Finite Group Databasebeta

← C2×C3×#4747 #11864771

Subgroups — C2×C3×#4747 #11864771

Order 1152.

1781 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1004C211yesmaximal, characteristic, complemented
2C2×C3×#940C211yesmaximal, characteristic, complemented
3C2×C3×#908C212yesmaximal, complemented
4C2×C32×#100C211yesmaximal, characteristic, complemented
5C2×C3×#825C212yesmaximal, complemented
6C3×#4747C218yesmaximal, direct product factor
7C2×C3×#453—31nomaximal
8C2×#4747C311yesmaximal, characteristic, direct product factor
9C22×C32×D4C2211yescharacteristic
10C22×C3×#67—22no—
11C22×C3×#67—22no—
12C3×#1004—22no—
13C3×#1004—24no—
14C3×#1004C2214yescomplemented
15C2×C3×#237—22no—
16C2×C3×D3×D4—22no—
17C2×C3×#235C2212yes—
18C22×C3×D12C2211yescharacteristic, complemented
19C2×C32×M16C2212yescomplemented
20C3×#940—22no—
21C3×#940C2214yescomplemented
22C2×C3×C4×Dic3C2211yescharacteristic, complemented
23C3×#908—24no—
24C3×#908C2218yescomplemented
25C2×C3×D24—22no—
26C2×C3×#204—22no—
27C32×#100C2214yescomplemented
28C3×#825C2218yescomplemented
29C2×C3×H(5,𝔽2)—31no—
30C2×#1004C2×C311yescharacteristic, complemented
31C2×C3×AGL(1,ℤ/8)—32no—
32C2×C3×#127—31no—
33C2×#940C2×C311yescharacteristic, complemented
34C2×#908C2×C312yescomplemented
35C3×#453—38no—
36C2×C3×#104—32no—
37C2×C3×#100—21no—
38C2×C3×#100D311yescharacteristic, complemented
39C2×C3×#100C2×C311yescharacteristic, complemented
40C2×#825C2×C312yescomplemented
41#4747C2×C318yesdirect product factor
42C24×C32D412yes—
43C23×C3×D3—22no—
44C23×C3×D3—21no—
45C2×C32×D4—22no—
46C2×C32×D4—22no—
47C2×C32×D4D412yes—
48C2×C32×D4C2312yes—
49C22×C32×C4C2311yescharacteristic
50C2×C3×#67—24no—
51C2×C3×#67—28no—
52C2×C3×#67—22no—
53C2×C3×#67—28no—
54C2×C3×#67—28no—
55C2×C3×#67—48no—
56C22×C3×Dic3—22no—
57C22×C3×Dic3—21no—
58C3×#237—28no—
59C3×#237—28no—
60C3×D3×D4—28no—
61C3×D3×D4—28no—
62C3×#235—22no—
63C3×#235—22no—
64C3×#235D414yescomplemented
65C3×#235C2314yes—
66C2×C3×D12—22no—
67C2×C3×D12—22no—
68C2×C3×D12—21no—
69C2×C3×D12D412yescomplemented
70C2×C3×D12C2312yescomplemented
71C2×C3×C4×D3—22no—
72C2×C3×Dic6D412yescomplemented
73C32×M16—22no—
74C32×M16C2314yescomplemented
75C2×C32×C8—22no—
76C3×C4×Dic3—21no—
77C3×C4×Dic3C2312yescomplemented
78C3×D24—28no—
79C3×#204—28no—
80C2×#453—31noSylow
81C3×H(5,𝔽2)—34no—
82C3×H(5,𝔽2)—64no—
83C3×H(5,𝔽2)—62no—
84C2×C3×#41—32no—
85C2×C3×#41—62no—
86C22×C3×D4—21no—
87C22×C3×D4—31no—
88C22×C3×D4—62no—
89C22×C3×D4—62no—
90C22×C3×D4—62no—
91C22×C3×D4C2×D311yescharacteristic
92C22×C3×D4C22×C311yescharacteristic
93C22×#67—22no—
94C22×#67—22no—
95#1004—22no—
96#1004—24no—
97#1004C22×C314yescomplemented
