Finite Group Databasebeta

← C32×C4×#239 #13178950

Subgroups — C32×C4×#239 #13178950

Order 1728.

2140 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#239C211yesmaximal, characteristic
2C33×C4×Q8C211yesmaximal, characteristic, complemented
3C32×Q8×Dic3C211yesmaximal, characteristic, complemented
4C32×#893C213yesmaximal, complemented
5C32×#892C213yesmaximal, complemented
6C32×C4×D12C213yesmaximal, complemented
7C32×C42×D3C213yesmaximal, complemented
8C32×C4×#41—31nomaximal
9C3×C4×#239C314yesmaximal, direct product factor
10C2×C33×Q8C2211yescharacteristic
11C32×#239C418yesdirect product factor
12C2×C32×D12C2213yes—
13C2×C32×C4×D3C2213yes—
14C2×C32×C4×D3C2213yes—
15C2×C32×C4×D3C2213yes—
16C33×C4⋊C4C2213yes—
17C33×C42C2213yes—
18C32×(C2×D3)⋊C4C2216yes—
19C32×C4⋊Dic3C2213yes—
20C32×C4×Dic3C2213yes—
21C32×C4×Dic3C2211yescharacteristic
22C32×C4×Dic3C2213yes—
23C2×C32×#41—31no—
24C2×C3×#239C2×C314yes—
25C32×C4×Q8—24no—
26C32×C4×Q8—31no—
27C32×C4×Q8D311yescharacteristic, complemented
28C32×C4×Q8C2×C314yescomplemented
29C32×C4×D4—33no—
30C32×C4×D4—33no—
31C32×#117—33no—
32C2×C32×C42—33no—
33C3×Q8×Dic3C2×C314yescomplemented
34C3×#893C2×C3112yescomplemented
35C3×#892C2×C3112yescomplemented
36C3×C4×D12C2×C3112yescomplemented
37C3×C42×D3C2×C3112yescomplemented
38C22×C32×D3C2313yes—
39C33×Q8C2×C414yescomplemented
40C2×C33×C4C2313yes—
41C2×C33×C4C2313yes—
42C2×C33×C4C2311yescharacteristic
43C2×C32×Dic3C2313yes—
44C2×C32×Dic3C2311yescharacteristic
45C2×C32×Dic3C2311yescharacteristic
46C32×D12C2×C4112yescomplemented
47C32×C4×D3—26no—
48C32×C4×D3—26no—
49C32×C4×D3C2×C4112yescomplemented
50C3×C4×#41—34no—
51C4×#239C3211yescharacteristic, direct product factor
52C32×#41—38no—
53C2×C32×Q8—24no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C22×C314yes—
56C2×C32×D4—33no—
57C22×C32×C4—33no—
58C22×C32×C4—33no—
59C22×C32×C4—33no—
60C3×#239C3×C4132yesdirect product factor
61C2×C3×D12C22×C3112yes—
62C2×C3×C4×D3C22×C3112yes—
63C2×C3×C4×D3C22×C3112yes—
64C2×C3×C4×D3C22×C3112yes—
65C32×C4⋊C4—212no—
66C32×C4⋊C4—33no—
67C32×C4⋊C4C2×D313yes—
68C32×C4⋊C4C22×C3112yes—
69C32×C22⋊C4—36no—
70C32×C42—212no—
71C32×C42—33no—
72C32×C42—33no—
73C32×C42—31no—
74C32×C42C2×D313yes—
75C32×C42C22×C3112yes—
76C3×(C2×D3)⋊C4C22×C3124yes—
77C3×C4⋊Dic3C22×C3112yes—
78C3×C4×Dic3C22×C3112yes—
79C3×C4×Dic3C22×C3112yes—
80C3×C4×Dic3C22×C314yes—
81C22×C33C2411yescharacteristic
82C2×C32×D3—23no—
83C2×C32×D3—23no—
84C2×C32×D3C22×C416yes—
85C33×C4—23no—
