Finite Group Databasebeta

← C2×C3×#4786 #11865428

Subgroups — C2×C3×#4786 #11865428

Order 1152.

1397 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#940C211yesmaximal, characteristic, complemented
2C22×C3×#213C211yesmaximal, characteristic, complemented
3C22×C3×D24C211yesmaximal, characteristic, complemented
4C2×C3×#893C211yesmaximal, characteristic, complemented
5C2×C32×#102C211yesmaximal, characteristic, complemented
6C2×C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C2×C3×#809C211yesmaximal, characteristic, complemented
8C3×#4786C218yesmaximal, direct product factor
9C2×C3×#459—31nomaximal
10C2×#4786C311yesmaximal, characteristic, direct product factor
11C22×C32×D4C2211yescharacteristic
12C22×C3×#67—21no—
13C22×C3×#67—21no—
14C22×C3×D12C2211yescharacteristic
15C22×C3×D12C2211yescharacteristic
16C22×C3×C4×D3—21no—
17C22×C32×C8C2211yescharacteristic
18C2×C32×C4⋊C4C2211yescharacteristic
19C3×#940—22no—
20C3×#940C2214yescomplemented
21C2×C3×#213—24no—
22C2×C3×#213C2214yescomplemented
23C2×C3×(C2×D3)⋊C4—21no—
24C2×C3×C4×Dic3C2211yescharacteristic
25C22×C3×C3⋊C8C2211yescharacteristic
26C2×C3×D24—24no—
27C2×C3×D24C2214yescomplemented
28C3×#893—22no—
29C3×#893C2214yescomplemented
30C32×#102C2214yescomplemented
31C3×Dic3⋊C8C2214yescomplemented
32C3×#809C2214yescomplemented
33C22×C3×D8—31no—
34C22×C3×D8—31no—
35C2×C3×#127—31no—
36C2×C3×C4×D4—31no—
37C2×#940C2×C311yescharacteristic, complemented
38C22×#213C2×C311yescharacteristic, complemented
39C22×D24C2×C311yescharacteristic, complemented
40C2×#893C2×C311yescharacteristic, complemented
41C3×#459—38no—
42C2×C3×C4⋊C8—31no—
43C2×C3×#102—21no—
44C2×C3×#102—31no—
45C2×C3×#102D311yescharacteristic, complemented
46C2×C3×#102C2×C311yescharacteristic, complemented
47C2×Dic3⋊C8C2×C311yescharacteristic, complemented
48C2×#809C2×C311yescharacteristic, complemented
49#4786C2×C318yesdirect product factor
50C24×C32—21no—
51C23×C3×D3—21no—
52C23×C3×D3—21no—
53C2×C32×D4—22no—
54C2×C32×D4—21no—
55C2×C32×D4—22no—
56C2×C32×D4C2312yes—
57C22×C32×C4—21no—
58C22×C32×C4C2311yescharacteristic
59C2×C3×#67—24no—
60C2×C3×#67—24no—
61C2×C3×#67—44no—
62C2×C3×#67—44no—
63C22×C3×Dic3D411yescharacteristic
64C22×C3×Dic3D411yescharacteristic
65C2×C3×D12—21no—
66C2×C3×D12—22no—
67C2×C3×D12—22no—
68C2×C3×D12—22no—
69C2×C3×D12—21no—
70C2×C3×D12D414yescomplemented
71C2×C3×D12C2312yes—
72C2×C3×D12C2312yes—
73C2×C3×C4×D3—24no—
74C2×C3×C4×D3—44no—
75C2×C32×C8—22no—
76C2×C32×C8C2312yes—
77C32×C4⋊C4—21no—
78C32×C4⋊C4C2312yes—
79C3×#213—28no—
80C3×(C2×D3)⋊C4—24no—
81C3×C4×Dic3D412yes—
82C3×C4×Dic3C2312yes—
83C2×C3×C3⋊C8—22no—
84C2×C3×C3⋊C8C2312yes—
85C3×D24—28no—
86C2×#459—31noSylow
87C22×C3×D4—21no—
88C22×C3×D4—31no—
89C22×C3×D4—31no—
90C22×C3×D4—61no—
91C22×C3×D4—61no—
92C22×C3×D4C2×D311yescharacteristic
93C22×C3×D4C22×C311yescharacteristic
94C23×C3×C4—61no—
