Finite Group Databasebeta

← C3×#10596432 #13180691

Subgroups — C3×#10596432 #13180691

Order 1728.

1068 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 86 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×D3×S4C212yesmaximal, complemented
2C3×A4×D12C211yesmaximal, characteristic, complemented
3C3×C4×C3⋊S4C211yesmaximal, characteristic, complemented
4C32×C4⋊S4C211yesmaximal, characteristic, complemented
5C3×#64209C212yesmaximal, complemented
6C3×D4×S4—31nomaximal
7#10596432C311yesmaximal, characteristic, direct product factor
8C3×#5609—31nomaximal
9C2×C3×D3×A4C2212yescomplemented
10C2×C3×C3⋊S4C2211yescharacteristic
11C2×C32×S4C2212yescomplemented
12C3×D3×S4—24no—
13C32×C4×A4C2211yescharacteristic, complemented
14C3×#3802—41nomaximal
15C3×C3⋊SL(2,ℤ/4)C2211yescharacteristic, complemented
16C22×C3×S4—32no—
17C2×D3×S4C2×C312yescomplemented
18C2×C3×D3×D4—31no—
19C2×C3×D3×D4—32no—
20C22×C3×D12D311yescharacteristic, complemented
21C3×D4×A4—31no—
22A4×D12—21no—
23A4×D12C2×C311yescharacteristic, complemented
24C4×C3⋊S4C2×C311yescharacteristic, complemented
25C3×GL(2,ℤ/4)—32no—
26C3×C4⋊S4—21no—
27C3×C4⋊S4D311yescharacteristic, complemented
28C3×C4⋊S4C2×C311yescharacteristic, complemented
29C3×C4×S4—31no—
30C3×C4×S4—61no—
31#64209C2×C312yescomplemented
32C32×#121—31no—
33C3×#940—31no—
34C3×#937—32no—
35C3×C4×#67—31no—
36C3×#895—31no—
37C3×#893—31no—
38C3×#884—32no—
39C3×#882—32no—
40C2×C32×A4C2311yescharacteristic
41C2×C3×D32—42no—
42C3×D3×A4—22no—
43C3×C3⋊S4D412yescomplemented
44C32×S4—22no—
45C3×C4×C3⋊D3—41no—
46C32×D12—41no—
47C32×D12—41no—
48C3×#625—42no—
49D4×S4—31no—
50C3×D42—91no—
51#5609—31no—
52C23×C3×D3—32no—
53C23×C3×D3C2×D312yescomplemented
54C22×C3×A4—32no—
55C2×D3×A4—22no—
56C2×D3×A4C22×C312yescomplemented
57C2×C3⋊S4C22×C311yescharacteristic
58C2×C3×S4—22no—
59C2×C3×S4—31no—
60C2×C3×S4—64no—
61C2×C3×S4—61no—
62C2×C3×S4C2×D312yescomplemented
63C2×C3×S4C22×C312yescomplemented
64D3×S4—24no—
65C2×C32×D4—31no—
66C2×C32×D4—32no—
67C22×C32×C4C2×D311yescharacteristic, complemented
68C2×C3×#67—31no—
69C2×C3×#67—32no—
70C2×C3×#67—32no—
71C2×C3×#67—32no—
72C3×D3×D4—62no—
73C3×D3×D4—61no—
74C3×D3×D4—64no—
75C3×D3×D4—64no—
76C3×D3×D4—61no—
77C3×D3×D4—121no—
78C2×C3×D12—31no—
79C2×C3×D12—32no—
80C2×C3×D12—31no—
81C2×C3×D12—32no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—31no—
84C3×C4×A4—21no—
85C3×C4×A4—21no—