86C33×C4C22×C416yes—
87C33×C4#4112yescomplemented
88C32×Dic3—23no—
89C32×Dic3C22×C412yes—
90C32×Dic3#4112yescomplemented
91C2×C3×#41—34no—
92C2×#239C2×C3211yescharacteristic
93C3×C4×Q8—24no—
94C3×C4×Q8—34no—
95C3×C4×Q8C3×D314yescomplemented
96C3×C4×Q8C2×C3211yescharacteristic, complemented
97C3×C4×D4—312no—
98C3×C4×D4—312no—
99C3×#117—312no—
100C2×C3×C42—312no—
101Q8×Dic3C2×C3211yescharacteristic, complemented
102#893C2×C3213yescomplemented
103#892C2×C3213yescomplemented
104C4×D12C2×C3213yescomplemented
105C42×D3C2×C3213yescomplemented
106C23×C32—33no—
107C22×C3×D3C23×C3112yes—
108C32×Q8—216no—
109C32×Q8C4×D314yescomplemented
110C32×Q8C2×C3×C4116yescomplemented
111C32×D4—312no—
112C2×C32×C4—24no—
113C2×C32×C4—212no—
114C2×C32×C4—212no—
115C2×C32×C4—31no—
116C2×C32×C4—31no—
117C2×C32×C4—312no—
118C2×C32×C4—33no—
119C2×C32×C4—66no—
120C2×C32×C4—66no—
121C2×C32×C4C22×D313yes—
122C2×C32×C4C22×D313yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C23×C3112yes—
125C2×C32×C4C23×C3112yes—
126C2×C32×C4C23×C314yes—
127C2×C3×Dic3C23×C3112yes—
128C2×C3×Dic3C23×C314yes—
129C2×C3×Dic3C23×C314yes—
130C3×D12C2×C3×C4148yescomplemented
131C3×C4×D3—224no—
132C3×C4×D3—224no—
133C3×C4×D3C2×C3×C4148yescomplemented
134C4×#41—31noSylow
135C2×C33C23×C411yescharacteristic
136C2×C33C2×#4111yescharacteristic
137C2×C33C2×#4111yescharacteristic
138C32×D3—26no—
139C3×#41—332no—
140C2×C3×Q8—24no—
141C2×C3×Q8C2×C3×D314yes—
142C2×C3×Q8C22×C3211yescharacteristic
143C2×C3×D4—312no—
144C22×C3×C4—312no—
145C22×C3×C4—312no—
146C22×C3×C4—312no—
147#239C32×C418yesdirect product factor
148C2×D12C22×C3213yes—
149C2×C4×D3C22×C3213yes—
150C2×C4×D3C22×C3213yes—
151C2×C4×D3C22×C3213yes—
152C3×C4⋊C4—212no—
153C3×C4⋊C4—312no—
154C3×C4⋊C4C2×C3×D3112yes—
155C3×C4⋊C4C22×C3213yes—
156C3×C22⋊C4—324no—
157C3×C42—212no—
158C3×C42—34no—
159C3×C42—312no—
160C3×C42—312no—
161C3×C42C2×C3×D3112yes—
162C3×C42C22×C3213yes—
163(C2×D3)⋊C4C22×C3216yes—
164C4⋊Dic3C22×C3213yes—
165C4×Dic3C22×C3211yescharacteristic
166C4×Dic3C22×C3213yes—
167C4×Dic3C22×C3213yes—
168C22×C32—24no—
169C22×C32—36no—
170C22×C32—63no—
171C22×C32—63no—
172C22×C32C23×D311yescharacteristic
173C22×C32C24×C314yes—
174C2×C3×D3—212no—
175C2×C3×D3—212no—
176C2×C3×D3C22×C3×C4124yes—
177C32×C4—23no—
178C32×C4—224no—
179C32×C4—28no—
180C32×C4—212no—
181C32×C4—32no—
182C32×C4—32no—
183C32×C4—412no—
184C32×C4—63no—
185C32×C4C2×C4×D316yes—
186C32×C4#23912yesdirect product factor
187C32×C4C22×C3×C4124yes—
188C32×C4C3×#4118yescomplemented