95C22×#67—21no—
96C22×#67—21no—
97C22×D12C22×C311yescharacteristic
98C22×D12C22×C311yescharacteristic
99C22×C4×D3—21no—
100C2×C3×D8—34no—
101C2×C3×D8—34no—
102C2×C3×D8—64no—
103C2×C3×D8—64no—
104C22×C3×C8—21no—
105C22×C3×C8—31no—
106C22×C3×C8C2×D311yescharacteristic
107C22×C3×C8C22×C311yescharacteristic
108C3×#127—34no—
109C3×#127—62no—
110C3×C4×D4—34no—
111C3×C4×D4—62no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C4⋊C4C2×D311yescharacteristic
114C2×C3×C4⋊C4C22×C311yescharacteristic
115C2×C3×C22⋊C4—61no—
116C2×C3×C42—31no—
117#940—22no—
118#940C22×C314yescomplemented
119C2×#213—24no—
120C2×#213C22×C314yescomplemented
121C2×(C2×D3)⋊C4—21no—
122C2×C4×Dic3C22×C311yescharacteristic
123C22×C3⋊C8C22×C311yescharacteristic
124C2×D24—24no—
125C2×D24C22×C314yescomplemented
126#893—22no—
127#893C22×C314yescomplemented
128C3×C4⋊C8—34no—
129C3×#102—24no—
130C3×#102—34no—
131C3×#102C2×D314yescomplemented
132C3×#102C22×C314yescomplemented
133Dic3⋊C8C22×C314yescomplemented
134#809C22×C314yescomplemented
135C23×C32—22no—
136C23×C32—42no—
137C23×C32—42no—
138C23×C32—42no—
139C23×C32C2×D411yescharacteristic
140C22×C3×D3—24no—
141C22×C3×D3—22no—
142C22×C3×D3—22no—
143C22×C3×D3—42no—
144C22×C3×D3—42no—
145C22×C3×D3—42no—
146C22×C3×D3—42no—
147C22×C3×D3—42no—
148C32×D4—22no—
149C32×D4—22no—
150C32×D4—42no—
151C2×C32×C4—22no—
152C2×C32×C4—42no—
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4C2×D412yes—
155C2×C32×C4#4112yes—
156C2×C32×C4C2411yescharacteristic
157C3×#67—48no—
158C3×#67—48no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3D814yescomplemented
161C2×C3×Dic3C2×D412yes—
162C2×C3×Dic3C2×D412yes—
163C3×D12—24no—
164C3×D12—22no—
165C3×D12—22no—
166C3×D12—22no—
167C3×D12—42no—
168C3×D12C2×D414yescomplemented
169C3×C4×D3—48no—
170C32×C8—22no—
171C3×C3⋊C8—22no—
172C22×D8—31no—
173C22×D8—31no—
174C2×#127—31no—
175C2×C4×D4—31no—
176#459—38no—
177C2×C4⋊C8—31no—
178C2×#102—31no—
179C2×#102C3×D311yescharacteristic, complemented
180C24×C3—21no—
181C24×C3—21no—
182C24×C3—41no—
183C24×C3—61no—
184C24×C3—61no—
185C23×D3—21no—
186C23×D3—21no—
187C2×C3×D4—24no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—21no—
191C2×C3×D4—22no—
192C2×C3×D4—21no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—34no—
196C2×C3×D4—32no—
197C2×C3×D4—32no—
198C2×C3×D4—42no—
199C2×C3×D4—41no—
200C2×C3×D4—64no—
201C2×C3×D4—64no—
202C2×C3×D4—62no—
203C2×C3×D4—62no—
204C2×C3×D4—61no—
205C2×C3×D4—62no—
206C2×C3×D4—61no—
207C2×C3×D4—124no—
208C2×C3×D4—124no—
209C2×C3×D4C22×D312yes—
210C2×C3×D4C23×C312yes—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—31no—
215C22×C3×C4—31no—
216C22×C3×C4—41no—
217C22×C3×C4—64no—
218C22×C3×C4—124no—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C23×C311yescharacteristic
221C2×#67—24no—
222C2×#67—24no—
223C2×#67—44no—
224C2×#67—44no—
225C22×Dic3C3×D411yescharacteristic
