Finite Group Databasebeta

← C2×D3×#489 #11664393

Subgroups — C2×D3×#489 #11664393

Order 768.

2486 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×D3×#124C211yesmaximal, characteristic
3C2×C3×#489C211yesmaximal, characteristic, complemented
4C2×#5089C211yesmaximal, characteristic, complemented
5C2×#5055C211yesmaximal, characteristic, complemented
6D3×#489C2116yesmaximal, direct product factor
7C2×D3×#103C212yesmaximal, complemented
8C2×#4794C212yesmaximal, complemented
9C2×D3×#102C212yesmaximal
10C2×#4765C212yesmaximal, complemented
11C2×#4714C211yesmaximal, characteristic, complemented
12C2×D3×#97C211yesmaximal, characteristic, complemented
13C22×#489—31nomaximal, Sylow
14C22×D3×Q8—21no—
15C22×D3×Q8C2211yescharacteristic
16C22×D3×D4C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic, complemented
18C2×C3×#124C2211yescharacteristic
19C2×#947C2211yescharacteristic
20C2×#944—21no—
21C2×#936C2211yescharacteristic
22C22×D3×C8C2212yes—
23D3×C4⋊Q8—24no—
24D3×C4⋊Q8C2218yescomplemented
25D3×#124—24no—
26D3×#124C2218yes—
27C2×D3×C4⋊C4—21no—
28C2×D3×C4⋊C4C2212yes—
29C2×#888—21no—
30C2×#888C2212yescomplemented
31C2×#885—21no—
32C2×#885—21no—
33C2×D3×C22⋊C4—21no—
34C2×D3×C22⋊C4—21no—
35C2×#875C2211yescharacteristic
36C2×C42×D3C2211yescharacteristic
37C2×#869C2211yescharacteristic, complemented
38C3×#489C2218yescomplemented
39C2×C3×#103C2212yescomplemented
40C2×C3×#102C2212yes—
41C2×C3×#97C2211yescharacteristic, complemented
42C2×#835C2212yescomplemented
43C2×#832C2212yes—
44C2×#821C2212yes—
45C2×#816C2212yescomplemented
46C2×#815C2212yescomplemented
47#5089C2218yescomplemented
48#5055C2218yescomplemented
49C2×#810C2212yescomplemented
50C2×#809C2212yes—
51C2×#803C2211yescharacteristic, complemented
52D3×#103C22116yescomplemented
53#4794C22116yescomplemented
54D3×#102C22116yes—
55#4765C22116yescomplemented
56#4714C2218yescomplemented
57D3×#97C2218yescomplemented
58C22×C4⋊Q8—31no—
59C22×#124—31no—
60C2×#489—316no—
61C2×#489—32no—
62C2×#489—32no—
63C2×#489—31no—
64C2×#489—31no—
65C2×#489—31no—
66C2×#489D311yescharacteristic, direct product factor
67C22×#103—32no—
68C22×#102—32no—
69C22×#97—31no—
70C24×D3—21no—
71C22×C3×Q8—21no—
72C22×C3×Q8C2311yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C22×#67—21no—
75C2×D3×Q8—22no—
76C2×D3×Q8—28no—
77C2×D3×Q8—28no—
78C2×D3×Q8—48no—
79C2×D3×Q8C2314yes—
80C2×D3×D4—22no—
81C2×D3×D4—28no—
82C2×D3×D4C2314yes—
83C22×D12C2311yescharacteristic
84C22×C4×D3—22no—
85C22×C4×D3—21no—
86C22×C4×D3D411yescharacteristic
87C22×C4×D3D411yescharacteristic
88C22×C4×D3C2311yescharacteristic
89C22×Dic6—21no—
90C22×Dic6—22no—
91C22×Dic6—21no—
92C22×Dic6C2311yescharacteristic
93C22×C3×C8C2312yes—
94C3×C4⋊Q8—22no—
95C3×C4⋊Q8C2314yescomplemented
96C3×#124—22no—
97C3×#124C2314yes—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C42C2311yescharacteristic
