Finite Group Databasebeta

← C2×C3×#64366 #13181741

Subgroups — C2×C3×#64366 #13181741

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
2C3×#64366C2116yesmaximal, direct product factor
3C2×C3×#3832C212yesmaximal, complemented
4C2×C3×D4×C3⋊D3C216yesmaximal, complemented
5C2×C3×#3828C213yesmaximal, complemented
6C22×C3×C4⋊(C3⋊D3)C213yesmaximal, complemented
7C2×#64366C311yesmaximal, characteristic, direct product factor
8C2×C3×#1009—34nomaximal
9C23×C3×C3⋊D3C2216yescomplemented
10C33×#41C2218yescomplemented
11C2×C33×Q8C2211yescharacteristic, complemented
12C2×C33×D4C2213yescomplemented
13C22×C33×C4C2213yescomplemented
14C2×C3×#638C2216yescomplemented
15C3×#3832C22116yescomplemented
16C3×D4×C3⋊D3C22148yescomplemented
17C3×#3828C22124yescomplemented
18C2×C3×C4⋊(C3⋊D3)C2213yescomplemented
19C2×C3×C4⋊(C3⋊D3)C2216yescomplemented
20C2×C3×C4⋊(C3⋊D3)C22124yescomplemented
21C2×C3×C4×C3⋊D3C2216yescomplemented
22C2×C3×#634C2211yescharacteristic, complemented
23C2×C32×#41—24no—
24C2×C32×#41D314yescomplemented
25C2×C32×#41C2×C311yescharacteristic, complemented
26#64366C2×C3116yesdirect product factor
27C2×#3832C2×C312yescomplemented
28C2×D4×C3⋊D3C2×C316yescomplemented
29C2×#3828C2×C313yescomplemented
30C22×C4⋊(C3⋊D3)C2×C313yescomplemented
31C3×#1009—364no—
32C2×C3×#239—38no—
33C2×C3×D3×D4—324no—
34C2×C3×#235—312no—
35C22×C3×D12—312no—
36C23×C33C2313yes—
37C22×C3×C3⋊D3—412no—
38C22×C3×C3⋊D3C2316yes—
39C22×C3×C3⋊D3C23124yescomplemented
40C33×Q8C2314yescomplemented
41C33×D4C23112yescomplemented
42C2×C33×C4C2311yescharacteristic
43C2×C33×C4C2313yes—
44C2×C33×C4C23112yescomplemented
45C3×#638C23124yescomplemented
46C2×C3×C3⋊Dic3C2312yes—
47C3×C4⋊(C3⋊D3)C23112yescomplemented
48C3×C4⋊(C3⋊D3)C23124yescomplemented
49C3×C4×C3⋊D3C23124yescomplemented
50C3×#634C2314yescomplemented
51C2×C3×H(5,𝔽2)—91no—
52C2×#1009—34no—
53C23×C3⋊D3C22×C316yescomplemented
54C23×C3×D3—324no—
55C32×#41—232no—
56C32×#41C2×D3132yescomplemented
57C32×#41C22×C318yescomplemented
58C2×C32×Q8—24no—
59C2×C32×Q8C2×D314yescomplemented
60C2×C32×Q8C22×C311yescharacteristic, complemented
61C2×C32×D4—212no—
62C2×C32×D4C2×D3112yescomplemented
63C2×C32×D4C22×C313yescomplemented
64C22×C32×C4—212no—
65C22×C32×C4C2×D3112yescomplemented
66C22×C32×C4C22×C313yescomplemented
67C2×#638C22×C316yescomplemented
68#3832C22×C3116yescomplemented
69D4×C3⋊D3C22×C3148yescomplemented
70#3828C22×C3124yescomplemented
