Finite Group Databasebeta

← C2×C3×C4×#239 #11899956

Subgroups — C2×C3×C4×#239 #11899956

Order 1152.

3605 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#239C211yesmaximal, characteristic
2C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
3C2×C3×Q8×Dic3C211yesmaximal, characteristic, complemented
4C3×C4×#239C2116yesmaximal, direct product factor
5C2×C3×#893C213yesmaximal, complemented
6C2×C3×#892C213yesmaximal, complemented
7C2×C3×C4×D12C213yesmaximal, complemented
8C2×C3×C42×D3C213yesmaximal, complemented
9C2×C3×C4×#41—31nomaximal
10C2×C4×#239C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C2×C3×#239C4116yesdirect product factor
13C2×C3×#239C2218yes—
14C22×C3×D12C2213yes—
15C22×C3×C4×D3C2213yes—
16C22×C3×C4×D3C2213yes—
17C22×C3×C4×D3C2213yes—
18C32×C4×Q8C2218yescomplemented
19C2×C32×C4⋊C4C2213yes—
20C2×C32×C42C2213yes—
21C3×Q8×Dic3C2218yescomplemented
22C2×C3×(C2×D3)⋊C4C2216yes—
23C2×C3×C4⋊Dic3C2213yes—
24C2×C3×C4×Dic3C2213yes—
25C2×C3×C4×Dic3C2213yes—
26C2×C3×C4×Dic3C2211yescharacteristic
27C3×#893C22124yescomplemented
28C3×#892C22124yescomplemented
29C3×C4×D12C22124yescomplemented
30C3×C42×D3C22124yescomplemented
31C22×C3×#41—31no—
32C22×#239C2×C311yescharacteristic
33C3×C4×#41—316no—
34C2×C3×C4×Q8—21no—
35C2×C3×C4×Q8—31no—
36C2×C3×C4×Q8D311yescharacteristic, complemented
37C2×C3×C4×Q8C2×C311yescharacteristic, complemented
38C2×C3×C4×D4—33no—
39C2×C3×C4×D4—33no—
40C2×C3×#117—33no—
41C22×C3×C42—33no—
42C2×Q8×Dic3C2×C311yescharacteristic, complemented
43C4×#239C2×C3116yesdirect product factor
44C2×#893C2×C313yescomplemented
45C2×#892C2×C313yescomplemented
46C2×C4×D12C2×C313yescomplemented
47C2×C42×D3C2×C313yescomplemented
48C23×C3×D3C2313yes—
49C2×C32×Q8C2×C418yescomplemented
50C2×C32×Q8C2314yes—
51C22×C32×C4C2313yes—
52C22×C32×C4C2313yes—
53C22×C32×C4C2311yescharacteristic
54C22×C3×Dic3C2313yes—
55C22×C3×Dic3C2311yescharacteristic
56C22×C3×Dic3C2311yescharacteristic
57C3×#239C2×C4164yesdirect product factor
58C2×C3×D12C2×C4124yescomplemented
59C2×C3×D12C23112yes—
60C2×C3×C4×D3—212no—
61C2×C3×C4×D3—212no—
62C2×C3×C4×D3C2×C4124yescomplemented
63C2×C3×C4×D3C23112yes—
64C2×C3×C4×D3C23112yes—
65C2×C3×C4×D3C23112yes—
66C32×C4⋊C4C23112yes—
67C32×C42C23112yes—
68C3×(C2×D3)⋊C4C23124yes—
69C3×C4⋊Dic3C23112yes—
70C3×C4×Dic3C23112yes—
71C3×C4×Dic3C23112yes—
72C3×C4×Dic3C2314yes—
73C2×C4×#41—31noSylow
74C2×C3×#41—316no—
75C2×C3×#41—38no—
76C22×C3×Q8—21no—
77C22×C3×Q8C2×D311yescharacteristic
78C22×C3×Q8C22×C311yescharacteristic
79C22×C3×D4—33no—
80C23×C3×C4—33no—
81C23×C3×C4—33no—
