Finite Group Databasebeta

← C2×C3×#64207 #13180724

Subgroups — C2×C3×#64207 #13180724

Order 1728.

1179 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1179 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C22×C3×C3⋊S4C211yesmaximal, characteristic, complemented
2C22×C32×S4C211yesmaximal, characteristic, complemented
3C2×C3×Dic3×A4C211yesmaximal, characteristic, complemented
4C3×#64207C214yesmaximal, direct product factor
5C2×C3×C4⋊S4—31nomaximal
6C2×#64207C311yesmaximal, characteristic, direct product factor
7C2×C3×#939—31nomaximal
8C22×C32×A4C2211yescharacteristic
9C2×C3×C3⋊S4—22no—
10C2×C3×C3⋊S4C2212yescomplemented
11C2×C32×S4—22no—
12C2×C32×S4C2212yescomplemented
13C2×C3×#626—41nomaximal
14C3×Dic3×A4C2212yescomplemented
15C22×C3⋊S4C2×C311yescharacteristic, complemented
16C22×C3×S4—21no—
17C22×C3×S4—31no—
18C22×C3×S4—61no—
19C22×C3×S4D311yescharacteristic, complemented
20C22×C3×S4C2×C311yescharacteristic, complemented
21C22×C32×D4—31no—
22C22×C3×#67—31no—
23C22×C3×#67—31no—
24C23×C3×Dic3D311yescharacteristic, complemented
25C2×C3×C4×A4—31no—
26C2×Dic3×A4—21no—
27C2×Dic3×A4C2×C311yescharacteristic, complemented
28C3×C4⋊S4—34no—
29#64207C2×C314yesdirect product factor
30C2×C3×#217—31no—
31C3×#939—34no—
32C3×#939—34no—
33C2×C3×(C2×D3)⋊C4—31no—
34C2×C3×Dic3⋊C4—31no—
35C22×C3×C3⋊D3—41no—
36C22×C32×D3—41no—
37C2×C32×A4D412yescomplemented
38C2×C32×A4C2311yescharacteristic
39C3×C3⋊S4—22no—
40C32×S4—22no—
41C2×C32×Dic3—41no—
42C3×#626—44no—
43C2×C4⋊S4—31no—
44C2×C3×#121—91no—
45C2×#939—31no—
46C24×C32—31no—
47C24×C32C2×D311yescharacteristic
48C23×C3×D3—31no—
49C22×C3×A4—21no—
50C22×C3×A4—21no—
51C22×C3×A4—21no—
52C22×C3×A4—41no—
53C22×C3×A4C2×D311yescharacteristic
54C22×C3×A4C22×C311yescharacteristic
55C2×C3⋊S4—22no—
56C2×C3⋊S4C22×C312yescomplemented
57C2×C3×S4—24no—
58C2×C3×S4—22no—
59C2×C3×S4—22no—
60C2×C3×S4—22no—
61C2×C3×S4—32no—
62C2×C3×S4—62no—
63C2×C3×S4—64no—
64C2×C3×S4—62no—
65C2×C3×S4C2×D312yescomplemented
66C2×C3×S4C22×C312yescomplemented
67C2×C32×D4—32no—
68C2×C32×D4—32no—
69C2×C32×D4—62no—
70C2×C32×D4—62no—
71C22×C32×C4—31no—
72C2×C3×#67—34no—
73C2×C3×#67—32no—
74C2×C3×#67—32no—
75C2×C3×#67—61no—
76C2×C3×#67—62no—
77C2×C3×#67—62no—
78C2×C3×#67—62no—
79C2×C3×#67—61no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—32no—
83C22×C3×Dic3—31no—
84C22×C3×Dic3C2×D312yescomplemented
85C2×C3×D12—121no—
86C3×C4×A4—32no—
87C2×#626—41no—