86C3×C4×A4—21no—
87C3×C4×A4—41no—
88C3×C4×A4C2×D311yescharacteristic, complemented
89C3×C4×A4C22×C311yescharacteristic, complemented
90#3802—41no—
91C3⋊SL(2,ℤ/4)C22×C311yescharacteristic, complemented
92C3×SL(2,ℤ/4)—31no—
93C3×SL(2,ℤ/4)—61no—
94C32×C4⋊C4—31no—
95C32×C22⋊C4—32no—
96C3×#217—31no—
97C3×(C2×D3)⋊C4—31no—
98C3×(C2×D3)⋊C4—32no—
99C3×(C2×D3)⋊C4—32no—
100C3×Dic3⋊C4—31no—
101C3×C4×Dic3—31no—
102C2×C3×C3⋊D3—41no—
103C2×C32×D3—42no—
104C2×C32×D3—42no—
105C32×A4C2×D411yescharacteristic, complemented
106C3×D32—84no—
107C33×C4—41no—
108C3×C3⋊Dic3—41no—
109C22×S4—32no—
110C22×C3×D4—31no—
111C22×C3×D4—92no—
112C22×C3×D4—91no—
113C2×D3×D4—31no—
114C2×D3×D4—32no—
115C22×D12C3×D311yescharacteristic, complemented
116D4×A4—31no—
117D4×A4—61no—
118GL(2,ℤ/4)—32no—
119C4⋊S4C3×D311yescharacteristic, complemented
120C4×S4—31no—
121C4×S4—61no—
122C3×#127—91no—
123C3×#121—31no—
124C3×#121—31no—
125C3×#121—61no—
126C3×#121—92no—
127C3×#121—91no—
128C3×#120—92no—
129C3×#120—92no—
130C3×C4×D4—91no—
131C3×C4×D4—91no—
132#940—31no—
133#937—32no—
134C4×#67—31no—
135#895—31no—
136#893—31no—
137#884—32no—
138#882—32no—
139C23×C32—32no—
140C23×C32C22×D311yescharacteristic
141C22×C3×D3—22no—
142C22×C3×D3—31no—
143C22×C3×D3—32no—
144C22×C3×D3—32no—
145C22×C3×D3—64no—
146C22×C3×D3—62no—
147C22×C3×D3—62no—
148C22×C3×D3—62no—
149C22×C3×D3—122no—
150C22×C3×D3—122no—
151C2×C3×A4—21no—
152C2×C3×A4—21no—
153C2×C3×A4—21no—
154C2×C3×A4—41no—
155C2×C3×A4—62no—
156C2×C3×A4C22×D311yescharacteristic
157C2×C3×A4C23×C311yescharacteristic
158C2×D32—42no—
159D3×A4—22no—
160D3×A4—42no—
161C3⋊S4C3×D412yescomplemented
162C3×S4—22no—
163C3×S4—22no—
164C3×S4—32no—
165C3×S4—42no—
166C3×S4—62no—
167C32×D4—61no—
168C32×D4—62no—
169C32×D4—62no—
170C32×D4—61no—
171C32×D4—121no—
172C2×C32×C4—31no—
173C2×C32×C4—31no—
174C2×C32×C4—32no—
175C4×C3⋊D3—41no—
176C3×#67—32no—
177C3×#67—32no—
178C3×#67—64no—
179C3×#67—62no—
180C3×#67—62no—
181C3×#67—122no—
182C3×#67—122no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C3×D12—32no—
187C3×D12—31no—
188C3×D12—41no—
189C3×D12—41no—
190C3×D12—41no—
191C3×D12—61no—
192C3×D12—64no—
193C3×D12—81no—
194C3×D12—81no—
195C3×D12S411yescharacteristic, complemented
196C3×C4×D3—64no—
197C3×C4×D3—61no—
198C3×C4×D3—61no—
199C3×C4×D3—121no—
200C3×C4×D3—241no—
201#625—42no—
202D42—91noSylow
203C2×C33—41no—
204C3×C3⋊D3—42no—
205C32×D3—82no—
