Finite Group Databasebeta

← C2×C3×#85645 #72274511

Subgroups — C2×C3×#85645 #72274511

Order 2304.

14778 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 14778 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C22⋊(C22⋊A4)C211yesmaximal, characteristic, complemented
2C3×#85645C212yesmaximal, direct product factor
3C2×#85645C313yesmaximal, direct product factor
4C23×C3×#120C311yesmaximal, characteristic, complemented
5C3×C22⋊(C22⋊A4)C2211yescharacteristic, complemented
6C22×C3×C22⋊A4—41nomaximal
7C2×C3×#120⋊C3—44nomaximal
8C27×C3C2×C311yescharacteristic, complemented
9C2×C22⋊(C22⋊A4)C2×C313yescomplemented
10C24×C3×D4—31no—
11C22×C3×#120—34no—
12C22×C3×#120—38no—
13C22×C3×#120—34no—
14C22×C3×#120—32no—
15C22×C3×#120C2×C312yescomplemented
16C23×C3×C22⋊C4—31no—
17#85645C2×C316yesdirect product factor
18C2×C3×C22⋊A4—41no—
19C2×C3×C22⋊A4—42no—
20C2×C3×C22⋊A4—44no—
21C2×C3×C22⋊A4—88no—
22C3×#120⋊C3—48no—
23C23×#120C3211yescharacteristic, complemented, Sylow
24C26×C3—31no—
25C26×C3—34no—
26C26×C3—34no—
27C26×C3—31no—
28C26×C3—68no—
29C26×C3—68no—
30C26×C3A411yescharacteristic, complemented
31C26×C3C22×C311yescharacteristic, complemented
32C22⋊(C22⋊A4)C22×C313yescomplemented
33C22×C22⋊A4—43no—
34C23×C3×D4—312no—
35C23×C3×D4—36no—
36C23×C3×D4—36no—
37C23×C3×D4—32no—
38C23×C3×D4—32no—
39C23×C3×D4—68no—
40C23×C3×D4—68no—
41C24×C3×C4—31no—
42C2×C3×#120—324no—
43C2×C3×#120—324no—
44C2×C3×#120—34no—
45C2×C3×#120—324no—
46C2×C3×#120—324no—
47C2×C3×#120—312no—
48C2×C3×#120—38no—
49C2×C3×#120—38no—
50C2×C3×#120—38no—
51C2×C3×#120—38no—
52C2×C3×#120—34no—
53C2×C3×#120A414yescomplemented
54C22×C3×C22⋊C4—32no—
55C22×C3×C22⋊C4—312no—
56C22×C3×C22⋊C4—312no—
57C22×C3×C22⋊C4—32no—
58C2×#120⋊C3—412no—
59C3×C22⋊A4—41no—
60C3×C22⋊A4—44no—
61C3×C22⋊A4—88no—
62C22×C3×A4—164no—
63C22×C3×A4—161no—
64C27C2×C3211yescharacteristic, complemented
65C24×D4—31no—
66C22×#120—32no—
67C22×#120—34no—
68C22×#120—34no—
69C22×#120—38no—
70C22×#120C2×C3212yescomplemented
71C23×C22⋊C4—31no—
72C25×C3—28no—
73C25×C3—36no—
74C25×C3—36no—
75C25×C3—38no—
76C25×C3—38no—
77C25×C3—31no—
78C25×C3—31no—
79C25×C3—312no—
80C25×C3—312no—
81C25×C3—34no—
82C25×C3—32no—
83C25×C3—31no—
84C25×C3—34no—
85C25×C3—34no—
86C25×C3—31no—
87C25×C3—68no—
88C25×C3—68no—
89C25×C3—64no—
90C25×C3—68no—
91C25×C3—68no—
92C25×C3—648no—
93C25×C3—648no—
94C25×C3—68no—
95C25×C3—616no—
96C25×C3—696no—
97C25×C3—64no—
98C25×C3—68no—
99C25×C3—68no—