88Dic3×A4—22no—
89Dic3×A4C22×C312yescomplemented
90C3×#217—34no—
91C3×(C2×D3)⋊C4—34no—
92C3×Dic3⋊C4—34no—
93C22×C33—41no—
94C2×C3×C3⋊D3—42no—
95C2×C3×C3⋊D3—82no—
96C2×C32×D3—42no—
97C2×C32×D3—82no—
98C32×A4C2×D411yescharacteristic, complemented
99C32×Dic3—42no—
100C22×S4—31no—
101C22×S4—61no—
102C22×S4C3×D311yescharacteristic, complemented
103C22×C3×D4—31no—
104C22×C3×D4—31no—
105C22×C3×D4—61no—
106C22×C3×D4—91no—
107C22×C3×D4—91no—
108C23×C3×C4—31no—
109C22×#67—31no—
110C22×#67—31no—
111C23×Dic3C3×D311yescharacteristic, complemented
112C2×C4×A4—31no—
113C2×C4×A4—61no—
114C4⋊S4—34no—
115C3×#121—94no—
116C3×#121—94no—
117C2×C3×C4⋊C4—91no—
118C2×C3×C22⋊C4—91no—
119C2×C3×C22⋊C4—91no—
120C2×#217—31no—
121#939—34no—
122#939—34no—
123C2×(C2×D3)⋊C4—31no—
124C2×Dic3⋊C4—31no—
125C23×C32—31no—
126C23×C32—31no—
127C23×C32—32no—
128C23×C32—32no—
129C23×C32—62no—
130C23×C32—62no—
131C23×C32—61no—
132C23×C32—61no—
133C23×C32D1212yescomplemented
134C23×C32C22×D311yescharacteristic
135C22×C3⋊D3—41no—
136C22×C3×D3—32no—
137C22×C3×D3—41no—
138C22×C3×D3—41no—
139C22×C3×D3—62no—
140C22×C3×D3—62no—
141C22×C3×D3—61no—
142C22×C3×D3—61no—
143C22×C3×D3—81no—
144C22×C3×D3—121no—
145C22×C3×D3—241no—
146C2×C3×A4—22no—
147C2×C3×A4—22no—
148C2×C3×A4—21no—
149C2×C3×A4—21no—
150C2×C3×A4—22no—
151C2×C3×A4—21no—
152C2×C3×A4—42no—
153C2×C3×A4—41no—
154C2×C3×A4#6712yescomplemented
155C2×C3×A4C3×D412yescomplemented
156C2×C3×A4C22×D311yescharacteristic
157C2×C3×A4C23×C311yescharacteristic
158C3⋊S4—22no—
159C3×S4—24no—
160C3×S4—22no—
161C3×S4—22no—
162C3×S4—64no—
163C3×S4—62no—
164C32×D4—64no—
165C32×D4—62no—
166C32×D4—62no—
167C2×C32×C4—32no—
168C2×C32×C4—62no—
169C2×C32×C4—121no—
170C3×#67—64no—
171C3×#67—64no—
172C3×#67—62no—
173C3×#67—62no—
174C3×#67—64no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—41no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—81no—
184C2×C3×Dic3S411yescharacteristic, complemented
185C3×D12—124no—
186#626—44no—
187C2×#121—91noSylow
188C2×C33—42no—
189C2×C33—41no—
190C3×C3⋊D3—82no—
191C32×D3—82no—
192C24×C3—21no—
193C24×C3—31no—
194C24×C3—31no—
195C24×C3—61no—
196C24×C3—91no—
197C24×C3D3211yescharacteristic
198C24×C3C2×C3×D311yescharacteristic
199C23×D3—31no—
200C22×A4—21no—
201C22×A4—21no—
202C22×A4—41no—
203C22×A4C2×C3×D311yescharacteristic
204C2×S4—22no—
205C2×S4—32no—
206C2×S4—62no—
207C2×S4—64no—
208C2×S4—62no—
209C2×S4C2×C3×D312yescomplemented