103C2×#217—21no—
104C2×#217—21no—
105#947—22no—
106#947C2314yes—
107#944—28no—
108#936—22no—
109#936C2314yes—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3C2312yes—
116C2×Dic3⋊C4—22no—
117C2×Dic3⋊C4—22no—
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2312yes—
122C2×D3×C8—28no—
123C2×D3×C8C2318yes—
124D3×C4⋊C4—24no—
125D3×C4⋊C4—28no—
126D3×C4⋊C4C2318yes—
127#888—24no—
128#888—28no—
129#888C2318yescomplemented
130#885—28no—
131#885—28no—
132D3×C22⋊C4—28no—
133D3×C22⋊C4—28no—
134#875—22no—
135#875C2314yes—
136C42×D3D414yes—
137C42×D3C2314yes—
138#869—22no—
139#869C2314yescomplemented
140C3×#103C2318yescomplemented
141C3×#102C2318yes—
142C3×#97C2314yescomplemented
143#835C2318yescomplemented
144#832C2318yes—
145#821C2318yes—
146#816C2318yescomplemented
147#815C2318yescomplemented
148#810C2318yescomplemented
149#809C2318yes—
150#803C2314yescomplemented
151C23×Q8—31no—
152C23×Q8—61no—
153C23×D4—31no—
154C23×C8—32no—
155C2×C4⋊Q8—38no—
156C2×C4⋊Q8—32no—
157C2×C4⋊Q8—31no—
158C2×C4⋊Q8—61no—
159C2×C4⋊Q8—64no—
160C2×C4⋊Q8—61no—
161C2×C4⋊Q8C2×D311yescharacteristic, complemented
162C2×#124—31no—
163C2×#124—31no—
164C2×#124—38no—
165C2×#124—31no—
166C2×#124—61no—
167C2×#124—61no—
168C2×#124—64no—
169C2×#124C2×D311yescharacteristic
170C22×C4⋊C4—32no—
171C22×C4⋊C4—61no—
172C22×C22⋊C4—61no—
173C22×C22⋊C4—61no—
174C22×C42—31no—
175#489—38no—
176#489—38no—
177#489—316no—
178#489—316no—
179#489—38no—
180#489C2×D318yesdirect product factor
181C2×#103—32no—
182C2×#103—32no—
183C2×#103—32no—
184C2×#103—316no—
185C2×#103C2×D312yescomplemented
186C2×#102—32no—
187C2×#102—32no—
188C2×#102—32no—
189C2×#102—316no—
190C2×#102C2×D312yes—
191C2×#97—32no—
192C2×#97—31no—
193C2×#97—38no—
194C2×#97C2×D311yescharacteristic, complemented
195C24×C3—21no—
196C23×D3—21no—
197C23×D3—24no—
198C23×D3—44no—
199C23×D3—44no—
200C23×D3—44no—
201C23×D3C2×D411yescharacteristic
202C2×C3×Q8—24no—
203C2×C3×Q8—21no—
204C2×C3×Q8—24no—
205C2×C3×Q8—44no—
206C2×C3×Q8C2412yes—
207C2×C3×D4—21no—
208C2×C3×D4—24no—
209C2×C3×D4C2412yes—
210C22×C3×C4—22no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C2×#67—24no—
216C2×#67—44no—
217C22×Dic3—22no—
218C22×Dic3—21no—
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222D3×Q8—216no—
223D3×Q8—432no—
224D3×Q8—48no—
225D3×D4—216no—
226D3×D4—48no—
227C2×D12—24no—
228C2×D12—21no—
229C2×D12C2412yes—
230C2×C4×D3—24no—
231C2×C4×D3—24no—
232C2×C4×D3—24no—
233C2×C4×D3—28no—
234C2×C4×D3—44no—
235C2×C4×D3—48no—
236C2×C4×D3C2×D414yes—
237C2×C4×D3C2×D414yes—
238C2×C4×D3C2×D412yes—
239C2×C4×D3#4118yes—
240C2×C4×D3C2412yes—
241C2×Dic6—24no—
242C2×Dic6—24no—
243C2×Dic6—21no—
244C2×Dic6—24no—
245C2×Dic6—28no—
246C2×Dic6—48no—
247C2×Dic6—44no—
248C2×Dic6—44no—
249C2×Dic6C2412yes—
250C2×C3×C8—24no—