98C2×#237—22no—
99C2×D3×D4—22no—
100C2×#235C22×C312yes—
101C22×D12C22×C311yescharacteristic, complemented
102C3×AGL(1,ℤ/8)—38no—
103C3×AGL(1,ℤ/8)—64no—
104C2×C3×SD16—62no—
105C2×C3×D8—62no—
106C2×C3×M16—22no—
107C2×C3×M16C2×D312yescomplemented
108C2×C3×M16C22×C312yescomplemented
109C3×#127—34no—
110C3×#127—62no—
111C2×C3×C42—31no—
112#940—22no—
113#940C22×C314yescomplemented
114C2×C4×Dic3C22×C311yescharacteristic, complemented
115#908—24no—
116#908C22×C318yescomplemented
117C2×D24—22no—
118C2×#204—22no—
119C3×#104—38no—
120C3×#100—24no—
121C3×#100C2×D314yescomplemented
122C3×#100C22×C314yescomplemented
123#825C22×C318yescomplemented
124C23×C32—22no—
125C23×C32—22no—
126C23×C32—44no—
127C23×C32C2×D411yescharacteristic
128C23×C32C2×D414yes—
129C22×C3×D3—22no—
130C22×C3×D3—28no—
131C22×C3×D3—22no—
132C22×C3×D3—44no—
133C22×C3×D3—41no—
134C22×C3×D3—41no—
135C22×C3×D3—84no—
136C32×D4—24no—
137C32×D4—24no—
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D412yes—
140C2×C32×C4C2×D412yes—
141C2×C32×C4C2411yescharacteristic
142C3×#67—216no—
143C3×#67—24no—
144C3×#67—24no—
145C3×#67—416no—
146C2×C3×Dic3—28no—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C3×D12—22no—
151C3×D12—24no—
152C3×D12—42no—
153C3×D12C2×D414yescomplemented
154C3×C4×D3—24no—
155C3×C4×D3—24no—
156C3×Dic6—22no—
157C3×Dic6C2×D414yescomplemented
158C32×C8—24no—
159C2×H(5,𝔽2)—31no—
160C2×AGL(1,ℤ/8)—32no—
161C2×#127—31no—
162#453—38no—
163C2×#104—32no—
164C2×#100C3×D311yescharacteristic, complemented
165C24×C3—22no—
166C24×C3—62no—
167C24×C3—61no—
168C24×C3D1212yes—
169C24×C3C3×D412yes—
170C23×D3—22no—
171C23×D3—21no—
172C3×#41—34no—
173C3×#41—34no—
174C3×#41—68no—
175C3×#41—68no—
176C3×#41—62no—
177C3×#41—62no—
178C2×C3×Q8—32no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—32no—
186C2×C3×D4—32no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—68no—
190C2×C3×D4—68no—
191C2×C3×D4—64no—
192C2×C3×D4—68no—
193C2×C3×D4—62no—
194C2×C3×D4—68no—
195C2×C3×D4—68no—
196C2×C3×D4—61no—
197C2×C3×D4—62no—
198C2×C3×D4—62no—
199C2×C3×D4—128no—
200C2×C3×D4D1212yes—
201C2×C3×D4C3×D412yes—
202C2×C3×D4C22×D312yes—
203C2×C3×D4C23×C312yes—
204C22×C3×C4—21no—
205C22×C3×C4—62no—
206C22×C3×C4—61no—
207C22×C3×C4—62no—
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C23×C311yescharacteristic
210C2×#67—24no—
211C2×#67—28no—
212C2×#67—22no—
213C2×#67—28no—
214C2×#67—28no—
215C2×#67—48no—
216C22×Dic3—22no—
217C22×Dic3—21no—
218#237—28no—
219#237—28no—
220D3×D4—28no—
221D3×D4—28no—
222#235—22no—
223#235—22no—
224#235C3×D414yescomplemented
225#235C23×C314yes—
226C2×D12—22no—
227C2×D12—22no—
228C2×D12—21no—