210C2×C3×D4—32no—
211C2×C3×D4—32no—
212C2×C3×D4—32no—
213C2×C3×D4—32no—
214C2×C3×D4—62no—
215C2×C3×D4—62no—
216C2×C3×D4—64no—
217C2×C3×D4—62no—
218C2×C3×D4—62no—
219C2×C3×D4—64no—
220C2×C3×D4—62no—
221C2×C3×D4—62no—
222C2×C3×D4—94no—
223C2×C3×D4—92no—
224C2×C3×D4—92no—
225C2×C3×D4—181no—
226C2×C3×D4—182no—
227C2×C3×D4—181no—
228C2×C3×D4—182no—
229C2×C3×D4—182no—
230C22×C3×C4—31no—
231C22×C3×C4—31no—
232C22×C3×C4—32no—
233C22×C3×C4—61no—
234C22×C3×C4—91no—
235C22×C3×C4—91no—
236C22×C3×C4—92no—
237C22×C3×C4—91no—
238C2×#67—34no—
239C2×#67—32no—
240C2×#67—32no—
241C2×#67—61no—
242C2×#67—62no—
243C2×#67—62no—
244C2×#67—62no—
245C2×#67—61no—
246C22×Dic3—31no—
247C22×Dic3—31no—
248C22×Dic3—32no—
249C22×Dic3—31no—
250C22×Dic3C2×C3×D312yescomplemented
251C2×D12—121no—
252C4×A4—32no—
253C4×A4—62no—
254C3×C4⋊C4—94no—
255C3×C22⋊C4—94no—
256C3×C22⋊C4—94no—
257#217—34no—
258(C2×D3)⋊C4—34no—
259Dic3⋊C4—34no—
260C22×C32—32no—
261C22×C32—31no—
262C22×C32—32no—
263C22×C32—32no—
264C22×C32—31no—
265C22×C32—31no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—62no—
269C22×C32—61no—
270C22×C32—62no—
271C22×C32—62no—
272C22×C32—81no—
273C22×C32—81no—
274C22×C32—81no—
275C22×C32—122no—
276C22×C32—122no—
277C22×C32—161no—
278C22×C32C2×D1211yescharacteristic, complemented
279C22×C32C2×S411yescharacteristic
280C2×C3⋊D3—42no—
281C2×C3⋊D3—82no—
282C2×C3×D3—42no—
283C2×C3×D3—42no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—82no—
288C2×C3×D3—82no—
289C2×C3×D3—82no—
290C2×C3×D3—84no—
291C2×C3×D3—122no—
292C2×C3×D3—122no—
293C2×C3×D3—122no—
294C2×C3×D3—242no—
295C2×C3×D3—242no—
296C2×C3×D3—244no—
297C3×A4—21no—
298C3×A4—21no—
299C3×A4—21no—
300C3×A4—41no—
301C3×A4C2×#6711yescharacteristic, complemented
302C3×A4C2×C3×D411yescharacteristic, complemented
303C32×C4—62no—
304C32×C4—122no—
305C3×Dic3—32no—
306C3×Dic3—42no—
307C3×Dic3—62no—
308C3×Dic3—82no—
309C3×Dic3C2×S412yescomplemented
310C22×D4—31no—
311C22×D4—91no—
312C22×D4—91no—
313C23×C4—31no—
314#121—94no—
315#121—94no—
316C2×C4⋊C4—91no—
317C2×C22⋊C4—91no—
318C2×C22⋊C4—91no—
319C33—41noSylow
320C23×C3—22no—
321C23×C3—21no—
322C23×C3—31no—
323C23×C3—31no—
324C23×C3—31no—
325C23×C3—32no—
326C23×C3—31no—
327C23×C3—32no—
328C23×C3—32no—
329C23×C3—32no—
330C23×C3—61no—
331C23×C3—61no—
332C23×C3—62no—
333C23×C3—62no—
334C23×C3—61no—
335C23×C3—62no—
336C23×C3—62no—
337C23×C3—61no—
338C23×C3—62no—
339C23×C3—62no—
340C23×C3—64no—