100C25×C3—632no—
101C25×C3—616no—
102C25×C3—696no—
103C25×C3C2×A411yescharacteristic
104C25×C3C2×A412yescomplemented
105C25×C3C2×A414yescomplemented
106C2×C22⋊A4—43no—
107C2×C22⋊A4—46no—
108C2×C22⋊A4—412no—
109C2×C22⋊A4—824no—
110C22×C3×D4—324no—
111C22×C3×D4—312no—
112C22×C3×D4—312no—
113C22×C3×D4—312no—
114C22×C3×D4—324no—
115C22×C3×D4—312no—
116C22×C3×D4—312no—
117C22×C3×D4—34no—
118C22×C3×D4—616no—
119C22×C3×D4—616no—
120C22×C3×D4—624no—
121C22×C3×D4—624no—
122C22×C3×D4—624no—
123C22×C3×D4—696no—
124C22×C3×D4—624no—
125C23×C3×C4—36no—
126C23×C3×C4—36no—
127C23×C3×C4—32no—
128C23×C3×C4—64no—
129C23×C3×C4—64no—
130C3×#120—348no—
131C3×#120—324no—
132C3×#120—38no—
133C3×#120—324no—
134C3×#120—324no—
135C3×#120—324no—
136C3×#120—38no—
137C3×#120—38no—
138C3×#120C2×A418yescomplemented
139C2×C3×C22⋊C4—324no—
140C2×C3×C22⋊C4—324no—
141C2×C3×C22⋊C4—312no—
142C2×C3×C22⋊C4—324no—
143C2×C3×C22⋊C4—324no—
144C2×C3×C22⋊C4—34no—
145#120⋊C3—424no—
146C2×C3×A4—168no—
147C2×C3×A4—164no—
148C2×C3×A4—161no—
149C2×C3×A4—162no—
150C2×C3×A4—328no—
151C26—31no—
152C26—34no—
153C26—31no—
154C26—34no—
155C26—68no—
156C26—68no—
157C26C3×A411yescharacteristic, complemented
158C26C22×C3211yescharacteristic, complemented
159C23×D4—312no—
160C23×D4—36no—
161C23×D4—36no—
162C23×D4—32no—
163C23×D4—32no—
164C23×D4—68no—
165C23×D4—68no—
166C24×C4—31no—
167C2×#120—324no—
168C2×#120—324no—
169C2×#120—34no—
170C2×#120—324no—
171C2×#120—324no—
172C2×#120—38no—
173C2×#120—312no—
174C2×#120—34no—
175C2×#120—38no—
176C2×#120—38no—
177C2×#120—38no—
178C2×#120C3×A414yescomplemented
179C22×C22⋊C4—312no—
180C22×C22⋊C4—32no—
181C22×C22⋊C4—312no—
182C22×C22⋊C4—32no—
183C24×C3—28no—
184C24×C3—31no—
185C24×C3—34no—
186C24×C3—34no—
187C24×C3—31no—
188C24×C3—312no—
189C24×C3—36no—
190C24×C3—36no—
191C24×C3—324no—
192C24×C3—348no—
193C24×C3—324no—
194C24×C3—38no—
195C24×C3—38no—
196C24×C3—34no—
197C24×C3—312no—
198C24×C3—34no—
199C24×C3—32no—
200C24×C3—34no—
201C24×C3—32no—
202C24×C3—31no—
203C24×C3—32no—
204C24×C3—32no—
205C24×C3—44no—
206C24×C3—68no—
207C24×C3—632no—
208C24×C3—68no—
209C24×C3—68no—
210C24×C3—624no—
211C24×C3—648no—
212C24×C3—624no—
213C24×C3—648no—
214C24×C3—624no—
215C24×C3—624no—
216C24×C3—68no—
217C24×C3—68no—
218C24×C3—616no—
219C24×C3—68no—
220C24×C3—648no—
221C24×C3—648no—
222C24×C3—624no—
223C24×C3—648no—
224C24×C3—696no—
225C24×C3—648no—
226C24×C3—696no—
227C24×C3—624no—
228C24×C3—66no—