229C2×D12C3×D412yescomplemented
230C2×D12C23×C312yescomplemented
231C2×C4×D3—22no—
232C2×Dic6C3×D412yescomplemented
233C3×SD16—68no—
234C3×D8—68no—
235C3×M16—22no—
236C3×M16—22no—
237C3×M16—24no—
238C3×M16—42no—
239C3×M16C22×D314yescomplemented
240C3×M16C23×C314yescomplemented
241C2×C3×C8—22no—
242C2×C3×C8—22no—
243C2×C3×C8—42no—
244C3×C42—32no—
245C3×C42—61no—
246C4×Dic3—21no—
247C4×Dic3C23×C312yescomplemented
248D24—28no—
249#204—28no—
250C22×C32—22no—
251C22×C32—24no—
252C22×C32—44no—
253C22×C32—44no—
254C22×C32—82no—
255C22×C32#12011yescharacteristic
256C22×C32#12011yescharacteristic
257C22×C32C22×D411yescharacteristic
258C2×C3×D3—24no—
259C2×C3×D3—44no—
260C2×C3×D3—44no—
261C2×C3×D3—42no—
262C2×C3×D3—88no—
263C2×C3×D3—82no—
264C2×C3×D3—82no—
265C32×C4#12012yes—
266C32×C4C22×D412yes—
267C3×Dic3—24no—
268C3×Dic3—24no—
269H(5,𝔽2)—34no—
270H(5,𝔽2)—64no—
271H(5,𝔽2)—62no—
272C2×#41—32no—
273C2×#41—62no—
274C22×D4—31no—
275C22×D4—62no—
276C22×D4—62no—
277C22×D4—62no—
278C22×D4C2×C3×D311yescharacteristic
279AGL(1,ℤ/8)—38no—
280AGL(1,ℤ/8)—64no—
281C2×SD16—62no—
282C2×D8—62no—
283C2×M16C2×C3×D312yescomplemented
284#127—34no—
285#127—62no—
286C2×C42—31no—
287#104—38no—
288#100C2×C3×D314yescomplemented
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—21no—
294C23×C3—24no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—44no—
298C23×C3—44no—
299C23×C3—68no—
300C23×C3—62no—
301C23×C3—62no—
302C23×C3—84no—
303C23×C3—124no—
304C23×C3—121no—
305C23×C3—121no—
306C23×C3—244no—
307C23×C3C2×D1211yescharacteristic
308C23×C3C2×D1214yes—
309C23×C3C2×C3×D411yescharacteristic
310C23×C3C2×C3×D414yes—
311C22×D3—22no—
312C22×D3—28no—
313C22×D3—22no—
314C22×D3—44no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—84no—
318C3×Q8—34no—
319C3×Q8—62no—
320C3×D4—24no—
321C3×D4—24no—
322C3×D4—24no—
323C3×D4—24no—
324C3×D4—34no—
325C3×D4—44no—
326C3×D4—44no—
327C3×D4—616no—
328C3×D4—64no—
329C3×D4—64no—
330C3×D4—62no—
331C3×D4—64no—
332C3×D4—1216no—
333C3×D4—122no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—68no—
339C2×C3×C4—64no—
340C2×C3×C4—62no—
341C2×C3×C4—62no—
342C2×C3×C4—64no—
343C2×C3×C4—64no—
344C2×C3×C4C2×D1211yescharacteristic
345C2×C3×C4D3×D412yes—
346C2×C3×C4D3×D412yes—
347C2×C3×C4C2×C3×D411yescharacteristic
348C2×C3×C4C2×C3×D412yes—
349C2×C3×C4C2×C3×D412yes—
350C2×C3×C4C23×D311yescharacteristic
351C2×C3×C4C24×C311yescharacteristic
352#67—216no—
353#67—24no—
354#67—24no—
355#67—416no—
356C2×Dic3—28no—
357C2×Dic3—24no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360D12—22no—
361D12—24no—
362D12—42no—
363D12C2×C3×D414yescomplemented
364C4×D3—24no—
365C4×D3—24no—
366Dic6—22no—