71C2×C4⋊(C3⋊D3)C22×C313yescomplemented
72C2×C4⋊(C3⋊D3)C22×C316yescomplemented
73C2×C4⋊(C3⋊D3)C22×C3124yescomplemented
74C2×C4×C3⋊D3C22×C316yescomplemented
75C2×#634C22×C311yescharacteristic, complemented
76C2×C3×#67—324no—
77C3×#239—364no—
78C3×D3×D4—3192no—
79C3×#235—396no—
80C2×C3×D12—312no—
81C2×C3×D12—324no—
82C2×C3×D12—396no—
83C2×C3×C4×D3—324no—
84C2×C3×Dic6—34no—
85C22×C33—26no—
86C22×C33C2416yes—
87C22×C33C2411yescharacteristic
88C2×C3×C3⋊D3—212no—
89C2×C3×C3⋊D3—424no—
90C2×C3×C3⋊D3C24112yes—
91C33×C4C2412yes—
92C33×C4C2416yes—
93C3×C3⋊Dic3C2414yes—
94C3×H(5,𝔽2)—916no—
95C2×C3×#41—24no—
96C2×C3×#41—92no—
97C2×C3×#41—93no—
98C2×C3×#41C3×D314yescomplemented
99C2×C3×#41C3⋊D311yescharacteristic, complemented
100C22×C3×D4—96no—
101C22×C3×D4—93no—
102#1009—364no—
103C2×#239—38no—
104C2×D3×D4—324no—
105C2×#235—312no—
106C22×D12—312no—
107C23×C32—212no—
108C23×C32C22×D3112yes—
109C23×C32C23×C313yes—
110C22×C3⋊D3—412no—
111C22×C3⋊D3C23×C316yes—
112C22×C3⋊D3C23×C3124yescomplemented
113C22×C3×D3—324no—
114C22×C3×D3—396no—
115C22×C3×D3—1248no—
116C32×Q8—216no—
117C32×Q8C22×D3116yescomplemented
118C32×Q8C23×C314yescomplemented
119C32×D4—248no—
120C32×D4C22×D3148yescomplemented
121C32×D4C23×C3112yescomplemented
122C2×C32×C4—24no—
123C2×C32×C4—212no—
124C2×C32×C4—248no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D3112yes—
127C2×C32×C4C22×D3148yescomplemented
128C2×C32×C4C23×C311yescharacteristic
129C2×C32×C4C23×C313yes—
130C2×C32×C4C23×C3112yescomplemented
131#638C23×C3124yescomplemented
132C2×C3⋊Dic3C23×C312yes—
133C4⋊(C3⋊D3)C23×C3112yescomplemented
134C4⋊(C3⋊D3)C23×C3124yescomplemented
135C4×C3⋊D3C23×C3124yescomplemented
136#634C23×C314yescomplemented
137C3×#67—396no—
138C2×C3×Dic3—38no—
139C3×D12—348no—
140C3×D12—396no—
141C3×C4×D3—396no—
142C3×Dic6—316no—
143C2×H(5,𝔽2)—91noSylow
144C2×C33—26no—
145C2×C33H(5,𝔽2)12yescomplemented
146C2×C33C2511yescharacteristic
147C3×C3⋊D3—212no—
148C24×C3—96no—
149C23×D3—324no—
150C3×#41—232no—
151C3×#41—916no—
152C3×#41—924no—
153C3×#41C2×C3×D3132yescomplemented
154C3×#41C2×C3⋊D318yescomplemented
155C2×C3×Q8—24no—
156C2×C3×Q8—91no—
157C2×C3×Q8C2×C3×D314yescomplemented
158C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
159C2×C3×D4—212no—
160C2×C3×D4—96no—
161C2×C3×D4—948no—
162C2×C3×D4—96no—
163C2×C3×D4—93no—
164C2×C3×D4—924no—
165C2×C3×D4C2×C3×D3112yescomplemented
166C2×C3×D4C2×C3⋊D313yescomplemented