206C32×D3—82no—
207C24×C3—32no—
208C24×C3—92no—
209C23×D3—32no—
210C23×D3C2×C3×D312yescomplemented
211C22×A4—32no—
212C22×A4—62no—
213C2×S4—31no—
214C2×S4—61no—
215C2×S4—64no—
216C2×S4C2×C3×D312yescomplemented
217C2×C3×D4—31no—
218C2×C3×D4—31no—
219C2×C3×D4—32no—
220C2×C3×D4—32no—
221C2×C3×D4—62no—
222C2×C3×D4—61no—
223C2×C3×D4—91no—
224C2×C3×D4—92no—
225C2×C3×D4—91no—
226C2×C3×D4—92no—
227C2×C3×D4—91no—
228C2×C3×D4—92no—
229C2×C3×D4—92no—
230C2×C3×D4—92no—
231C2×C3×D4—181no—
232C2×C3×D4—184no—
233C2×C3×D4—182no—
234C2×C3×D4—184no—
235C2×C3×D4—181no—
236C22×C3×C4—21no—
237C22×C3×C4—91no—
238C22×C3×C4—92no—
239C22×C3×C4D3211yescharacteristic, complemented
240C22×C3×C4C2×C3×D311yescharacteristic, complemented
241C2×#67—31no—
242C2×#67—32no—
243C2×#67—32no—
244C2×#67—32no—
245D3×D4—62no—
246D3×D4—61no—
247D3×D4—64no—
248D3×D4—64no—
249D3×D4—61no—
250D3×D4—121no—
251C2×D12—31no—
252C2×D12—32no—
253C2×D12—31no—
254C2×D12—32no—
255C2×C4×D3—32no—
256C2×C4×D3—31no—
257C4×A4—21no—
258C4×A4—21no—
259C4×A4—41no—
260C4×A4C2×C3×D311yescharacteristic, complemented
261SL(2,ℤ/4)—31no—
262SL(2,ℤ/4)—61no—
263C3×C4⋊C4—31no—
264C3×C4⋊C4—31no—
265C3×C4⋊C4—61no—
266C3×C4⋊C4—91no—
267C3×C22⋊C4—32no—
268C3×C22⋊C4—32no—
269C3×C22⋊C4—62no—
270C3×C22⋊C4—91no—
271C3×C22⋊C4—92no—
272C3×C22⋊C4—92no—
273C3×C22⋊C4—91no—
274C3×C42—91no—
275#217—31no—
276(C2×D3)⋊C4—31no—
277(C2×D3)⋊C4—32no—
278(C2×D3)⋊C4—32no—
279Dic3⋊C4—31no—
280C4×Dic3—31no—
281C22×C32—31no—
282C22×C32—31no—
283C22×C32—62no—
284C22×C32—62no—
285C22×C32—62no—
286C22×C32—122no—
287C22×C32D3×D411yescharacteristic, complemented
288C2×C3⋊D3—41no—
289C2×C3×D3—32no—
290C2×C3×D3—32no—
291C2×C3×D3—42no—
292C2×C3×D3—42no—
293C2×C3×D3—42no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C2×C3×D3—82no—
301C2×C3×D3—82no—
302C2×C3×D3—121no—
303C2×C3×D3—124no—
304C2×C3×D3—124no—
305C2×C3×D3—241no—
306C2×C3×D3—244no—
307C2×C3×D3C2×S412yescomplemented
308C3×A4—21no—
309C3×A4—21no—
310C3×A4—21no—
311C3×A4—41no—
312C3×A4D3×D411yescharacteristic, complemented
313C3×A4C2×C3×D411yescharacteristic, complemented
314D32—84no—
315C32×C4—31no—
316C32×C4—41no—
317C32×C4—41no—
318C32×C4—62no—
319C32×C4—81no—
320C32×C4—81no—
321C32×C4—81no—
322C32×C4—161no—
323C32×C4C2×S411yescharacteristic, complemented
324C3⋊Dic3—41no—
325C3×Dic3—32no—
326C3×Dic3—61no—
327C3×Dic3—61no—
328C3×Dic3—121no—
329C3×Dic3—241no—
330C22×D4—31no—