226C22×Dic3C3×D411yescharacteristic
227C2×D12—21no—
228C2×D12—22no—
229C2×D12—22no—
230C2×D12—22no—
231C2×D12—21no—
232C2×D12C3×D414yescomplemented
233C2×D12C23×C312yes—
234C2×D12C23×C312yes—
235C2×C4×D3—24no—
236C2×C4×D3—44no—
237C3×D8—68no—
238C3×D8—68no—
239C2×C3×C8—22no—
240C2×C3×C8—22no—
241C2×C3×C8—22no—
242C2×C3×C8—32no—
243C2×C3×C8—42no—
244C2×C3×C8—62no—
245C2×C3×C8C22×D312yes—
246C2×C3×C8C23×C312yes—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—41no—
251C3×C4⋊C4C22×D312yes—
252C3×C4⋊C4C23×C312yes—
253C3×C22⋊C4—64no—
254C3×C42—32no—
255C3×C42—32no—
256#213—28no—
257(C2×D3)⋊C4—24no—
258C4×Dic3C3×D412yes—
259C4×Dic3C23×C312yes—
260C2×C3⋊C8—22no—
261C2×C3⋊C8C23×C312yes—
262D24—28no—
263C22×C32—42no—
264C22×C32—42no—
265C22×C32—42no—
266C22×C32—84no—
267C22×C32#12112yes—
268C22×C32C2×D812yes—
269C22×C32AGL(1,ℤ/8)12yes—
270C22×C32C22×D411yescharacteristic
271C2×C3×D3—24no—
272C2×C3×D3—42no—
273C2×C3×D3—42no—
274C2×C3×D3—48no—
275C2×C3×D3—42no—
276C2×C3×D3—42no—
277C2×C3×D3—42no—
278C2×C3×D3—84no—
279C32×C4—42no—
280C32×C4#12112yes—
281C32×C4C22×D411yescharacteristic
282C32×C4C2×#4111yescharacteristic
283C3×Dic3—22no—
284C3×Dic3C2×D814yescomplemented
285C22×D4—31no—
286C22×D4—31no—
287C22×D4—61no—
288C22×D4—61no—
289C22×D4C2×C3×D311yescharacteristic
290C23×C4—61no—
291C2×D8—34no—
292C2×D8—34no—
293C2×D8—64no—
294C2×D8—64no—
295C22×C8—31no—
296C22×C8C2×C3×D311yescharacteristic
297#127—34no—
298#127—62no—
299C4×D4—34no—
300C4×D4—62no—
301C2×C4⋊C4C2×C3×D311yescharacteristic
302C2×C22⋊C4—61no—
303C2×C42—31no—
304C4⋊C8—34no—
305#102—34no—
306#102C2×C3×D314yescomplemented
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—21no—
310C23×C3—42no—
311C23×C3—44no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—64no—
319C23×C3—62no—
320C23×C3—62no—
321C23×C3—82no—
322C23×C3—82no—
323C23×C3—122no—
324C23×C3—122no—
325C23×C3—122no—
326C23×C3—122no—
327C23×C3—122no—
328C23×C3D3×D411yescharacteristic
329C23×C3C2×C3×D411yescharacteristic
330C22×D3—24no—
331C22×D3—22no—
332C22×D3—22no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—42no—
338C3×D4—24no—
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4—34no—
344C3×D4—42no—
345C3×D4—44no—
346C3×D4—42no—
347C3×D4—42no—
348C3×D4—64no—
349C3×D4—62no—
350C3×D4—62no—
351C3×D4—62no—
352C3×D4—128no—
353C3×D4—128no—
354C3×D4—122no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—34no—
362C2×C3×C4—32no—
363C2×C3×C4—32no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—62no—
368C2×C3×C4—82no—
369C2×C3×C4—128no—
370C2×C3×C4#23512yes—
371C2×C3×C4D3×D411yescharacteristic
372C2×C3×C4D3×D412yes—
373C2×C3×C4C2×C3×D411yescharacteristic
374C2×C3×C4C2×C3×D412yes—
375C2×C3×C4C3×#4112yes—
376C2×C3×C4C23×D311yescharacteristic
377C2×C3×C4C24×C311yescharacteristic
378#67—48no—
379#67—48no—
380C2×Dic3—22no—