167C22×C3×C4—212no—
168C22×C3×C4—96no—
169C22×C3×C4C2×C3×D3112yescomplemented
170C22×C3×C4C2×C3⋊D313yescomplemented
171C2×#67—324no—
172#239—364no—
173D3×D4—3192no—
174#235—396no—
175C2×D12—312no—
176C2×D12—324no—
177C2×D12—396no—
178C2×C4×D3—324no—
179C2×Dic6—34no—
180C22×C32—26no—
181C22×C32—24no—
182C22×C32—224no—
183C22×C32—224no—
184C22×C32—424no—
185C22×C32C23×D3124yes—
186C22×C32C23×D314yes—
187C22×C32C24×C316yes—
188C22×C32C24×C311yescharacteristic
189C2×C3⋊D3—212no—
190C2×C3⋊D3—424no—
191C2×C3⋊D3C24×C3112yes—
192C2×C3×D3—348no—
193C2×C3×D3—648no—
194C2×C3×D3—1296no—
195C32×C4—28no—
196C32×C4—224no—
197C32×C4C23×D318yes—
198C32×C4C23×D3124yes—
199C32×C4C24×C312yes—
200C32×C4C24×C316yes—
201C3⋊Dic3C24×C314yes—
202C3×Dic3—316no—
203H(5,𝔽2)—916no—
204C2×#41—92no—
205C2×#41—93no—
206C2×#41C3×C3⋊D311yescharacteristic, complemented
207C22×D4—96no—
208C22×D4—93no—
209C33C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
210C23×C3—212no—
211C23×C3—96no—
212C23×C3—924no—
213C23×C3—3612no—
214C23×C3C22×C3×D3112yes—
215C23×C3C22×C3⋊D313yes—
216C22×D3—324no—
217C22×D3—396no—
218C22×D3—1248no—
219C3×Q8—216no—
220C3×Q8—94no—
221C3×Q8C22×C3×D3116yescomplemented
222C3×Q8C22×C3⋊D314yescomplemented
223C3×D4—248no—
224C3×D4—924no—
225C3×D4—924no—
226C3×D4—912no—
227C3×D4C22×C3×D3148yescomplemented
228C3×D4C22×C3⋊D3112yescomplemented
229C2×C3×C4—24no—
230C2×C3×C4—212no—
231C2×C3×C4—248no—
232C2×C3×C4—92no—
233C2×C3×C4—924no—
234C2×C3×C4C22×C3×D314yes—
235C2×C3×C4C22×C3×D3112yes—
236C2×C3×C4C22×C3×D3148yescomplemented
237C2×C3×C4C22×C3⋊D313yes—
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×C3×C4C22×C3⋊D3112yescomplemented
240#67—396no—
241C2×Dic3—38no—
242D12—348no—
243D12—396no—
244C4×D3—396no—
245Dic6—316no—
246C2×C32—26no—
247C2×C32—28no—
248C2×C32—24no—
249C2×C32—224no—
250C2×C32—424no—
251C2×C32#100918yescomplemented
252C2×C32C3×H(5,𝔽2)12yescomplemented
253C2×C32C24×D314yes—
254C2×C32C25×C311yescharacteristic
255C3⋊D3—212no—
256C3×D3—648no—
257C24—96no—
258#41—916no—
259#41—924no—
260#41C2×C3×C3⋊D318yescomplemented
261C2×Q8—91no—
262C2×Q8C2×C3×C3⋊D311yescharacteristic, complemented
263C2×D4—96no—
264C2×D4—948no—
265C2×D4—96no—
266C2×D4—93no—
267C2×D4—924no—
268C2×D4C2×C3×C3⋊D313yescomplemented
269C22×C4—96no—
270C22×C4C2×C3×C3⋊D313yescomplemented
271C22×C3—224no—
272C22×C3—26no—
273C22×C3—24no—
274C22×C3—224no—