82C23×C3×C4—33no—
83C2×#239C3×C4116yesdirect product factor
84C2×#239C22×C318yes—
85C22×D12C22×C313yes—
86C22×C4×D3C22×C313yes—
87C22×C4×D3C22×C313yes—
88C22×C4×D3C22×C313yes—
89C3×C4×Q8—28no—
90C3×C4×Q8—38no—
91C3×C4×Q8C2×D318yescomplemented
92C3×C4×Q8C22×C318yescomplemented
93C3×C4×D4—324no—
94C3×C4×D4—324no—
95C3×#117—324no—
96C2×C3×C4⋊C4—23no—
97C2×C3×C4⋊C4—33no—
98C2×C3×C4⋊C4C2×D313yes—
99C2×C3×C4⋊C4C22×C313yes—
100C2×C3×C22⋊C4—36no—
101C2×C3×C42—23no—
102C2×C3×C42—33no—
103C2×C3×C42—33no—
104C2×C3×C42—31no—
105C2×C3×C42—324no—
106C2×C3×C42C2×D313yes—
107C2×C3×C42C22×C313yes—
108Q8×Dic3C22×C318yescomplemented
109C2×(C2×D3)⋊C4C22×C316yes—
110C2×C4⋊Dic3C22×C313yes—
111C2×C4×Dic3C22×C311yescharacteristic
112C2×C4×Dic3C22×C313yes—
113C2×C4×Dic3C22×C313yes—
114#893C22×C3124yescomplemented
115#892C22×C3124yescomplemented
116C4×D12C22×C3124yescomplemented
117C42×D3C22×C3124yescomplemented
118C23×C32C2411yescharacteristic
119C22×C3×D3—26no—
120C22×C3×D3—26no—
121C22×C3×D3C22×C4112yes—
122C22×C3×D3C2416yes—
123C32×Q8C22×C4116yescomplemented
124C2×C32×C4—26no—
125C2×C32×C4C22×C4112yes—
126C2×C32×C4#4114yescomplemented
127C2×C32×C4C2416yes—
128C2×C32×C4C2416yes—
129C2×C32×C4C2412yes—
130C2×C3×Dic3—26no—
131C2×C3×Dic3C22×C414yes—
132C2×C3×Dic3#4114yescomplemented
133C2×C3×Dic3C2416yes—
134C2×C3×Dic3C2412yes—
135C2×C3×Dic3C2412yes—
136C3×D12C22×C4148yescomplemented
137C3×C4×D3—224no—
138C3×C4×D3—224no—
139C3×C4×D3C22×C4148yescomplemented
140C22×#41—31no—
141C4×#41—316no—
142C2×C4×Q8—31no—
143C2×C4×Q8C3×D311yescharacteristic, complemented
144C2×C4×D4—33no—
145C2×C4×D4—33no—
146C2×#117—33no—
147C22×C42—33no—
148C24×C3—33no—
149C23×D3C23×C313yes—
150C3×#41—364no—
151C2×C3×Q8—28no—
152C2×C3×Q8—24no—
153C2×C3×Q8C4×D318yescomplemented
154C2×C3×Q8C2×C3×C418yescomplemented
155C2×C3×Q8C22×D314yes—
156C2×C3×Q8C23×C314yes—
157C2×C3×D4—324no—
158C2×C3×D4—312no—
159C22×C3×C4—21no—
160C22×C3×C4—23no—
161C22×C3×C4—23no—
162C22×C3×C4—31no—
163C22×C3×C4—31no—
164C22×C3×C4—324no—
165C22×C3×C4—312no—
166C22×C3×C4—33no—
167C22×C3×C4—312no—
168C22×C3×C4—312no—
169C22×C3×C4—612no—
170C22×C3×C4—612no—
171C22×C3×C4C22×D313yes—
172C22×C3×C4C22×D313yes—
173C22×C3×C4C22×D311yescharacteristic
174C22×C3×C4C23×C313yes—
175C22×C3×C4C23×C313yes—
176C22×C3×C4C23×C311yescharacteristic
177C22×Dic3C23×C313yes—
178C22×Dic3C23×C311yescharacteristic
179C22×Dic3C23×C311yescharacteristic
180#239C2×C3×C4164yesdirect product factor
181C2×D12C2×C3×C4124yescomplemented
182C2×D12C23×C3112yes—
183C2×C4×D3—212no—
184C2×C4×D3—212no—