381C2×Dic3C3×D814yescomplemented
382C2×Dic3C2×C3×D412yes—
383C2×Dic3C2×C3×D412yes—
384D12—24no—
385D12—22no—
386D12—22no—
387D12—22no—
388D12—42no—
389D12C2×C3×D414yescomplemented
390C4×D3—48no—
391C3×C8—22no—
392C3×C8—22no—
393C3×C8—42no—
394C3×C8—62no—
395C3⋊C8—22no—
396C2×C32—82no—
397C2×C32#45914yescomplemented
398C2×C32C2×#12111yescharacteristic
399C2×C32C22×D811yescharacteristic
400C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
401C3×D3—44no—
402C3×D3—82no—
403C24—21no—
404C24—61no—
405C24—61no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—32no—
410C2×D4—32no—
411C2×D4—34no—
412C2×D4—64no—
413C2×D4—64no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—61no—
417C2×D4—62no—
418C2×D4—61no—
419C2×D4—124no—
420C2×D4—124no—
421C2×D4C22×C3×D312yes—
422C22×C4—21no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—64no—
426C22×C4—124no—
427C22×C4C22×C3×D311yescharacteristic
428D8—68no—
429D8—68no—
430C2×C8—22no—
431C2×C8—32no—
432C2×C8—62no—
433C2×C8C22×C3×D312yes—
434C4⋊C4—21no—
435C4⋊C4C22×C3×D312yes—
436C22⋊C4—64no—
437C42—32no—
438C42—32no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—21no—
443C22×C3—44no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—64no—
451C22×C3—82no—
452C22×C3—82no—
453C22×C3—88no—
454C22×C3—84no—
455C22×C3—84no—
456C22×C3—122no—
457C22×C3—122no—
458C22×C3—128no—
459C22×C3—122no—
460C22×C3—122no—
461C22×C3—122no—
462C22×C3—244no—
463C22×C3#88412yes—
464C22×C3D3×D812yes—
465C22×C3#91412yes—
466C22×C3C3×#12112yes—
467C22×C3C2×C3×D812yes—
468C22×C3C3×AGL(1,ℤ/8)12yes—
469C22×C3C2×D3×D411yescharacteristic
470C22×C3C22×C3×D411yescharacteristic
471C2×D3—24no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—48no—
475C2×D3—42no—
476C2×D3—42no—
477C2×D3—42no—
478C2×D3—84no—
479C3×C4—22no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—34no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—62no—
486C3×C4—82no—
487C3×C4#88412yes—
488C3×C4C3×#12112yes—
489C3×C4C2×#23511yescharacteristic
490C3×C4C2×D3×D411yescharacteristic
491C3×C4C22×C3×D411yescharacteristic
492C3×C4C2×C3×#4111yescharacteristic
493Dic3—22no—
494Dic3C2×C3×D814yescomplemented
495C32C2×#45911yescharacteristic, complemented, Sylow
496C23—22no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—64no—
501C23—62no—
502C23—62no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23C3×D3×D411yescharacteristic
509D4—22no—
510D4—22no—
511D4—34no—
512D4—42no—
513D4—64no—
514D4—62no—
515D4—62no—
516D4—62no—
517D4—128no—
518D4—128no—
519D4—122no—
520C2×C4—22no—
521C2×C4—34no—
522C2×C4—32no—
523C2×C4—32no—
524C2×C4—42no—
525C2×C4—62no—
526C2×C4—128no—
527C2×C4C3×#23512yes—
528C2×C4C3×D3×D411yescharacteristic
529C2×C4C3×D3×D412yes—
530C2×C4C23×C3×D311yescharacteristic
531C8—22no—
532C8—62no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—24no—
537C2×C3—84no—
538C2×C3—82no—