251C2×C3×C8C2414yes—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—22no—
254C3×C4⋊C4C2414yes—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—24no—
257C3×C42C2×D412yes—
258C3×C42C2412yes—
259#217—24no—
260#217—24no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—24no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—24no—
265C4⋊Dic3—22no—
266C4⋊Dic3—24no—
267C4⋊Dic3C2414yes—
268Dic3⋊C4—28no—
269Dic3⋊C4—28no—
270C4×Dic3C2×D412yes—
271C4×Dic3C2×D414yes—
272C4×Dic3C2×D414yes—
273C4×Dic3C2412yes—
274C2×C3⋊C8—24no—
275C2×C3⋊C8C2414yes—
276D3×C8—216no—
277C25—61no—
278C22×Q8—21no—
279C22×Q8—34no—
280C22×Q8—31no—
281C22×Q8—61no—
282C22×Q8—62no—
283C22×Q8—62no—
284C22×Q8—68no—
285C22×Q8—68no—
286C22×Q8—61no—
287C22×Q8—128no—
288C22×Q8C22×D311yescharacteristic
289C22×D4—34no—
290C22×D4—31no—
291C22×D4—68no—
292C22×D4—62no—
293C22×D4—61no—
294C22×D4C22×D311yescharacteristic
295C23×C4—31no—
296C23×C4—31no—
297C23×C4—31no—
298C23×C4—61no—
299C23×C4—62no—
300C22×C8—32no—
301C22×C8—38no—
302C22×C8—68no—
303C22×C8C22×D312yes—
304C4⋊Q8—22no—
305C4⋊Q8—38no—
306C4⋊Q8—34no—
307C4⋊Q8—68no—
308C4⋊Q8—64no—
309C4⋊Q8—68no—
310C4⋊Q8—62no—
311C4⋊Q8C22×D314yescomplemented
312#124—22no—
313#124—34no—
314#124—34no—
315#124—34no—
316#124—68no—
317#124—68no—
318#124—62no—
319#124—62no—
320#124—62no—
321#124C22×D314yes—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—38no—
325C2×C4⋊C4—64no—
326C2×C4⋊C4—68no—
327C2×C4⋊C4—62no—
328C2×C4⋊C4—62no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4C22×D312yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—68no—
334C2×C22⋊C4—68no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—31no—
344C2×C42—34no—
345C2×C42—34no—
346C2×C42C22×D311yescharacteristic
347#103—38no—
348#103—38no—
349#103—38no—
350#103C22×D318yescomplemented
351#102—38no—
352#102—38no—
353#102—38no—
354#102C22×D318yes—
355#97—38no—
356#97—34no—
357#97C22×D314yescomplemented
358C23×C3—22no—
359C23×C3—42no—
360C23×C3—42no—
361C23×C3—42no—
362C23×C3C22×D411yescharacteristic
363C22×D3—22no—
364C22×D3—42no—
365C22×D3—42no—
366C22×D3—42no—
367C22×D3—48no—
368C22×D3—48no—
369C22×D3—48no—
370C22×D3—816no—
371C22×D3#12414yes—
372C22×D3AGL(1,ℤ/8)14yes—
373C22×D3#13714yes—
374C22×D3C22×D412yes—
375C3×Q8—24no—
376C3×Q8—48no—
377C3×Q8—42no—
378C3×D4—24no—
379C3×D4—42no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—24no—
383C2×C3×C4—22no—
384C2×C3×C4—42no—
385C2×C3×C4—44no—
386C2×C3×C4C22×D412yes—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4114yes—
390C2×C3×C4C2511yescharacteristic
391#67—48no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—24no—
395C2×Dic3—22no—
396C2×Dic3—42no—
397C2×Dic3—44no—
398C2×Dic3#12414yes—
399C2×Dic3C22×D412yes—
400C2×Dic3C22×D412yes—
401C2×Dic3C22×D412yes—
402D12—24no—
403D12—42no—