185C2×C4×D3C2×C3×C4124yescomplemented
186C2×C4×D3C23×C3112yes—
187C2×C4×D3C23×C3112yes—
188C2×C4×D3C23×C3112yes—
189C3×C4⋊C4—212no—
190C3×C4⋊C4—312no—
191C3×C4⋊C4C22×D3112yes—
192C3×C4⋊C4C23×C3112yes—
193C3×C22⋊C4—324no—
194C3×C42—212no—
195C3×C42—34no—
196C3×C42—312no—
197C3×C42—312no—
198C3×C42C22×D3112yes—
199C3×C42C23×C3112yes—
200(C2×D3)⋊C4C23×C3124yes—
201C4⋊Dic3C23×C3112yes—
202C4×Dic3C23×C314yes—
203C4×Dic3C23×C3112yes—
204C4×Dic3C23×C3112yes—
205C22×C32C23×C412yes—
206C22×C32C2×#4112yes—
207C22×C32C2×#4112yes—
208C22×C32C2511yescharacteristic
209C2×C3×D3—224no—
210C2×C3×D3—26no—
211C2×C3×D3—26no—
212C2×C3×D3C23×C4112yes—
213C32×C4—26no—
214C32×C4C23×C4112yes—
215C32×C4C2×#4114yescomplemented
216C3×Dic3—26no—
217C3×Dic3C23×C414yes—
218C3×Dic3C2×#4114yescomplemented
219C2×#41—316no—
220C2×#41—38no—
221C22×Q8C2×C3×D311yescharacteristic
222C22×D4—33no—
223C23×C4—33no—
224C23×C4—33no—
225C23×C4—33no—
226C4×Q8—38no—
227C4×Q8C2×C3×D318yescomplemented
228C4×D4—324no—
229C4×D4—324no—
230#117—324no—
231C2×C4⋊C4—33no—
232C2×C4⋊C4C2×C3×D313yes—
233C2×C22⋊C4—36no—
234C2×C42—33no—
235C2×C42—33no—
236C2×C42—31no—
237C2×C42—324no—
238C2×C42C2×C3×D313yes—
239C23×C3—21no—
240C23×C3—312no—
241C23×C3—36no—
242C23×C3—66no—
243C23×C3—66no—
244C23×C3C23×D311yescharacteristic
245C23×C3C24×C311yescharacteristic
246C22×D3—26no—
247C22×D3—26no—
248C22×D3C22×C3×C4112yes—
249C22×D3C24×C316yes—
250C3×Q8—216no—
251C3×Q8C2×C4×D3116yescomplemented
252C3×Q8C22×C3×C4116yescomplemented
253C3×D4—348no—
254C2×C3×C4—26no—
255C2×C3×C4—26no—
256C2×C3×C4—22no—
257C2×C3×C4—26no—
258C2×C3×C4—26no—
259C2×C3×C4—212no—
260C2×C3×C4—24no—
261C2×C3×C4—34no—
262C2×C3×C4—34no—
263C2×C3×C4—32no—
264C2×C3×C4—32no—
265C2×C3×C4—348no—
266C2×C3×C4—36no—
267C2×C3×C4—46no—
268C2×C3×C4—624no—
269C2×C3×C4—66no—
270C2×C3×C4—624no—
271C2×C3×C4C2×C4×D3112yes—
272C2×C3×C4#23914yesdirect product factor
273C2×C3×C4C22×C3×C4112yes—
274C2×C3×C4C3×#4114yescomplemented
275C2×C3×C4C23×D316yes—
276C2×C3×C4C23×D316yes—
277C2×C3×C4C23×D312yes—
278C2×C3×C4C24×C316yes—
279C2×C3×C4C24×C316yes—
280C2×C3×C4C24×C312yes—
281C2×Dic3—26no—
282C2×Dic3C22×C3×C414yes—
283C2×Dic3C3×#4114yescomplemented
284C2×Dic3C24×C316yes—
285C2×Dic3C24×C312yes—
286C2×Dic3C24×C312yes—
287D12C22×C3×C4148yescomplemented
288C4×D3—224no—
289C4×D3—224no—
290C4×D3C22×C3×C4148yescomplemented
291C2×C32C4×#4114yescomplemented
292C2×C32C24×C411yescharacteristic
293C2×C32C22×#4111yescharacteristic
294C2×C32C22×#4111yescharacteristic
295C3×D3—212no—
296C24—33no—
297#41—364no—
