Finite Group Databasebeta

← C2×C3×#4866 #11869429

Subgroups — C2×C3×#4866 #11869429

Order 1152.

1001 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#895C211yesmaximal, characteristic
2C2×C3×#892C211yesmaximal, characteristic, complemented
3C2×C32×#106C211yesmaximal, characteristic, complemented
4C2×C3×#821C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6C2×C3×#809C211yesmaximal, characteristic, complemented
7C2×C3×#807C211yesmaximal, characteristic, complemented
8C3×#4866C218yesmaximal, direct product factor
9C2×C3×#482—31nomaximal
10C2×#4866C311yesmaximal, characteristic, direct product factor
11C22×C3×D12—21no—
12C22×C3×D12C2211yescharacteristic
13C22×C3×C4×D3C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C3×(C2×D3)⋊C4—21no—
18C2×C3×(C2×D3)⋊C4—21no—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×C4×Dic3—21no—
21C22×C3×C3⋊C8C2211yescharacteristic
22C3×#895—22no—
23C3×#895C2214yes—
24C3×#892—22no—
25C3×#892C2214yescomplemented
26C32×#106C2214yescomplemented
27C3×#821C2214yescomplemented
28C3×(C2×D3)⋊C8C2214yescomplemented
29C3×#809C2214yescomplemented
30C3×#807C2214yescomplemented
31C2×C3×#121—31no—
32C2×C3×#117—31no—
33C2×#895C2×C311yescharacteristic
34C2×#892C2×C311yescharacteristic, complemented
35C3×#482—38no—
36C2×C3×#107—31no—
37C2×C3×#106—21no—
38C2×C3×#106D311yescharacteristic, complemented
39C2×C3×#106C2×C311yescharacteristic, complemented
40C2×C3×#102—31no—
41C2×C3×#102—31no—
42C2×C3×C22⋊C8—31no—
43C2×#821C2×C311yescharacteristic, complemented
44C2×(C2×D3)⋊C8C2×C311yescharacteristic, complemented
45C2×#809C2×C311yescharacteristic, complemented
46C2×#807C2×C311yescharacteristic, complemented
47#4866C2×C318yesdirect product factor
48C23×C3×D3—21no—
49C23×C3×D3D411yescharacteristic
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4C2311yescharacteristic
53C22×C3×Dic3—21no—
54C22×C3×Dic3D411yescharacteristic
55C2×C3×D12—22no—
56C2×C3×D12—22no—
57C2×C3×D12—24no—
58C2×C3×D12—21no—
59C2×C3×D12—44no—
60C2×C3×D12C2312yes—
61C2×C3×C4×D3—22no—
62C2×C3×C4×D3—22no—
63C2×C3×C4×D3D412yes—
64C2×C3×C4×D3C2312yes—
65C2×C32×C8—22no—
66C2×C32×C8C2312yes—
67C32×C4⋊C4—21no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4C2312yes—
70C32×C4⋊C4C2312yes—
71C3×(C2×D3)⋊C4—24no—
72C3×(C2×D3)⋊C4—24no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3C2312yes—
75C3×C4×Dic3—24no—
76C2×C3×C3⋊C8—22no—
77C2×C3×C3⋊C8C2312yes—
78C2×#482—31noSylow
79C22×C3×D4—31no—
80C22×C3×D4—61no—
81C23×C3×C4—31no—
82C22×D12—21no—
83C22×D12C22×C311yescharacteristic
84C22×C4×D3C22×C311yescharacteristic
85C22×C3×C8—21no—
86C22×C3×C8—31no—
87C22×C3×C8C2×D311yescharacteristic
88C22×C3×C8C22×C311yescharacteristic
89C3×#121—34no—
90C3×#121—62no—
91C3×#117—34no—
92C3×#117—62no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—31no—
96C2×C3×C4⋊C4C2×D311yescharacteristic
97C2×C3×C4⋊C4C2×D311yescharacteristic
98C2×C3×C4⋊C4C22×C311yescharacteristic
99C2×C3×C4⋊C4C22×C311yescharacteristic
100C2×C3×C22⋊C4—61no—
101C2×C3×C22⋊C4—61no—
102C2×C3×C42—61no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3C22×C311yescharacteristic
106C2×C4×Dic3—21no—
107C22×C3⋊C8C22×C311yescharacteristic
108#895—22no—
109#895C22×C314yes—
110#892—22no—
111#892C22×C314yescomplemented
112C3×#107—34no—
113C3×#106—24no—
114C3×#106C2×D314yescomplemented
115C3×#106C22×C314yescomplemented
116C3×#102—34no—
117C3×#102—34no—
118C3×C22⋊C8—34no—
119#821C22×C314yescomplemented
120(C2×D3)⋊C8C22×C314yescomplemented
121#809C22×C314yescomplemented
122#807C22×C314yescomplemented
123C23×C32C2×D411yescharacteristic
124C22×C3×D3—22no—
125C22×C3×D3—22no—
126C22×C3×D3—22no—
127C22×C3×D3—22no—
128C22×C3×D3—42no—
129C22×C3×D3—42no—
130C22×C3×D3—42no—
