirect
35. C22×C3×D12C2×C3semidirect
39. C22×C32×D3C23semidirect
40. C33×Q8C23semidirect
41. C33×D4C23semidirect
44. C2×C33×C4C23semidirect
45. C32×#67C23semidirect
47. C32×D12C23semidirect
48. C32×D12C23semidirect
49. C32×C4×D3C23semidirect
50. C32×Dic6C23semidirect
52. C2×#1009C32direct (N × H)
54. C23×C3×D3C22×C3semidirect
58. C32×#41C2×D3semidirect
59. C32×#41C22×C3semidirect
62. C2×C32×Q8C2×D3semidirect
63. C2×C32×Q8C22×C3semidirect
70. C2×C32×D4C2×D3semidirect
71. C2×C32×D4C22×C3semidirect
74. C22×C32×C4C2×D3semidirect
75. C22×C32×C4C22×C3semidirect
76. C2×C3×#67C22×C3semidirect
77. C3×#239C22×C3semidirect
78. C3×D3×D4C22×C3semidirect
79. C3×#235C22×C3semidirect
80. C2×C3×D12C22×C3semidirect
81. C2×C3×D12C22×C3semidirect
82. C2×C3×D12C22×C3semidirect
83. C2×C3×C4×D3C22×C3semidirect
84. C2×C3×Dic6C22×C3semidirect
98. C2×C3×#41C3×D3semidirect
99. C2×C3×#41C2×C32semidirect
102. #1009C2×C32direct (N × H)
103. C2×#239C2×C32semidirect
104. C2×D3×D4C2×C32semidirect
105. C2×#235C2×C32semidirect
106. C22×D12C2×C32semidirect
115. C22×C3×D3C23×C3semidirect
118. C32×Q8C22×D3semidirect
119. C32×Q8C23×C3semidirect
124. C32×D4C22×D3semidirect
125. C32×D4C23×C3semidirect
133. C2×C32×C4C22×D3semidirect
136. C2×C32×C4C23×C3semidirect
137. C3×#67C23×C3semidirect
139. C3×D12C23×C3semidirect
140. C3×D12C23×C3semidirect
141. C3×C4×D3C23×C3semidirect
142. C3×Dic6C23×C3semidirect
145. C2×C33H(5,𝔽2)semidirect
149. C23×D3C22×C32semidirect
153. C3×#41C2×C3×D3semidirect
154. C3×#41C22×C32semidirect
157. C2×C3×Q8C2×C3×D3semidirect
158. C2×C3×Q8C22×C32semidirect
165. C2×C3×D4C2×C3×D3semidirect
166. C2×C3×D4C22×C32semidirect
169. C22×C3×C4C2×C3×D3semidirect
170. C22×C3×C4C22×C32semidirect
171. C2×#67C22×C32semidirect
172. #239C22×C32semidirect
173. D3×D4C22×C32semidirect
174. #235C22×C32semidirect
175. C2×D12C22×C32semidirect
176. C2×D12C22×C32semidirect
177. C2×D12C22×C32semidirect
178. C2×C4×D3C22×C32semidirect
179. C2×Dic6C22×C32semidirect
206. C2×#41C32×D3semidirect
209. C33C2×H(5,𝔽2)semidirect
218. C22×D3C23×C32semidirect
221. C3×Q8C22×C3×D3semidirect
222. C3×Q8C23×C32semidirect
227. C3×D4C22×C3×D3semidirect
228. C3×D4C23×C32semidirect
236. C2×C3×C4C22×C3×D3semidirect
239. C2×C3×C4C23×C32semidirect
240. #67C23×C32semidirect
242. D12C23×C32semidirect
243. D12C23×C32semidirect
244. C4×D3C23×C32semidirect
245. Dic6C23×C32semidirect
252. C2×C32#1009direct (N × H)
253. C2×C32C3×H(5,𝔽2)semidirect
260. #41C2×C32×D3semidirect
262. C2×Q8C2×C32×D3semidirect
268. C2×D4C2×C32×D3semidirect
270. C22×C4C2×C32×D3semidirect
295. C32C2×#1009direct (N × H)
296. C32C2×C3×H(5,𝔽2)semidirect
302. Q8C22×C32×D3semidirect
306. D4C22×C32×D3semidirect
311. C2×C4C22×C32×D3semidirect
318. C2×C3C3×#1009direct (N × H)
319. C2×C3C32×H(5,𝔽2)semidirect
333. C3C2×C3×#1009direct (N × H)
334. C3C2×C32×H(5,𝔽2)semidirect
337. C2C32×#1009direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×#412
C32×#10092
C2×C32×#2392
C2×C32×D3×D42
C2×C32×#2352
C22×C32×D122
C2×C32×H(5,𝔽2)3
C2×C3×#10093