275C22×C3—424no—
276C22×C3—912no—
277C22×C3—1812no—
278C22×C3—3624no—
279C22×C3C23×C3×D3124yes—
280C22×C3C23×C3×D314yes—
281C22×C3C23×C3⋊D316yes—
282C22×C3C23×C3⋊D311yescharacteristic
283C2×D3—348no—
284C2×D3—648no—
285C2×D3—1296no—
286C3×C4—28no—
287C3×C4—224no—
288C3×C4—94no—
289C3×C4C23×C3×D318yes—
290C3×C4C23×C3×D3124yes—
291C3×C4C23×C3⋊D316yes—
292C3×C4C23×C3⋊D312yes—
293Dic3—316no—
294C32—24no—
295C32C2×#100914yescomplemented
296C32C2×C3×H(5,𝔽2)11yescharacteristic, complemented
297C23—96no—
298C23—924no—
299C23—3612no—
300C23C22×C3×C3⋊D313yes—
301Q8—94no—
302Q8C22×C3×C3⋊D314yescomplemented
303D4—924no—
304D4—924no—
305D4—912no—
306D4C22×C3×C3⋊D3112yescomplemented
307C2×C4—92no—
308C2×C4—924no—
309C2×C4C22×C3×C3⋊D313yes—
310C2×C4C22×C3×C3⋊D311yescharacteristic
311C2×C4C22×C3×C3⋊D3112yescomplemented
312C2×C3—224no—
313C2×C3—26no—
314C2×C3—28no—
315C2×C3—24no—
316C2×C3—424no—
317C2×C3—1812no—
318C2×C3C3×#100918yescomplemented
319C2×C3#6436612yesdirect product factor
320C2×C3C24×C3×D314yes—
321C2×C3C24×C3⋊D311yescharacteristic
322D3—648no—
323C22—26no—
324C22—912no—
325C22—1812no—
326C22—3624no—
327C22C23×C3×C3⋊D316yes—
328C22C23×C3×C3⋊D311yescharacteristic
329C4—94no—
330C4C23×C3×C3⋊D316yes—
331C4C23×C3×C3⋊D312yes—
332C3—24no—
333C3C2×C3×#100914yescomplemented
334C3C2×#6436611yescharacteristic, direct product factor
335C2—26no—
336C2—1812no—
337C2C3×#6436612yesdirect product factor
338C2C24×C3×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC2×C32C25×C3
FrattiniC2C24×C3×C3⋊D3
FittingC2×C33×#41C2
socleC22×C33C24
hypercenterC2×C3×#41C3⋊D3
hypocenterC32C2×C3×H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. C3×#64366C2direct (N × H)
3. C2×C3×#3832C2semidirect
4. C2×C3×D4×C3⋊D3C2semidirect
5. C2×C3×#3828C2semidirect
6. C22×C3×C4⋊(C3⋊D3)C2semidirect
7. C2×#64366C3direct (N × H)
9. C23×C3×C3⋊D3C22semidirect
10. C33×#41C22semidirect
11. C2×C33×Q8C22semidirect
12. C2×C33×D4C22semidirect
13. C22×C33×C4C22semidirect
14. C2×C3×#638C22semidirect
15. C3×#3832C22semidirect
16. C3×D4×C3⋊D3C22semidirect
17. C3×#3828C22semidirect
18. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C2×C3×C4⋊(C3⋊D3)C22semidirect
21. C2×C3×C4×C3⋊D3C22semidirect
22. C2×C3×#634C22semidirect
24. C2×C32×#41D3semidirect
25. C2×C32×#41C2×C3semidirect
26. #64366C2×C3direct (N × H)
27. C2×#3832C2×C3semidirect
28. C2×D4×C3⋊D3C2×C3semidirect
29. C2×#3828C2×C3semidirect