229C24×C3—66no—
230C24×C3—66no—
231C24×C3—624no—
232C24×C3—66no—
233C24×C3—632no—
234C24×C3—68no—
235C24×C3—68no—
236C24×C3—68no—
237C24×C3—632no—
238C24×C3—696no—
239C24×C3—696no—
240C24×C3—696no—
241C24×C3—696no—
242C24×C3—696no—
243C24×C3—696no—
244C24×C3—6192no—
245C24×C3—696no—
246C24×C3—6192no—
247C24×C3—616no—
248C24×C3—68no—
249C24×C3—124no—
250C24×C3—124no—
251C24×C3—1212no—
252C24×C3C22×A411yescharacteristic
253C24×C3C22×A414yescomplemented
254C24×C3C22⋊A411yescharacteristic, complemented
255C22⋊A4—43no—
256C22⋊A4—412no—
257C22⋊A4—824no—
258C22×A4—1612no—
259C22×A4—163no—
260C2×C3×D4—324no—
261C2×C3×D4—324no—
262C2×C3×D4—38no—
263C2×C3×D4—38no—
264C2×C3×D4—68no—
265C2×C3×D4—68no—
266C2×C3×D4—624no—
267C2×C3×D4—648no—
268C2×C3×D4—696no—
269C2×C3×D4—648no—
270C2×C3×D4—648no—
271C2×C3×D4—648no—
272C2×C3×D4—696no—
273C2×C3×D4—624no—
274C22×C3×C4—312no—
275C22×C3×C4—312no—
276C22×C3×C4—34no—
277C22×C3×C4—612no—
278C22×C3×C4—624no—
279C22×C3×C4—612no—
280C22×C3×C4—68no—
281C3×C22⋊C4—324no—
282C3×C22⋊C4—38no—
283C3×C22⋊C4—324no—
284C3×C22⋊C4—38no—
285C22×C32—641no—
286C3×A4—164no—
287C3×A4—161no—
288C3×A4—328no—
289C25—28no—
290C25—36no—
291C25—36no—
292C25—38no—
293C25—38no—
294C25—31no—
295C25—31no—
296C25—312no—
297C25—312no—
298C25—34no—
299C25—32no—
300C25—31no—
301C25—34no—
302C25—34no—
303C25—31no—
304C25—68no—
305C25—68no—
306C25—64no—
307C25—68no—
308C25—68no—
309C25—648no—
310C25—648no—
311C25—68no—
312C25—616no—
313C25—696no—
314C25—64no—
315C25—68no—
316C25—68no—
317C25—632no—
318C25—616no—
319C25—696no—
320C25C2×C3×A411yescharacteristic
321C25C2×C3×A412yescomplemented
322C25C2×C3×A414yescomplemented
323C22×D4—324no—
324C22×D4—312no—
325C22×D4—312no—
326C22×D4—312no—
327C22×D4—324no—
328C22×D4—312no—
329C22×D4—312no—
330C22×D4—34no—
331C22×D4—616no—
332C22×D4—616no—
333C22×D4—624no—
334C22×D4—624no—
335C22×D4—624no—
336C22×D4—696no—
337C22×D4—624no—
338C23×C4—36no—
339C23×C4—36no—
340C23×C4—32no—
341C23×C4—64no—
342C23×C4—64no—
343#120—348no—
344#120—324no—
345#120—38no—
346#120—324no—
347#120—324no—
348#120—324no—
349#120—38no—
350#120—38no—
351#120C2×C3×A418yescomplemented
352C2×C22⋊C4—324no—
353C2×C22⋊C4—324no—
354C2×C22⋊C4—312no—
355C2×C22⋊C4—324no—
356C2×C22⋊C4—324no—
357C2×C22⋊C4—34no—
358C23×C3—28no—
359C23×C3—36no—
360C23×C3—312no—
361C23×C3—36no—
362C23×C3—312no—
363C23×C3—34no—
364C23×C3—34no—
365C23×C3—34no—
366C23×C3—31no—
367C23×C3—31no—