341C23×C3—61no—
342C23×C3—61no—
343C23×C3—92no—
344C23×C3—121no—
345C23×C3—121no—
346C23×C3—122no—
347C23×C3—181no—
348C23×C3—182no—
349C23×C3—182no—
350C23×C3—181no—
351C23×C3#62612yescomplemented
352C23×C3C3×D1212yescomplemented
353C23×C3C2×D3211yescharacteristic
354C23×C3C22×C3×D311yescharacteristic
355C22×D3—32no—
356C22×D3—41no—
357C22×D3—62no—
358C22×D3—61no—
359C22×D3—62no—
360C22×D3—61no—
361C22×D3—121no—
362C22×D3—241no—
363C2×A4—22no—
364C2×A4—22no—
365C2×A4—21no—
366C2×A4—21no—
367C2×A4—42no—
368C2×A4—41no—
369C2×A4C3×#6712yescomplemented
370C2×A4C22×C3×D311yescharacteristic
371S4—22no—
372S4—64no—
373S4—62no—
374C3×D4—64no—
375C3×D4—64no—
376C3×D4—62no—
377C3×D4—64no—
378C3×D4—62no—
379C3×D4—64no—
380C3×D4—68no—
381C3×D4—62no—
382C3×D4—62no—
383C3×D4—184no—
384C3×D4—184no—
385C3×D4—184no—
386C3×D4—182no—
387C3×D4—182no—
388C2×C3×C4—31no—
389C2×C3×C4—32no—
390C2×C3×C4—32no—
391C2×C3×C4—62no—
392C2×C3×C4—62no—
393C2×C3×C4—62no—
394C2×C3×C4—64no—
395C2×C3×C4—92no—
396C2×C3×C4—92no—
397C2×C3×C4—91no—
398C2×C3×C4—92no—
399C2×C3×C4—92no—
400C2×C3×C4—121no—
401C2×C3×C4—181no—
402C2×C3×C4—182no—
403C2×C3×C4—241no—
404#67—64no—
405#67—64no—
406#67—62no—
407#67—62no—
408#67—64no—
409C2×Dic3—31no—
410C2×Dic3—32no—
411C2×Dic3—32no—
412C2×Dic3—32no—
413C2×Dic3—32no—
414C2×Dic3—61no—
415C2×Dic3—62no—
416C2×Dic3C3×S411yescharacteristic, complemented
417D12—124no—
418C2×C32—31no—
419C2×C32—31no—
420C2×C32—32no—
421C2×C32—41no—
422C2×C32—41no—
423C2×C32—42no—
424C2×C32—42no—
425C2×C32—82no—
426C2×C32—81no—
427C2×C32—81no—
428C2×C32—81no—
429C2×C32—82no—
430C2×C32—82no—
431C2×C32—122no—
432C2×C32—162no—
433C2×C32—161no—
434C2×C32C4⋊S412yescomplemented
435C2×C32C22×S411yescharacteristic
436C3⋊D3—82no—
437C3×D3—82no—
438C3×D3—82no—
439C3×D3—84no—
440C3×D3—122no—
441C3×D3—242no—
442C3×D3—244no—
443C24—31no—
444C24—91no—
445C24C3×D3211yescharacteristic
446C2×D4—32no—
447C2×D4—32no—
448C2×D4—62no—
449C2×D4—62no—
450C2×D4—94no—
451C2×D4—92no—
452C2×D4—92no—
453C2×D4—181no—
454C2×D4—182no—
455C2×D4—181no—
456C2×D4—182no—
457C2×D4—182no—
458C22×C4—31no—
459C22×C4—32no—
460C22×C4—91no—
461C22×C4—91no—
462C22×C4—92no—
463C22×C4—91no—
464C4⋊C4—94no—
465C22⋊C4—94no—
466C22⋊C4—94no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—32no—
470C22×C3—31no—
471C22×C3—32no—
472C22×C3—31no—
473C22×C3—31no—
474C22×C3—31no—
475C22×C3—31no—
476C22×C3—31no—
477C22×C3—32no—