30. C22×C4⋊(C3⋊D3)C2×C3semidirect
39. C22×C3×C3⋊D3C23semidirect
40. C33×Q8C23semidirect
41. C33×D4C23semidirect
44. C2×C33×C4C23semidirect
45. C3×#638C23semidirect
47. C3×C4⋊(C3⋊D3)C23semidirect
48. C3×C4⋊(C3⋊D3)C23semidirect
49. C3×C4×C3⋊D3C23semidirect
50. C3×#634C23semidirect
53. C23×C3⋊D3C22×C3semidirect
56. C32×#41C2×D3semidirect
57. C32×#41C22×C3semidirect
59. C2×C32×Q8C2×D3semidirect
60. C2×C32×Q8C22×C3semidirect
62. C2×C32×D4C2×D3semidirect
63. C2×C32×D4C22×C3semidirect
65. C22×C32×C4C2×D3semidirect
66. C22×C32×C4C22×C3semidirect
67. C2×#638C22×C3semidirect
68. #3832C22×C3semidirect
69. D4×C3⋊D3C22×C3semidirect
70. #3828C22×C3semidirect
71. C2×C4⋊(C3⋊D3)C22×C3semidirect
72. C2×C4⋊(C3⋊D3)C22×C3semidirect
73. C2×C4⋊(C3⋊D3)C22×C3semidirect
74. C2×C4×C3⋊D3C22×C3semidirect
75. C2×#634C22×C3semidirect
98. C2×C3×#41C3×D3semidirect
99. C2×C3×#41C3⋊D3semidirect
112. C22×C3⋊D3C23×C3semidirect
117. C32×Q8C22×D3semidirect
118. C32×Q8C23×C3semidirect
120. C32×D4C22×D3semidirect
121. C32×D4C23×C3semidirect
127. C2×C32×C4C22×D3semidirect
130. C2×C32×C4C23×C3semidirect
131. #638C23×C3semidirect
133. C4⋊(C3⋊D3)C23×C3semidirect
134. C4⋊(C3⋊D3)C23×C3semidirect
135. C4×C3⋊D3C23×C3semidirect
136. #634C23×C3semidirect
145. C2×C33H(5,𝔽2)semidirect
153. C3×#41C2×C3×D3semidirect
154. C3×#41C2×C3⋊D3semidirect
157. C2×C3×Q8C2×C3×D3semidirect
158. C2×C3×Q8C2×C3⋊D3semidirect
165. C2×C3×D4C2×C3×D3semidirect
166. C2×C3×D4C2×C3⋊D3semidirect
169. C22×C3×C4C2×C3×D3semidirect
170. C22×C3×C4C2×C3⋊D3semidirect
206. C2×#41C3×C3⋊D3semidirect
209. C33C2×H(5,𝔽2)semidirect
221. C3×Q8C22×C3×D3semidirect
222. C3×Q8C22×C3⋊D3semidirect
227. C3×D4C22×C3×D3semidirect
228. C3×D4C22×C3⋊D3semidirect
236. C2×C3×C4C22×C3×D3semidirect
239. C2×C3×C4C22×C3⋊D3semidirect
251. C2×C32#1009semidirect
252. C2×C32C3×H(5,𝔽2)semidirect
260. #41C2×C3×C3⋊D3semidirect
262. C2×Q8C2×C3×C3⋊D3semidirect
268. C2×D4C2×C3×C3⋊D3semidirect
270. C22×C4C2×C3×C3⋊D3semidirect
295. C32C2×#1009semidirect
296. C32C2×C3×H(5,𝔽2)semidirect
302. Q8C22×C3×C3⋊D3semidirect
306. D4C22×C3×C3⋊D3semidirect
311. C2×C4C22×C3×C3⋊D3semidirect
318. C2×C3C3×#1009semidirect
319. C2×C3#64366direct (N × H)
333. C3C2×C3×#1009semidirect
334. C3C2×#64366direct (N × H)
337. C2C3×#64366direct (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
C3×#643662
C2×C3×#38322
C2×C3×D4×C3⋊D32
C2×C3×#38282
C22×C3×C4⋊(C3⋊D3)2
C2×#643663
C2×C3×#10093