131C22×C3×D3C2×D412yes—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—42no—
135C2×C32×C4—42no—
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4#4112yes—
138C2×C32×C4#4112yes—
139C2×C32×C4C2411yescharacteristic
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—42no—
143C2×C3×Dic3C2×D412yes—
144C3×D12—22no—
145C3×D12—22no—
146C3×D12—48no—
147C3×D12—42no—
148C3×C4×D3—24no—
149C3×C4×D3—22no—
150C3×C4×D3—22no—
151C32×C8—22no—
152C3×C3⋊C8—22no—
153C2×#121—31no—
154C2×#117—31no—
155#482—38no—
156C2×#107—31no—
157C2×#106C3×D311yescharacteristic, complemented
158C2×#102—31no—
159C2×#102—31no—
160C2×C22⋊C8—31no—
161C24×C3—31no—
162C24×C3—61no—
163C23×D3—21no—
164C23×D3C3×D411yescharacteristic
165C2×C3×D4—32no—
166C2×C3×D4—64no—
167C2×C3×D4—62no—
168C2×C3×D4—61no—
169C2×C3×D4—62no—
170C2×C3×D4—124no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—31no—
177C22×C3×C4—32no—
178C22×C3×C4—32no—
179C22×C3×C4—41no—
180C22×C3×C4—41no—
181C22×C3×C4—62no—
182C22×C3×C4—62no—
183C22×C3×C4—61no—
184C22×C3×C4C22×D311yescharacteristic
185C22×C3×C4C23×C311yescharacteristic
186C22×Dic3—21no—
187C22×Dic3C3×D411yescharacteristic
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—24no—
191C2×D12—21no—
192C2×D12—44no—
193C2×D12C23×C312yes—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3C3×D412yes—
197C2×C4×D3C23×C312yes—
198C2×C3×C8—24no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—32no—
203C2×C3×C8—62no—
204C2×C3×C8C22×D312yes—
205C2×C3×C8C23×C312yes—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—32no—
213C3×C4⋊C4—41no—
214C3×C4⋊C4—41no—
215C3×C4⋊C4—61no—
216C3×C4⋊C4C22×D312yes—
217C3×C4⋊C4C22×D312yes—
218C3×C4⋊C4C23×C312yes—
219C3×C4⋊C4C23×C312yes—
220C3×C22⋊C4—64no—
221C3×C22⋊C4—64no—
222C3×C42—64no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—24no—
225C4⋊Dic3—21no—
226C4⋊Dic3C23×C312yes—
227C4×Dic3—24no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8C23×C312yes—
230C22×C32#12312yes—
231C22×C32#13512yes—
232C22×C32AGL(1,ℤ/8)12yes—
233C22×C32C22×D411yescharacteristic
234C2×C3×D3—22no—
235C2×C3×D3—22no—
236C2×C3×D3—22no—
237C2×C3×D3—44no—
238C2×C3×D3—42no—
239C2×C3×D3—42no—
240C2×C3×D3—42no—
241C2×C3×D3—84no—
242C32×C4—42no—
243C32×C4—42no—
244C32×C4#12312yes—
245C32×C4C2×#4111yescharacteristic
246C32×C4C2×#4111yescharacteristic
247C3×Dic3—22no—
248C3×Dic3—42no—
249C22×D4—31no—
250C22×D4—61no—
251C23×C4—31no—
252C22×C8—31no—
253C22×C8C2×C3×D311yescharacteristic
254#121—34no—
255#121—62no—
256#117—34no—
257#117—62no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4C2×C3×D311yescharacteristic
260C2×C4⋊C4C2×C3×D311yescharacteristic
261C2×C22⋊C4—61no—
262C2×C22⋊C4—61no—
263C2×C42—61no—
264#107—34no—
265#106C2×C3×D314yescomplemented
266#102—34no—
267#102—34no—
268C22⋊C8—34no—
269C23×C3—21no—
270C23×C3—32no—
271C23×C3—62no—
272C23×C3—62no—
273C23×C3—62no—
274C23×C3—62no—
275C23×C3—122no—
276C23×C3—122no—
277C23×C3—122no—
278C23×C3D3×D411yescharacteristic
279C23×C3C2×C3×D411yescharacteristic
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3C2×C3×D412yes—
288C3×D4—62no—
289C3×D4—62no—
290C3×D4—128no—
291C3×D4—122no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—21no—
297C2×C3×C4—22no—
298C2×C3×C4—21no—
299C2×C3×C4—22no—
300C2×C3×C4—32no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—62no—
308C2×C3×C4—64no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—82no—
313C2×C3×C4—82no—
314C2×C3×C4—122no—
315C2×C3×C4#23512yes—
316C2×C3×C4D3×D411yescharacteristic
317C2×C3×C4#23912yes—
318C2×C3×C4C2×C3×D411yescharacteristic
319C2×C3×C4C3×#4112yes—
320C2×C3×C4C3×#4112yes—
321C2×C3×C4C23×D311yescharacteristic
322C2×C3×C4C24×C311yescharacteristic