367Dic6C2×C3×D414yescomplemented
368C3×C8—24no—
369C3×C8—24no—
370C3×C8—44no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—44no—
374C2×C32#45312yescomplemented
375C2×C32C2×#12011yescharacteristic
376C3×D3—44no—
377C3×D3—82no—
378C24—62no—
379C24—61no—
380C24C3×D1212yes—
381#41—34no—
382#41—34no—
383#41—68no—
384#41—68no—
385#41—62no—
386#41—62no—
387C2×Q8—32no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—32no—
391C2×D4—32no—
392C2×D4—68no—
393C2×D4—68no—
394C2×D4—64no—
395C2×D4—68no—
396C2×D4—62no—
397C2×D4—68no—
398C2×D4—68no—
399C2×D4—61no—
400C2×D4—62no—
401C2×D4—62no—
402C2×D4—128no—
403C2×D4C3×D1212yes—
404C2×D4C22×C3×D312yes—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4C22×C3×D311yescharacteristic
409SD16—68no—
410D8—68no—
411M16—22no—
412M16C22×C3×D314yescomplemented
413C2×C8—22no—
414C42—32no—
415C42—61no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—42no—
424C22×C3—44no—
425C22×C3—44no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3—64no—
430C22×C3—84no—
431C22×C3—84no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3—84no—
435C22×C3—124no—
436C22×C3—124no—
437C22×C3—122no—
438C22×C3—242no—
439C22×C3—248no—
440C22×C3—242no—
441C22×C3#88211yescharacteristic
442C22×C3#88211yescharacteristic
443C22×C3C3×#12011yescharacteristic
444C22×C3C3×#12011yescharacteristic
445C22×C3C22×D1211yescharacteristic
446C22×C3C22×C3×D411yescharacteristic
447C2×D3—24no—
448C2×D3—44no—
449C2×D3—44no—
450C2×D3—42no—
451C2×D3—88no—
452C2×D3—82no—
453C2×D3—82no—
454C3×C4—22no—
455C3×C4—22no—
456C3×C4—64no—
457C3×C4—64no—
458C3×C4#88212yes—
459C3×C4C3×#12012yes—
460C3×C4C2×D3×D412yes—
461C3×C4C22×C3×D412yes—
462Dic3—24no—
463Dic3—24no—
464C32C2×#45311yescharacteristic, complemented, Sylow
465C23—22no—
466C23—22no—
467C23—44no—
468C23—68no—
469C23—62no—
470C23—62no—
471C23—124no—
472C23—121no—
473C23—121no—
474C23—244no—
475C23C2×C3×D1211yescharacteristic
476C23C2×C3×D1214yes—
477Q8—34no—
478Q8—62no—
479D4—24no—
480D4—24no—
481D4—34no—
482D4—616no—
483D4—64no—
484D4—64no—
485D4—62no—
486D4—64no—
487D4—1216no—
488D4—122no—
489C2×C4—68no—
490C2×C4—64no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—64no—
494C2×C4—64no—
495C2×C4C2×C3×D1211yescharacteristic
496C2×C4C3×D3×D412yes—
497C2×C4C3×D3×D412yes—
498C2×C4C23×C3×D311yescharacteristic
499C8—24no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—22no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—44no—
509C2×C3—44no—
510C2×C3—84no—
511C2×C3—124no—
512C2×C3—242no—
513C2×C3#474712yesdirect product factor
514C2×C3C3×#45312yescomplemented
515C2×C3C2×#88211yescharacteristic