478C22×C3—32no—
479C22×C3—32no—
480C22×C3—32no—
481C22×C3—41no—
482C22×C3—62no—
483C22×C3—61no—
484C22×C3—62no—
485C22×C3—61no—
486C22×C3—62no—
487C22×C3—61no—
488C22×C3—61no—
489C22×C3—61no—
490C22×C3—62no—
491C22×C3—62no—
492C22×C3—62no—
493C22×C3—62no—
494C22×C3—64no—
495C22×C3—62no—
496C22×C3—62no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—122no—
500C22×C3—121no—
501C22×C3—122no—
502C22×C3—124no—
503C22×C3—122no—
504C22×C3—122no—
505C22×C3—124no—
506C22×C3—122no—
507C22×C3—122no—
508C22×C3—161no—
509C22×C3—182no—
510C22×C3—182no—
511C22×C3—182no—
512C22×C3—362no—
513C22×C3—362no—
514C22×C3C2×#62611yescharacteristic, complemented
515C22×C3C2×C3×D1211yescharacteristic, complemented
516C22×C3D3×S411yescharacteristic
517C22×C3C2×C3×S411yescharacteristic
518C2×D3—42no—
519C2×D3—62no—
520C2×D3—62no—
521C2×D3—62no—
522C2×D3—82no—
523C2×D3—122no—
524C2×D3—122no—
525C2×D3—122no—
526C2×D3—242no—
527C2×D3—242no—
528C2×D3—244no—
529A4—21no—
530A4—21no—
531A4—41no—
532A4C2×C3×#6711yescharacteristic, complemented
533C3×C4—32no—
534C3×C4—62no—
535C3×C4—62no—
536C3×C4—64no—
537C3×C4—92no—
538C3×C4—122no—
539C3×C4—182no—
540C3×C4—242no—
541Dic3—32no—
542Dic3—62no—
543Dic3C2×C3×S412yescomplemented
544C32—41no—
545C32—41no—
546C32—81no—
547C32—81no—
548C32—81no—
549C32—161no—
550C32C2×C4⋊S411yescharacteristic, complemented
551C23—31no—
552C23—31no—
553C23—32no—
554C23—32no—
555C23—62no—
556C23—61no—
557C23—62no—
558C23—61no—
559C23—92no—
560C23—182no—
561C23—181no—
562C23—182no—
563C23—181no—
564C23C3×#62612yescomplemented
565C23C2×C3×D3211yescharacteristic
566D4—64no—
567D4—62no—
568D4—62no—
569D4—184no—
570D4—184no—
571D4—184no—
572D4—182no—
573D4—182no—
574C2×C4—31no—
575C2×C4—32no—
576C2×C4—62no—
577C2×C4—92no—
578C2×C4—92no—
579C2×C4—91no—
580C2×C4—92no—
581C2×C4—92no—
582C2×C4—181no—
583C2×C4—182no—
584C2×C3—22no—
585C2×C3—21no—
586C2×C3—31no—
587C2×C3—31no—
588C2×C3—32no—
589C2×C3—31no—
590C2×C3—31no—
591C2×C3—32no—
592C2×C3—42no—
593C2×C3—41no—
594C2×C3—61no—
595C2×C3—61no—
596C2×C3—62no—
597C2×C3—82no—
598C2×C3—82no—
599C2×C3—81no—
600C2×C3—81no—
601C2×C3—124no—
602C2×C3—122no—
603C2×C3—122no—
604C2×C3—162no—
605C2×C3—161no—
606C2×C3—362no—
607C2×C3#6420712yesdirect product factor
608C2×C3C3×C4⋊S412yescomplemented
609C2×C3C2×D3×S411yescharacteristic
610C2×C3C22×C3×S411yescharacteristic
611D3—82no—
612D3—122no—
613D3—242no—
614D3—244no—
615C22—32no—