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—42no—
326C2×Dic3C2×C3×D412yes—
327D12—22no—
328D12—22no—
329D12—48no—
330D12—42no—
331C4×D3—24no—
332C4×D3—22no—
333C4×D3—22no—
334C3×C8—24no—
335C3×C8—22no—
336C3×C8—22no—
337C3×C8—62no—
338C3⋊C8—22no—
339C2×C32#48214yescomplemented
340C2×C32C2×#12311yescharacteristic
341C2×C32C2×#13511yescharacteristic
342C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
343C3×D3—42no—
344C3×D3—82no—
345C24—31no—
346C24—61no—
347C2×D4—32no—
348C2×D4—64no—
349C2×D4—62no—
350C2×D4—61no—
351C2×D4—62no—
352C2×D4—124no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—31no—
356C22×C4—32no—
357C22×C4—32no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—61no—
361C22×C4C22×C3×D311yescharacteristic
362C2×C8—22no—
363C2×C8—32no—
364C2×C8—62no—
365C2×C8C22×C3×D312yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—32no—
369C4⋊C4—61no—
370C4⋊C4C22×C3×D312yes—
371C4⋊C4C22×C3×D312yes—
372C22⋊C4—64no—
373C22⋊C4—64no—
374C42—64no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—21no—
379C22×C3—62no—
380C22×C3—62no—
381C22×C3—62no—
382C22×C3—124no—
383C22×C3—122no—
384C22×C3—122no—
385C22×C3—122no—
386C22×C3—244no—
387C22×C3#89412yes—
388C22×C3#91412yes—
389C22×C3#91612yes—
390C22×C3C3×#12312yes—
391C22×C3C3×#13512yes—
392C22×C3C3×AGL(1,ℤ/8)12yes—
393C22×C3C2×D3×D411yescharacteristic
394C22×C3C22×C3×D411yescharacteristic
395C2×D3—22no—
396C2×D3—22no—
397C2×D3—22no—
398C2×D3—44no—
399C2×D3—42no—
400C2×D3—42no—
401C2×D3—42no—
402C2×D3—84no—
403C3×C4—22no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—62no—
411C3×C4—82no—
412C3×C4—82no—
413C3×C4—122no—
414C3×C4#89412yes—
415C3×C4C3×#12312yes—
416C3×C4C2×#23511yescharacteristic
417C3×C4C2×#23911yescharacteristic
418C3×C4C2×C3×#4111yescharacteristic
419C3×C4C2×C3×#4111yescharacteristic
420Dic3—22no—
421Dic3—42no—
422C32C2×#48211yescharacteristic, complemented, Sylow
423C23—32no—
424C23—62no—
425C23—62no—
426C23—62no—
427C23—62no—
428C23—122no—
429C23—122no—
430C23—122no—
431C23C3×D3×D411yescharacteristic
432D4—62no—
433D4—62no—
434D4—128no—
435D4—122no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—32no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—62no—
442C2×C4—64no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—122no—
447C2×C4C3×#23512yes—
448C2×C4C3×D3×D411yescharacteristic
449C2×C4C3×#23912yes—
450C2×C4C23×C3×D311yescharacteristic
451C8—22no—
452C8—62no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—24no—
457C2×C3—122no—
458C2×C3—242no—
459C2×C3#486614yesdirect product factor
460C2×C3C3×#48214yescomplemented
461C2×C3C2×#89411yescharacteristic
462C2×C3C2×#91411yescharacteristic
463C2×C3C2×#91611yescharacteristic
464C2×C3C2×C3×#12311yescharacteristic
465C2×C3C2×C3×#13511yescharacteristic
466C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
467D3—42no—
468D3—82no—
469C22—62no—
470C22—62no—
471C22—62no—
472C22—124no—
473C22—122no—
474C22—122no—
475C22—122no—
476C22—244no—
477C22C3×#89412yes—
478C22C3×#91412yes—
479C22C3×#91612yes—
480C22C2×C3×D3×D411yescharacteristic
481C4—42no—
482C4—42no—
483C4—62no—
484C4—122no—
485C4C3×#89412yes—
486C4C2×C3×#23511yescharacteristic
487C4C2×C3×#23911yescharacteristic
488C3—21no—
489C3C2×#486611yescharacteristic, direct product factor
490C3C2×C3×#48211yescharacteristic, complemented
491C2—122no—
492C2—242no—
493C2C3×#486614yesdirect product factor
494C2C2×C3×#89411yescharacteristic
495C2C2×C3×#91411yescharacteristic
496C2C2×C3×#91611yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#106C2
socleC23×C32C2×D4
prime radicalC2×C3×#895C2
hypercenterC2×C3×#106D3
hypocenterC3C2×C3×#482
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#106