331C22×D4—92no—
332C22×D4—91no—
333#127—91no—
334#121—31no—
335#121—92no—
336#121—91no—
337#120—92no—
338#120—92no—
339C4×D4—91no—
340C4×D4—91no—
341C33—41noSylow
342C23×C3—21no—
343C23×C3—32no—
344C23×C3—32no—
345C23×C3—62no—
346C23×C3—62no—
347C23×C3—92no—
348C23×C3—91no—
349C23×C3—92no—
350C23×C3—182no—
351C23×C3—182no—
352C23×C3—184no—
353C23×C3—182no—
354C23×C3—362no—
355C23×C3C2×D3211yescharacteristic
356C23×C3C22×C3×D311yescharacteristic
357C22×D3—22no—
358C22×D3—31no—
359C22×D3—32no—
360C22×D3—32no—
361C22×D3—64no—
362C22×D3—62no—
363C22×D3—62no—
364C22×D3—62no—
365C22×D3—122no—
366C22×D3—122no—
367C2×A4—21no—
368C2×A4—21no—
369C2×A4—41no—
370C2×A4—62no—
371C2×A4—122no—
372C2×A4C22×C3×D311yescharacteristic
373S4—22no—
374S4—32no—
375S4—62no—
376C3×D4—31no—
377C3×D4—61no—
378C3×D4—61no—
379C3×D4—62no—
380C3×D4—62no—
381C3×D4—62no—
382C3×D4—62no—
383C3×D4—61no—
384C3×D4—61no—
385C3×D4—92no—
386C3×D4—92no—
387C3×D4—92no—
388C3×D4—91no—
389C3×D4—121no—
390C3×D4—121no—
391C3×D4—121no—
392C3×D4—122no—
393C3×D4—122no—
394C3×D4—181no—
395C3×D4—184no—
396C3×D4—184no—
397C3×D4—182no—
398C3×D4—182no—
399C3×D4—241no—
400C3×D4—362no—
401C2×C3×C4—31no—
402C2×C3×C4—31no—
403C2×C3×C4—31no—
404C2×C3×C4—31no—
405C2×C3×C4—32no—
406C2×C3×C4—32no—
407C2×C3×C4—61no—
408C2×C3×C4—61no—
409C2×C3×C4—62no—
410C2×C3×C4—91no—
411C2×C3×C4—91no—
412C2×C3×C4—91no—
413C2×C3×C4—181no—
414C2×C3×C4—184no—
415C2×C3×C4—181no—
416#67—32no—
417#67—32no—
418#67—64no—
419#67—62no—
420#67—62no—
421#67—122no—
422#67—122no—
423C2×Dic3—31no—
424C2×Dic3—31no—
425C2×Dic3—31no—
426D12—32no—
427D12—31no—
428D12—41no—
429D12—61no—
430D12—64no—
431D12C3×S411yescharacteristic, complemented
432C4×D3—64no—
433C4×D3—61no—
434C4×D3—61no—
435C4×D3—121no—
436C4×D3—241no—
437C2×C32—31no—
438C2×C32—31no—
439C2×C32—41no—
440C2×C32—41no—
441C2×C32—81no—
442C2×C32—81no—
443C2×C32—81no—
444C2×C32—122no—
445C2×C32—161no—
446C2×C32—242no—
447C2×C32C22×S411yescharacteristic
448C3⋊D3—42no—
449C3×D3—22no—
450C3×D3—62no—
451C3×D3—62no—
452C3×D3—82no—
453C3×D3—82no—
454C3×D3—82no—
455C3×D3—122no—
456C3×D3—162no—
457C3×D3—162no—
458C3×D3—242no—
459C24—32no—
460C24—92no—
461C2×D4—31no—
462C2×D4—32no—
463C2×D4—91no—
464C2×D4—92no—
465C2×D4—91no—
466C2×D4—92no—
467C2×D4—91no—
468C2×D4—92no—
469C2×D4—92no—
470C2×D4—92no—
471C2×D4—181no—
472C2×D4—184no—
473C2×D4—182no—
474C2×D4—184no—