189C3×Dic3—212no—
190C3×Dic3C22×C3×C418yes—
191C3×Dic3C3×#4118yescomplemented
192C2×#41—31no—
193C4×Q8—31no—
194C4×Q8C32×D311yescharacteristic, complemented
195C4×D4—33no—
196C4×D4—33no—
197#117—33no—
198C2×C42—33no—
199C33C4×#4111yescharacteristic, complemented, Sylow
200C23×C3—312no—
201C22×D3C23×C3213yes—
202C3×Q8—216no—
203C3×Q8C3×C4×D3116yescomplemented
204C3×Q8C2×C32×C414yescomplemented
205C3×D4—348no—
206C2×C3×C4—24no—
207C2×C3×C4—212no—
208C2×C3×C4—212no—
209C2×C3×C4—34no—
210C2×C3×C4—34no—
211C2×C3×C4—348no—
212C2×C3×C4—312no—
213C2×C3×C4—624no—
214C2×C3×C4—624no—
215C2×C3×C4C22×C3×D3112yes—
216C2×C3×C4C22×C3×D3112yes—
217C2×C3×C4C22×C3×D314yes—
218C2×C3×C4C23×C3213yes—
219C2×C3×C4C23×C3213yes—
220C2×C3×C4C23×C3211yescharacteristic
221C2×Dic3C23×C3213yes—
222C2×Dic3C23×C3211yescharacteristic
223C2×Dic3C23×C3211yescharacteristic
224D12C2×C32×C4112yescomplemented
225C4×D3—26no—
226C4×D3—26no—
227C4×D3C2×C32×C4112yescomplemented
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—66no—
232C2×C32C22×C4×D311yescharacteristic
233C2×C32C2×#23911yescharacteristic
234C2×C32D3×#4111yescharacteristic
235C2×C32C23×C3×C414yes—
236C2×C32C2×C3×#4114yes—
237C2×C32C2×C3×#4114yes—
238C3×D3—224no—
239#41—38no—
240C2×Q8C2×C32×D311yescharacteristic
241C2×D4—33no—
242C22×C4—33no—
243C22×C4—33no—
244C22×C4—33no—
245C4⋊C4—33no—
246C4⋊C4C2×C32×D313yes—
247C22⋊C4—36no—
248C42—31no—
249C42—33no—
250C42—33no—
251C42C2×C32×D313yes—
252C22×C3—24no—
253C22×C3—324no—
254C22×C3—612no—
255C22×C3—612no—
256C22×C3C23×C3×D314yes—
257C22×C3C24×C3211yescharacteristic
258C2×D3—23no—
259C2×D3—23no—
260C2×D3C22×C32×C416yes—
261C3×C4—212no—
262C3×C4—23no—
263C3×C4—224no—
264C3×C4—28no—
265C3×C4—38no—
266C3×C4—38no—
267C3×C4—412no—
268C3×C4—612no—
269C3×C4C2×C3×C4×D3124yes—
270C3×C4C3×#23918yesdirect product factor
271C3×C4C22×C32×C416yes—
272C3×C4C32×#4112yescomplemented
273Dic3—23no—
274Dic3C22×C32×C412yes—
275Dic3C32×#4112yescomplemented
276C32—24no—
277C32C4×#23911yescharacteristic, direct product factor
278C32C3×C4×#4114yescomplemented
279C23—33no—
280Q8C32×C4×D314yescomplemented
281D4—312no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—312no—
285C2×C4—33no—
286C2×C4—66no—
287C2×C4—66no—
288C2×C4C22×C32×D313yes—
289C2×C4C22×C32×D313yes—
290C2×C4C22×C32×D311yescharacteristic
291C2×C3—24no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—624no—
295C2×C3C22×C3×C4×D314yes—
296C2×C3C2×C3×#23914yes—
297C2×C3C3×D3×#4114yes—