368C23×C3—38no—
369C23×C3—348no—
370C23×C3—324no—
371C23×C3—324no—
372C23×C3—38no—
373C23×C3—34no—
374C23×C3—34no—
375C23×C3—32no—
376C23×C3—48no—
377C23×C3—68no—
378C23×C3—632no—
379C23×C3—68no—
380C23×C3—632no—
381C23×C3—61no—
382C23×C3—68no—
383C23×C3—648no—
384C23×C3—648no—
385C23×C3—624no—
386C23×C3—624no—
387C23×C3—696no—
388C23×C3—648no—
389C23×C3—648no—
390C23×C3—696no—
391C23×C3—624no—
392C23×C3—624no—
393C23×C3—68no—
394C23×C3—648no—
395C23×C3—68no—
396C23×C3—616no—
397C23×C3—624no—
398C23×C3—624no—
399C23×C3—648no—
400C23×C3—61no—
401C23×C3—63no—
402C23×C3—612no—
403C23×C3—612no—
404C23×C3—612no—
405C23×C3—612no—
406C23×C3—63no—
407C23×C3—68no—
408C23×C3—68no—
409C23×C3—616no—
410C23×C3—696no—
411C23×C3—696no—
412C23×C3—696no—
413C23×C3—696no—
414C23×C3—696no—
415C23×C3—696no—
416C23×C3—696no—
417C23×C3—6192no—
418C23×C3—6192no—
419C23×C3—632no—
420C23×C3—68no—
421C23×C3—68no—
422C23×C3—124no—
423C23×C3—128no—
424C23×C3—124no—
425C23×C3—1212no—
426C23×C3—128no—
427C23×C3—1224no—
428C23×C3—1212no—
429C23×C3#120⋊C314yescomplemented
430C23×C3C2×C22⋊A411yescharacteristic
431C23×C3C2×C22⋊A412yescomplemented
432C2×A4—1624no—
433C2×A4—163no—
434C2×A4—166no—
435C2×A4—1612no—
436C2×A4—3224no—
437C3×D4—616no—
438C3×D4—616no—
439C3×D4—648no—
440C3×D4—648no—
441C2×C3×C4—38no—
442C2×C3×C4—64no—
443C2×C3×C4—624no—
444C2×C3×C4—624no—
445C2×C3×C4—64no—
446C2×C32—642no—
447C2×C32—641no—
448C24—28no—
449C24—31no—
450C24—34no—
451C24—31no—
452C24—34no—
453C24—36no—
454C24—36no—
455C24—312no—
456C24—324no—
457C24—348no—
458C24—324no—
459C24—38no—
460C24—38no—
461C24—34no—
462C24—34no—
463C24—312no—
464C24—32no—
465C24—34no—
466C24—32no—
467C24—31no—
468C24—32no—
469C24—32no—
470C24—44no—
471C24—68no—
472C24—632no—
473C24—68no—
474C24—68no—
475C24—624no—
476C24—648no—
477C24—624no—
478C24—648no—
479C24—624no—
480C24—624no—
481C24—68no—
482C24—68no—
483C24—616no—
484C24—68no—
485C24—648no—
486C24—648no—
487C24—624no—
488C24—648no—
489C24—696no—
490C24—648no—
491C24—696no—
492C24—624no—
493C24—66no—
494C24—66no—
495C24—66no—
496C24—624no—
497C24—66no—
498C24—632no—
499C24—68no—
500C24—68no—
501C24—68no—
502C24—632no—
503C24—696no—
504C24—696no—
505C24—696no—
506C24—696no—
507C24—696no—
508C24—696no—
509C24—6192no—
510C24—696no—
511C24—6192no—
512C24—616no—
513C24—68no—
514C24—124no—
515C24—124no—
516C24—1212no—
517C24C22×C3×A411yescharacteristic
518C24C22×C3×A414yescomplemented