298C2×Q8C3×C4×D318yescomplemented
299C2×Q8C22×C3×D314yes—
300C2×D4—312no—
301C2×D4—324no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—312no—
305C22×C4—324no—
306C22×C4—33no—
307C22×C4—312no—
308C22×C4—312no—
309C22×C4—612no—
310C22×C4—612no—
311C22×C4C22×C3×D313yes—
312C22×C4C22×C3×D313yes—
313C22×C4C22×C3×D311yescharacteristic
314C4⋊C4—312no—
315C4⋊C4C22×C3×D3112yes—
316C22⋊C4—324no—
317C42—34no—
318C42—312no—
319C42—312no—
320C42C22×C3×D3112yes—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—21no—
325C22×C3—312no—
326C22×C3—66no—
327C22×C3—66no—
328C22×C3—624no—
329C22×C3C22×C4×D312yes—
330C22×C3C2×#23912yes—
331C22×C3D3×#4112yes—
332C22×C3C23×C3×C412yes—
333C22×C3C2×C3×#4112yes—
334C22×C3C2×C3×#4112yes—
335C22×C3C24×D311yescharacteristic
336C22×C3C25×C311yescharacteristic
337C2×D3—26no—
338C2×D3—26no—
339C2×D3—224no—
340C2×D3C23×C3×C4112yes—
341C3×C4—26no—
342C3×C4—26no—
343C3×C4—212no—
344C3×C4—24no—
345C3×C4—34no—
346C3×C4—34no—
347C3×C4—46no—
348C3×C4—66no—
349C3×C4C22×C4×D3112yes—
350C3×C4C2×#23914yesdirect product factor
351C3×C4C23×C3×C4112yes—
352C3×C4C2×C3×#4114yescomplemented
353Dic3—26no—
354Dic3C23×C3×C414yes—
355Dic3C2×C3×#4114yescomplemented
356C32C2×C4×#4111yescharacteristic, complemented, Sylow
357C23—312no—
358C23—36no—
359C23—66no—
360C23—66no—
361C23C23×C3×D311yescharacteristic
362Q8C2×C3×C4×D3116yescomplemented
363D4—348no—
364C2×C4—26no—
365C2×C4—34no—
366C2×C4—34no—
367C2×C4—32no—
368C2×C4—32no—
369C2×C4—348no—
370C2×C4—36no—
371C2×C4—624no—
372C2×C4—66no—
373C2×C4—624no—
374C2×C4C2×C3×C4×D3112yes—
375C2×C4C3×#23914yesdirect product factor
376C2×C4C23×C3×D316yes—
377C2×C4C23×C3×D316yes—
378C2×C4C23×C3×D312yes—
379C2×C3—24no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—612no—
384C2×C3C4×#23914yesdirect product factor
385C2×C3C3×C4×#4114yescomplemented
386C2×C3C23×C4×D311yescharacteristic
387C2×C3C22×#23911yescharacteristic
388C2×C3C2×D3×#4111yescharacteristic
389C2×C3C24×C3×C411yescharacteristic
390C2×C3C22×C3×#4111yescharacteristic
391C2×C3C22×C3×#4111yescharacteristic
392D3—212no—
393C22—312no—
394C22—624no—
395C22—66no—
396C22—66no—
397C22C22×C3×C4×D312yes—
398C22C2×C3×#23912yes—
399C22C3×D3×#4112yes—
400C22C24×C3×D311yescharacteristic
401C4—26no—
402C4—34no—
403C4—34no—
404C4—66no—
405C4C22×C3×C4×D3112yes—
406C4C2×C3×#23914yesdirect product factor
407C3—21no—
408C3C2×C4×#23911yescharacteristic, direct product factor
409C3C2×C3×C4×#4111yescharacteristic, complemented
410C2—612no—
411C2C3×C4×#23914yesdirect product factor
412C2C23×C3×C4×D311yescharacteristic