404C4×D3—28no—
405C4×D3—28no—
406C4×D3—48no—
407C4×D3—416no—
408C4×D3#12418yes—
409C4×D3C2×#4118yes—
410Dic6—24no—
411Dic6—416no—
412Dic6—48no—
413Dic6—42no—
414Dic6—48no—
415C3×C8—24no—
416C3⋊C8—24no—
417C24—21no—
418C24—31no—
419C24—64no—
420C24—61no—
421C24—124no—
422C24—124no—
423C24—124no—
424C2×Q8—21no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—32no—
428C2×Q8—44no—
429C2×Q8—64no—
430C2×Q8—68no—
431C2×Q8—616no—
432C2×Q8—61no—
433C2×Q8—64no—
434C2×Q8—64no—
435C2×Q8—124no—
436C2×Q8—124no—
437C2×Q8—128no—
438C2×Q8—128no—
439C2×Q8—1232no—
440C2×Q8C23×D312yes—
441C2×D4—21no—
442C2×D4—24no—
443C2×D4—32no—
444C2×D4—616no—
445C2×D4—64no—
446C2×D4—61no—
447C2×D4—64no—
448C2×D4—128no—
449C2×D4—124no—
450C2×D4C23×D312yes—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—31no—
454C22×C4—32no—
455C22×C4—38no—
456C22×C4—32no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—34no—
460C22×C4—34no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4—64no—
464C22×C4—68no—
465C22×C4—61no—
466C22×C4—62no—
467C22×C4—124no—
468C22×C4—128no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472C2×C8—24no—
473C2×C8—34no—
474C2×C8—616no—
475C2×C8—64no—
476C2×C8C23×D314yes—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—34no—
480C4⋊C4—68no—
481C4⋊C4—68no—
482C4⋊C4—62no—
483C4⋊C4—64no—
484C4⋊C4C23×D314yes—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—64no—
488C22⋊C4—64no—
489C22⋊C4—64no—
490C22⋊C4—64no—
491C22⋊C4—64no—
492C22⋊C4—64no—
493C42—34no—
494C42—34no—
495C42—32no—
496C42—32no—
497C42D3×D412yes—
498C42C23×D312yes—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—84no—
503C22×C3C2×#12412yes—
504C22×C3C2×AGL(1,ℤ/8)12yes—
505C22×C3C2×#13712yes—
506C22×C3C23×D411yescharacteristic
507C2×D3—42no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—84no—
511C2×D3—816no—
512C2×D3#489116yesdirect product factor
513C2×D3C2×#12414yes—
514C2×D3C2×AGL(1,ℤ/8)14yes—
515C2×D3C2×#13714yes—
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—42no—
519C3×C4—44no—
520C3×C4C2×#12412yes—
521C3×C4C22×#4112yes—
522Dic3—22no—
523Dic3—22no—
524Dic3—42no—
525Dic3—44no—
526Dic3C2×#12414yes—
527C23—22no—
528C23—34no—
529C23—32no—
530C23—34no—
531C23—34no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—62no—
536C23—128no—
537C23—128no—
538C23—128no—
539C23—122no—
540C23—122no—
541C23—122no—
542C23—2416no—
543C23C2×D3×D411yescharacteristic
544Q8—24no—
545Q8—48no—
546Q8—42no—
547Q8—64no—
548Q8—128no—
549Q8—128no—
550Q8—1216no—
551Q8—122no—
552D4—24no—
553D4—42no—
554D4—64no—
555D4—128no—
556D4—122no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—34no—
562C2×C4—32no—
563C2×C4—38no—
564C2×C4—38no—
565C2×C4—32no—
566C2×C4—32no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—68no—
570C2×C4—68no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—64no—