519C24C3×C22⋊A411yescharacteristic, complemented
520C2×D4—324no—
521C2×D4—324no—
522C2×D4—38no—
523C2×D4—38no—
524C2×D4—68no—
525C2×D4—68no—
526C2×D4—624no—
527C2×D4—648no—
528C2×D4—696no—
529C2×D4—648no—
530C2×D4—648no—
531C2×D4—648no—
532C2×D4—696no—
533C2×D4—624no—
534C22×C4—312no—
535C22×C4—312no—
536C22×C4—34no—
537C22×C4—612no—
538C22×C4—624no—
539C22×C4—612no—
540C22×C4—68no—
541C22⋊C4—324no—
542C22⋊C4—324no—
543C22⋊C4—38no—
544C22⋊C4—38no—
545C22×C3—28no—
546C22×C3—31no—
547C22×C3—34no—
548C22×C3—36no—
549C22×C3—36no—
550C22×C3—312no—
551C22×C3—312no—
552C22×C3—34no—
553C22×C3—31no—
554C22×C3—34no—
555C22×C3—38no—
556C22×C3—38no—
557C22×C3—41no—
558C22×C3—62no—
559C22×C3—68no—
560C22×C3—68no—
561C22×C3—616no—
562C22×C3—68no—
563C22×C3—696no—
564C22×C3—648no—
565C22×C3—648no—
566C22×C3—66no—
567C22×C3—68no—
568C22×C3—68no—
569C22×C3—68no—
570C22×C3—632no—
571C22×C3—616no—
572C22×C3—696no—
573C22×C3—68no—
574C22×C3—121no—
575C22×C3—128no—
576C22×C3—128no—
577C22×C3—128no—
578C22×C3—128no—
579C22×C3—123no—
580C22×C3—121no—
581C22×C3—643no—
582C22×C3C2×#120⋊C314yescomplemented
583C22×C3C22×C22⋊A411yescharacteristic
584A4—163no—
585A4—1612no—
586A4—3224no—
587C3×C4—68no—
588C32—641noSylow
589C23—28no—
590C23—36no—
591C23—312no—
592C23—36no—
593C23—312no—
594C23—34no—
595C23—31no—
596C23—31no—
597C23—34no—
598C23—34no—
599C23—38no—
600C23—324no—
601C23—324no—
602C23—348no—
603C23—38no—
604C23—34no—
605C23—34no—
606C23—32no—
607C23—48no—
608C23—68no—
609C23—632no—
610C23—68no—
611C23—632no—
612C23—61no—
613C23—68no—
614C23—648no—
615C23—624no—
616C23—624no—
617C23—648no—
618C23—624no—
619C23—624no—
620C23—696no—
621C23—648no—
622C23—648no—
623C23—696no—
624C23—68no—
625C23—648no—
626C23—68no—
627C23—616no—
628C23—624no—
629C23—624no—
630C23—648no—
631C23—61no—
632C23—63no—
633C23—612no—
634C23—612no—
635C23—612no—
636C23—612no—
637C23—63no—
638C23—68no—
639C23—68no—
640C23—616no—
641C23—696no—
642C23—696no—
643C23—696no—
644C23—696no—
645C23—696no—
646C23—696no—
647C23—6192no—
648C23—696no—
649C23—6192no—
650C23—632no—
651C23—68no—
652C23—68no—
653C23—124no—
654C23—128no—
655C23—124no—
656C23—1212no—
657C23—128no—
658C23—1224no—
659C23—1212no—
660C23C3×#120⋊C314yescomplemented
661C23C2×C3×C22⋊A411yescharacteristic
662C23C2×C3×C22⋊A412yescomplemented
663D4—616no—
664D4—616no—
665D4—648no—
666D4—648no—
667C2×C4—38no—
668C2×C4—64no—
669C2×C4—624no—
670C2×C4—624no—