Complements

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

NH ≅ G/Ntype
2. C2×C3×#892C2semidirect
3. C2×C32×#106C2semidirect
4. C2×C3×#821C2semidirect
5. C2×C3×(C2×D3)⋊C8C2semidirect
6. C2×C3×#809C2semidirect
7. C2×C3×#807C2semidirect
8. C3×#4866C2direct (N × H)
10. C2×#4866C3direct (N × H)
25. C3×#892C22semidirect
26. C32×#106C22semidirect
27. C3×#821C22semidirect
28. C3×(C2×D3)⋊C8C22semidirect
29. C3×#809C22semidirect
30. C3×#807C22semidirect
34. C2×#892C2×C3semidirect
38. C2×C3×#106D3semidirect
39. C2×C3×#106C2×C3semidirect
43. C2×#821C2×C3semidirect
44. C2×(C2×D3)⋊C8C2×C3semidirect
45. C2×#809C2×C3semidirect
46. C2×#807C2×C3semidirect
47. #4866C2×C3direct (N × H)
111. #892C22×C3semidirect
114. C3×#106C2×D3semidirect
115. C3×#106C22×C3semidirect
119. #821C22×C3semidirect
120. (C2×D3)⋊C8C22×C3semidirect
121. #809C22×C3semidirect
122. #807C22×C3semidirect
157. C2×#106C3×D3semidirect
265. #106C2×C3×D3semidirect
339. C2×C32#482semidirect
422. C32C2×#482semidirect
459. C2×C3#4866direct (N × H)
460. C2×C3C3×#482semidirect
489. C3C2×#4866direct (N × H)
490. C3C2×C3×#482semidirect
493. C2C3×#4866direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#8952
C2×C3×#8922
C2×C32×#1062
C2×C3×#8212
C2×C3×(C2×D3)⋊C82
C2×C3×#8092
C2×C3×#8072
C3×#48662
C2×C3×#4823
C2×#48663