616C22—31no—
617C22—32no—
618C22—32no—
619C22—31no—
620C22—31no—
621C22—61no—
622C22—62no—
623C22—62no—
624C22—62no—
625C22—122no—
626C22—122no—
627C22—182no—
628C22—182no—
629C22—182no—
630C22—362no—
631C22—362no—
632C22C2×C3×#62611yescharacteristic, complemented
633C22C3×D3×S411yescharacteristic
634C4—32no—
635C4—62no—
636C4—92no—
637C4—182no—
638C3—21no—
639C3—41no—
640C3—81no—
641C3—81no—
642C3—161no—
643C3C2×#6420711yescharacteristic, direct product factor
644C3C2×C3×C4⋊S411yescharacteristic, complemented
645C2—31no—
646C2—31no—
647C2—32no—
648C2—122no—
649C2—362no—
650C2C3×#6420712yesdirect product factor
651C2C2×C3×D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×S4
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2C2×C3×D3×S4
Fitting, socleC24×C32C2×D3
hypocenterC3×A4C2×C3×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×C3⋊S4C2semidirect
2. C22×C32×S4C2semidirect
3. C2×C3×Dic3×A4C2semidirect
4. C3×#64207C2direct (N × H)
6. C2×#64207C3direct (N × H)
10. C2×C3×C3⋊S4C22semidirect
12. C2×C32×S4C22semidirect
14. C3×Dic3×A4C22semidirect
15. C22×C3⋊S4C2×C3semidirect
19. C22×C3×S4D3semidirect
20. C22×C3×S4C2×C3semidirect
24. C23×C3×Dic3D3semidirect
27. C2×Dic3×A4C2×C3semidirect
29. #64207C2×C3direct (N × H)
37. C2×C32×A4D4semidirect
56. C2×C3⋊S4C22×C3semidirect
65. C2×C3×S4C2×D3semidirect
66. C2×C3×S4C22×C3semidirect
84. C22×C3×Dic3C2×D3semidirect
89. Dic3×A4C22×C3semidirect
98. C32×A4C2×D4semidirect
102. C22×S4C3×D3semidirect
111. C23×Dic3C3×D3semidirect
133. C23×C32D12semidirect
154. C2×C3×A4#67semidirect
155. C2×C3×A4C3×D4semidirect
184. C2×C3×Dic3S4semidirect
209. C2×S4C2×C3×D3semidirect
250. C22×Dic3C2×C3×D3semidirect
278. C22×C32C2×D12semidirect
301. C3×A4C2×#67semidirect
302. C3×A4C2×C3×D4semidirect
309. C3×Dic3C2×S4semidirect
351. C23×C3#626semidirect
352. C23×C3C3×D12semidirect
369. C2×A4C3×#67semidirect
416. C2×Dic3C3×S4semidirect
434. C2×C32C4⋊S4semidirect
514. C22×C3C2×#626semidirect
515. C22×C3C2×C3×D12semidirect
532. A4C2×C3×#67semidirect
543. Dic3C2×C3×S4semidirect
550. C32C2×C4⋊S4semidirect
564. C23C3×#626semidirect
607. C2×C3#64207direct (N × H)
608. C2×C3C3×C4⋊S4semidirect
632. C22C2×C3×#626semidirect
643. C3C2×#64207direct (N × H)
644. C3C2×C3×C4⋊S4semidirect
650. C2C3×#64207direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊S42
C22×C32×S42
C2×C3×Dic3×A42
C3×#642072
C2×C3×C4⋊S43
C2×#642073
C2×C3×#9393
C2×C3×#6264