575C2×C4—128no—
576C2×C4—1216no—
577C2×C4—122no—
578C2×C4—124no—
579C2×C4C2×D3×D412yes—
580C2×C4C2×D3×D412yes—
581C2×C4C2×D3×D411yescharacteristic
582C2×C4D3×#4114yes—
583C2×C4C24×D311yescharacteristic
584C8—24no—
585C8—64no—
586C2×C3—82no—
587C2×C3C2×#48914yescomplemented
588C2×C3C22×#12411yescharacteristic
589C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
590C2×C3C22×#13711yescharacteristic
591D3—82no—
592D3C2×#48918yesdirect product factor
593C22—316no—
594C22—34no—
595C22—34no—
596C22—34no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—84no—
601C22—122no—
602C22—122no—
603C22—122no—
604C22—2416no—
605C22—244no—
606C22D3×#12412yes—
607C22D3×AGL(1,ℤ/8)12yes—
608C22D3×#13712yes—
609C22C22×D3×D411yescharacteristic
610C4—22no—
611C4—22no—
612C4—34no—
613C4—42no—
614C4—44no—
615C4—62no—
616C4—62no—
617C4—122no—
618C4—124no—
619C4D3×#12412yes—
620C4C2×D3×#4112yes—
621C3C22×#48911yescharacteristic, complemented, Sylow
622C2—38no—
623C2—82no—
624C2—242no—
625C2D3×#48914yesdirect product factor
626C2C2×D3×#12411yescharacteristic
627C2C2×D3×AGL(1,ℤ/8)11yescharacteristic
628C2C2×D3×#13711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC22×D3×D4C22
hypercenterC2×#489D3
hypocenterC3C22×#489
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊Q8C2semidirect
3. C2×C3×#489C2semidirect
4. C2×#5089C2semidirect
5. C2×#5055C2semidirect
6. D3×#489C2direct (N × H)
7. C2×D3×#103C2semidirect
8. C2×#4794C2semidirect
10. C2×#4765C2semidirect
11. C2×#4714C2semidirect
12. C2×D3×#97C2semidirect
17. C2×C3×C4⋊Q8C22semidirect
24. D3×C4⋊Q8C22semidirect
30. C2×#888C22semidirect
37. C2×#869C22semidirect
38. C3×#489C22semidirect
39. C2×C3×#103C22semidirect
41. C2×C3×#97C22semidirect
42. C2×#835C22semidirect
45. C2×#816C22semidirect
46. C2×#815C22semidirect
47. #5089C22semidirect
48. #5055C22semidirect
49. C2×#810C22semidirect
51. C2×#803C22semidirect
52. D3×#103C22semidirect
53. #4794C22semidirect
55. #4765C22semidirect
56. #4714C22semidirect
57. D3×#97C22semidirect
66. C2×#489D3direct (N × H)
95. C3×C4⋊Q8C23semidirect
129. #888C23semidirect
139. #869C23semidirect
140. C3×#103C23semidirect
142. C3×#97C23semidirect
143. #835C23semidirect
146. #816C23semidirect
147. #815C23semidirect
148. #810C23semidirect
150. #803C23semidirect
161. C2×C4⋊Q8C2×D3semidirect
180. #489C2×D3direct (N × H)
185. C2×#103C2×D3semidirect
194. C2×#97C2×D3semidirect
311. C4⋊Q8C22×D3semidirect
350. #103C22×D3semidirect
357. #97C22×D3semidirect
512. C2×D3#489direct (N × H)
587. C2×C3C2×#489semidirect
592. D3C2×#489direct (N × H)
621. C3C22×#489semidirect
625. C2D3×#489direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
C2×D3×#1242
C2×C3×#4892
C2×#50892
C2×#50552
D3×#4892
C2×D3×#1032
C2×#47942
C2×D3×#1022
C2×#47652
C2×#47142
C2×D3×#972
C22×#4893