298C2×C3C23×C32×C411yescharacteristic
299C2×C3C2×C32×#4111yescharacteristic
300C2×C3C2×C32×#4111yescharacteristic
301D3—26no—
302C22—36no—
303C22—63no—
304C22—63no—
305C22C23×C32×D311yescharacteristic
306C4—23no—
307C4—32no—
308C4—32no—
309C4—63no—
310C4C2×C32×C4×D316yes—
311C4C32×#23912yesdirect product factor
312C3—24no—
313C3C3×C4×#23914yesdirect product factor
314C3C32×C4×#4111yescharacteristic, complemented
315C2—66no—
316C2C22×C32×C4×D311yescharacteristic
317C2C2×C32×#23911yescharacteristic
318C2C32×D3×#4111yescharacteristic
subgroupG/H
centerC2×C32×C4C22×D3
commutatorC2×C3C23×C32×C4
FrattiniC22C23×C32×D3
FittingC33×C4×Q8C2
socleC22×C33C24
prime radicalC2×C32×#239C2
cosocleC22×C3C24×C32
hypercenterC32×C4×Q8D3
hypocenterC3C32×C4×#41
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32×C4 ◁ C32×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C33×C4×Q8C2semidirect
3. C32×Q8×Dic3C2semidirect
4. C32×#893C2semidirect
5. C32×#892C2semidirect
6. C32×C4×D12C2semidirect
7. C32×C42×D3C2semidirect
9. C3×C4×#239C3direct (N × H)
11. C32×#239C4direct (N × H)
27. C32×C4×Q8D3semidirect
28. C32×C4×Q8C2×C3semidirect
33. C3×Q8×Dic3C2×C3semidirect
34. C3×#893C2×C3semidirect
35. C3×#892C2×C3semidirect
36. C3×C4×D12C2×C3semidirect
37. C3×C42×D3C2×C3semidirect
39. C33×Q8C2×C4semidirect
46. C32×D12C2×C4semidirect
49. C32×C4×D3C2×C4semidirect
51. C4×#239C32direct (N × H)
60. C3×#239C3×C4direct (N × H)
87. C33×C4#41semidirect
90. C32×Dic3#41semidirect
95. C3×C4×Q8C3×D3semidirect
96. C3×C4×Q8C2×C32semidirect
101. Q8×Dic3C2×C32semidirect
102. #893C2×C32semidirect
103. #892C2×C32semidirect
104. C4×D12C2×C32semidirect
105. C42×D3C2×C32semidirect
109. C32×Q8C4×D3semidirect
110. C32×Q8C2×C3×C4semidirect
130. C3×D12C2×C3×C4semidirect
133. C3×C4×D3C2×C3×C4semidirect
147. #239C32×C4direct (N × H)
186. C32×C4#239direct (N × H)
188. C32×C4C3×#41semidirect
191. C3×Dic3C3×#41semidirect
194. C4×Q8C32×D3semidirect
199. C33C4×#41semidirect
203. C3×Q8C3×C4×D3semidirect
204. C3×Q8C2×C32×C4semidirect
224. D12C2×C32×C4semidirect
227. C4×D3C2×C32×C4semidirect
270. C3×C4C3×#239direct (N × H)
272. C3×C4C32×#41semidirect
275. Dic3C32×#41semidirect
277. C32C4×#239direct (N × H)
278. C32C3×C4×#41semidirect
280. Q8C32×C4×D3semidirect
311. C4C32×#239direct (N × H)
313. C3C3×C4×#239direct (N × H)
314. C3C32×C4×#41semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
C33×C4×Q82
C32×Q8×Dic32
C32×#8932
C32×#8922
C32×C4×D122
C32×C42×D32
C32×C4×#413
C3×C4×#2393