413C2C22×C3×#23911yescharacteristic
414C2C2×C3×D3×#4111yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC2×C3C24×C3×C4
FrattiniC22C24×C3×D3
FittingC2×C32×C4×Q8C2
socleC23×C32C24
prime radicalC22×C3×#239C2
cosocleC22×C3C25×C3
hypercenterC2×C3×C4×Q8D3
hypocenterC3C2×C3×C4×#41
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4×Q8C2semidirect
3. C2×C3×Q8×Dic3C2semidirect
4. C3×C4×#239C2direct (N × H)
5. C2×C3×#893C2semidirect
6. C2×C3×#892C2semidirect
7. C2×C3×C4×D12C2semidirect
8. C2×C3×C42×D3C2semidirect
10. C2×C4×#239C3direct (N × H)
12. C2×C3×#239C4direct (N × H)
18. C32×C4×Q8C22semidirect
21. C3×Q8×Dic3C22semidirect
27. C3×#893C22semidirect
28. C3×#892C22semidirect
29. C3×C4×D12C22semidirect
30. C3×C42×D3C22semidirect
36. C2×C3×C4×Q8D3semidirect
37. C2×C3×C4×Q8C2×C3semidirect
42. C2×Q8×Dic3C2×C3semidirect
43. C4×#239C2×C3direct (N × H)
44. C2×#893C2×C3semidirect
45. C2×#892C2×C3semidirect
46. C2×C4×D12C2×C3semidirect
47. C2×C42×D3C2×C3semidirect
49. C2×C32×Q8C2×C4semidirect
57. C3×#239C2×C4direct (N × H)
58. C2×C3×D12C2×C4semidirect
62. C2×C3×C4×D3C2×C4semidirect
83. C2×#239C3×C4direct (N × H)
91. C3×C4×Q8C2×D3semidirect
92. C3×C4×Q8C22×C3semidirect
108. Q8×Dic3C22×C3semidirect
114. #893C22×C3semidirect
115. #892C22×C3semidirect
116. C4×D12C22×C3semidirect
117. C42×D3C22×C3semidirect
123. C32×Q8C22×C4semidirect
126. C2×C32×C4#41semidirect
132. C2×C3×Dic3#41semidirect
136. C3×D12C22×C4semidirect
139. C3×C4×D3C22×C4semidirect
143. C2×C4×Q8C3×D3semidirect
153. C2×C3×Q8C4×D3semidirect
154. C2×C3×Q8C2×C3×C4semidirect
180. #239C2×C3×C4direct (N × H)
181. C2×D12C2×C3×C4semidirect
185. C2×C4×D3C2×C3×C4semidirect
215. C32×C4C2×#41semidirect
218. C3×Dic3C2×#41semidirect
227. C4×Q8C2×C3×D3semidirect
251. C3×Q8C2×C4×D3semidirect
252. C3×Q8C22×C3×C4semidirect
272. C2×C3×C4#239direct (N × H)
274. C2×C3×C4C3×#41semidirect
283. C2×Dic3C3×#41semidirect
287. D12C22×C3×C4semidirect
290. C4×D3C22×C3×C4semidirect
291. C2×C32C4×#41semidirect
298. C2×Q8C3×C4×D3semidirect
350. C3×C4C2×#239direct (N × H)
352. C3×C4C2×C3×#41semidirect
355. Dic3C2×C3×#41semidirect
356. C32C2×C4×#41semidirect
362. Q8C2×C3×C4×D3semidirect
375. C2×C4C3×#239direct (N × H)
384. C2×C3C4×#239direct (N × H)
385. C2×C3C3×C4×#41semidirect
406. C4C2×C3×#239direct (N × H)
408. C3C2×C4×#239direct (N × H)
409. C3C2×C3×C4×#41semidirect
411. C2C3×C4×#239direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2392
C2×C32×C4×Q82
C2×C3×Q8×Dic32
C3×C4×#2392
C2×C3×#8932
C2×C3×#8922
C2×C3×C4×D122
C2×C3×C42×D32
C2×C3×C4×#413
C2×C4×#2393