671C2×C4—64no—
672C2×C3—31no—
673C2×C3—31no—
674C2×C3—34no—
675C2×C3—34no—
676C2×C3—42no—
677C2×C3—68no—
678C2×C3—68no—
679C2×C3—122no—
680C2×C3—646no—
681C2×C3—643no—
682C2×C3#8564511yescharacteristic, direct product factor
683C22—28no—
684C22—31no—
685C22—34no—
686C22—36no—
687C22—36no—
688C22—34no—
689C22—312no—
690C22—312no—
691C22—31no—
692C22—34no—
693C22—38no—
694C22—38no—
695C22—41no—
696C22—62no—
697C22—68no—
698C22—68no—
699C22—648no—
700C22—648no—
701C22—616no—
702C22—68no—
703C22—696no—
704C22—66no—
705C22—68no—
706C22—68no—
707C22—68no—
708C22—632no—
709C22—616no—
710C22—696no—
711C22—68no—
712C22—121no—
713C22—128no—
714C22—128no—
715C22—128no—
716C22—128no—
717C22—123no—
718C22—121no—
719C22C2×C3×#120⋊C314yescomplemented
720C22C22×C3×C22⋊A411yescharacteristic
721C4—68no—
722C3—643no—
723C3C2×#8564511yescharacteristic, direct product factor
724C2—31no—
725C2—34no—
726C2—31no—
727C2—34no—
728C2—42no—
729C2—68no—
730C2—68no—
731C2—122no—
732C2C3×#8564511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3#85645
commutator, cosocle, hypocenterC26C22×C32
FrattiniC22C22×C3×C22⋊A4
FittingC23×C3×#120C3
socleC25×C3C2×A4
LCSG ▷ C26
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊(C22⋊A4)C2semidirect
2. C3×#85645C2direct (N × H)
3. C2×#85645C3direct (N × H)
4. C23×C3×#120C3semidirect
5. C3×C22⋊(C22⋊A4)C22semidirect
8. C27×C3C2×C3semidirect
9. C2×C22⋊(C22⋊A4)C2×C3semidirect
15. C22×C3×#120C2×C3semidirect
17. #85645C2×C3direct (N × H)
23. C23×#120C32semidirect
30. C26×C3A4semidirect
31. C26×C3C22×C3semidirect
32. C22⋊(C22⋊A4)C22×C3semidirect
53. C2×C3×#120A4semidirect
64. C27C2×C32semidirect
70. C22×#120C2×C32semidirect
104. C25×C3C2×A4semidirect
105. C25×C3C2×A4semidirect
138. C3×#120C2×A4semidirect
157. C26C3×A4semidirect
158. C26C22×C32semidirect
178. C2×#120C3×A4semidirect
253. C24×C3C22×A4semidirect
254. C24×C3C22⋊A4semidirect
321. C25C2×C3×A4semidirect
322. C25C2×C3×A4semidirect
351. #120C2×C3×A4semidirect
429. C23×C3#120⋊C3semidirect
431. C23×C3C2×C22⋊A4semidirect
518. C24C22×C3×A4semidirect
519. C24C3×C22⋊A4semidirect
582. C22×C3C2×#120⋊C3semidirect
660. C23C3×#120⋊C3semidirect
662. C23C2×C3×C22⋊A4semidirect
682. C2×C3#85645direct (N × H)
719. C22C2×C3×#120⋊C3semidirect
723. C3C2×#85645direct (N × H)
732. C2C3×#85645direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊(C22⋊A4)2
C3×#856452
C2×#856453
C23×C3×#1203
C22×C3×C22⋊A44
C2×C3×#120⋊C34