539C2×C3—82no—
540C2×C3—124no—
541C2×C3—242no—
542C2×C3#478614yesdirect product factor
543C2×C3C3×#45914yescomplemented
544C2×C3C2×#88411yescharacteristic
545C2×C3C2×D3×D811yescharacteristic
546C2×C3C2×#91411yescharacteristic
547C2×C3C2×C3×#12111yescharacteristic
548C2×C3C22×C3×D811yescharacteristic
549C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
550D3—44no—
551D3—82no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—64no—
556C22—84no—
557C22—122no—
558C22—122no—
559C22—128no—
560C22—122no—
561C22—122no—
562C22—122no—
563C22—244no—
564C22C3×#88412yes—
565C22C3×D3×D812yes—
566C22C3×#91412yes—
567C22C2×C3×D3×D411yescharacteristic
568C4—34no—
569C4—42no—
570C4—62no—
571C4C3×#88412yes—
572C4C2×C3×#23511yescharacteristic
573C4C2×C3×D3×D411yescharacteristic
574C3—21no—
575C3C2×#478611yescharacteristic, direct product factor
576C3C2×C3×#45911yescharacteristic, complemented
577C2—82no—
578C2—124no—
579C2—242no—
580C2C3×#478614yesdirect product factor
581C2C2×C3×#88411yescharacteristic
582C2C2×C3×D3×D811yescharacteristic
583C2C2×C3×#91411yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
hypercenterC2×C3×#102D3
hypocenterC3C2×C3×#459
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#102

Complements

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

NH ≅ G/Ntype
1. C2×C3×#940C2semidirect
2. C22×C3×#213C2semidirect
3. C22×C3×D24C2semidirect
4. C2×C3×#893C2semidirect
5. C2×C32×#102C2semidirect
6. C2×C3×Dic3⋊C8C2semidirect
7. C2×C3×#809C2semidirect
8. C3×#4786C2direct (N × H)
10. C2×#4786C3direct (N × H)
20. C3×#940C22semidirect
22. C2×C3×#213C22semidirect
27. C2×C3×D24C22semidirect
29. C3×#893C22semidirect
30. C32×#102C22semidirect
31. C3×Dic3⋊C8C22semidirect
32. C3×#809C22semidirect
37. C2×#940C2×C3semidirect
38. C22×#213C2×C3semidirect
39. C22×D24C2×C3semidirect
40. C2×#893C2×C3semidirect
45. C2×C3×#102D3semidirect
46. C2×C3×#102C2×C3semidirect
47. C2×Dic3⋊C8C2×C3semidirect
48. C2×#809C2×C3semidirect
49. #4786C2×C3direct (N × H)
70. C2×C3×D12D4semidirect
118. #940C22×C3semidirect
120. C2×#213C22×C3semidirect
125. C2×D24C22×C3semidirect
127. #893C22×C3semidirect
131. C3×#102C2×D3semidirect
132. C3×#102C22×C3semidirect
133. Dic3⋊C8C22×C3semidirect
134. #809C22×C3semidirect
160. C2×C3×Dic3D8semidirect
168. C3×D12C2×D4semidirect
179. C2×#102C3×D3semidirect
232. C2×D12C3×D4semidirect
284. C3×Dic3C2×D8semidirect
306. #102C2×C3×D3semidirect
381. C2×Dic3C3×D8semidirect
389. D12C2×C3×D4semidirect
397. C2×C32#459semidirect
494. Dic3C2×C3×D8semidirect
495. C32C2×#459semidirect
542. C2×C3#4786direct (N × H)
543. C2×C3C3×#459semidirect
575. C3C2×#4786direct (N × H)
576. C3C2×C3×#459semidirect
580. C2C3×#4786direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9402
C22×C3×#2132
C22×C3×D242
C2×C3×#8932
C2×C32×#1022
C2×C3×Dic3⋊C82
C2×C3×#8092
C3×#47862
C2×C3×#4593
C2×#47863