516C2×C3C2×C3×#12011yescharacteristic
517D3—44no—
518D3—82no—
519C22—22no—
520C22—24no—
521C22—44no—
522C22—44no—
523C22—64no—
524C22—82no—
525C22—124no—
526C22—124no—
527C22—122no—
528C22—242no—
529C22—248no—
530C22—242no—
531C22C3×#88211yescharacteristic
532C22C3×#88211yescharacteristic
533C22C22×C3×D1211yescharacteristic
534C4—64no—
535C4—64no—
536C4C3×#88212yes—
537C4C2×C3×D3×D412yes—
538C3—21no—
539C3C2×#474711yescharacteristic, direct product factor
540C3C2×C3×#45311yescharacteristic, complemented
541C2—21no—
542C2—21no—
543C2—44no—
544C2—124no—
545C2—242no—
546C2C3×#474712yesdirect product factor
547C2C2×C3×#88211yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#100C2
socleC22×C32#120
hypercenterC2×C3×#100D3
hypocenterC3C2×C3×#453
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#100

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1004C2semidirect
2. C2×C3×#940C2semidirect
3. C2×C3×#908C2semidirect
4. C2×C32×#100C2semidirect
5. C2×C3×#825C2semidirect
6. C3×#4747C2direct (N × H)
8. C2×#4747C3direct (N × H)
14. C3×#1004C22semidirect
18. C22×C3×D12C22semidirect
19. C2×C32×M16C22semidirect
21. C3×#940C22semidirect
22. C2×C3×C4×Dic3C22semidirect
24. C3×#908C22semidirect
27. C32×#100C22semidirect
28. C3×#825C22semidirect
30. C2×#1004C2×C3semidirect
33. C2×#940C2×C3semidirect
34. C2×#908C2×C3semidirect
38. C2×C3×#100D3semidirect
39. C2×C3×#100C2×C3semidirect
40. C2×#825C2×C3semidirect
41. #4747C2×C3direct (N × H)
64. C3×#235D4semidirect
69. C2×C3×D12D4semidirect
70. C2×C3×D12C23semidirect
72. C2×C3×Dic6D4semidirect
74. C32×M16C23semidirect
77. C3×C4×Dic3C23semidirect
97. #1004C22×C3semidirect
101. C22×D12C22×C3semidirect
107. C2×C3×M16C2×D3semidirect
108. C2×C3×M16C22×C3semidirect
113. #940C22×C3semidirect
114. C2×C4×Dic3C22×C3semidirect
116. #908C22×C3semidirect
121. C3×#100C2×D3semidirect
122. C3×#100C22×C3semidirect
123. #825C22×C3semidirect
153. C3×D12C2×D4semidirect
157. C3×Dic6C2×D4semidirect
164. C2×#100C3×D3semidirect
224. #235C3×D4semidirect
229. C2×D12C3×D4semidirect
230. C2×D12C23×C3semidirect
232. C2×Dic6C3×D4semidirect
239. C3×M16C22×D3semidirect
240. C3×M16C23×C3semidirect
247. C4×Dic3C23×C3semidirect
283. C2×M16C2×C3×D3semidirect
288. #100C2×C3×D3semidirect
363. D12C2×C3×D4semidirect
367. Dic6C2×C3×D4semidirect
374. C2×C32#453semidirect
412. M16C22×C3×D3semidirect
464. C32C2×#453semidirect
513. C2×C3#4747direct (N × H)
514. C2×C3C3×#453semidirect
539. C3C2×#4747direct (N × H)
540. C3C2×C3×#453semidirect
546. C2C3×#4747direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#10042
C2×C3×#9402
C2×C3×#9082
C2×C32×#1002
C2×C3×#8252
C3×#47472
C2×C3×#4533
C2×#47473