475C2×D4—181no—
476C22×C4—91no—
477C22×C4—92no—
478C22×C4C3×D3211yescharacteristic, complemented
479C4⋊C4—31no—
480C4⋊C4—91no—
481C22⋊C4—32no—
482C22⋊C4—91no—
483C22⋊C4—92no—
484C22⋊C4—92no—
485C22⋊C4—91no—
486C42—91no—
487C22×C3—21no—
488C22×C3—31no—
489C22×C3—31no—
490C22×C3—31no—
491C22×C3—31no—
492C22×C3—32no—
493C22×C3—61no—
494C22×C3—61no—
495C22×C3—62no—
496C22×C3—62no—
497C22×C3—62no—
498C22×C3—62no—
499C22×C3—62no—
500C22×C3—62no—
501C22×C3—92no—
502C22×C3—92no—
503C22×C3—122no—
504C22×C3—122no—
505C22×C3—122no—
506C22×C3—122no—
507C22×C3—182no—
508C22×C3—182no—
509C22×C3—182no—
510C22×C3—182no—
511C22×C3—181no—
512C22×C3—181no—
513C22×C3—242no—
514C22×C3—364no—
515C22×C3—364no—
516C22×C3#380211yescharacteristic, complemented
517C22×C3C3×D3×D411yescharacteristic, complemented
518C2×D3—32no—
519C2×D3—32no—
520C2×D3—42no—
521C2×D3—62no—
522C2×D3—62no—
523C2×D3—61no—
524C2×D3—61no—
525C2×D3—62no—
526C2×D3—62no—
527C2×D3—121no—
528C2×D3—124no—
529C2×D3—124no—
530C2×D3—241no—
531C2×D3—244no—
532C2×D3C2×C3×S412yescomplemented
533A4—21no—
534A4—21no—
535A4—41no—
536A4C3×D3×D411yescharacteristic, complemented
537C3×C4—21no—
538C3×C4—31no—
539C3×C4—31no—
540C3×C4—41no—
541C3×C4—61no—
542C3×C4—62no—
543C3×C4—62no—
544C3×C4—81no—
545C3×C4—81no—
546C3×C4—92no—
547C3×C4—122no—
548C3×C4—161no—
549C3×C4—181no—
550C3×C4—181no—
551C3×C4D3×S411yescharacteristic, complemented
552C3×C4C2×C3×S411yescharacteristic, complemented
553Dic3—32no—
554Dic3—61no—
555Dic3—61no—
556Dic3—121no—
557Dic3—241no—
558C32—41no—
559C32—41no—
560C32—81no—
561C32—81no—
562C32—81no—
563C32—161no—
564C32D4×S411yescharacteristic, complemented
565C23—32no—
566C23—62no—
567C23—92no—
568C23—91no—
569C23—92no—
570C23—182no—
571C23—182no—
572C23—184no—
573C23—182no—
574C23—362no—
575C23C2×C3×D3211yescharacteristic
576D4—31no—
577D4—61no—
578D4—62no—
579D4—62no—
580D4—61no—
581D4—92no—
582D4—92no—
583D4—92no—
584D4—91no—
585D4—181no—
586D4—184no—
587D4—184no—
588D4—182no—
589D4—182no—
590D4—362no—
591C2×C4—31no—
592C2×C4—31no—
593C2×C4—32no—
594C2×C4—91no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—181no—
598C2×C4—184no—
599C2×C4—181no—
600C2×C3—21no—
601C2×C3—31no—
602C2×C3—31no—
603C2×C3—31no—
604C2×C3—31no—
605C2×C3—41no—
606C2×C3—61no—
607C2×C3—61no—
608C2×C3—62no—
609C2×C3—81no—
610C2×C3—81no—
611C2×C3—122no—
612C2×C3—122no—
613C2×C3—161no—
614C2×C3—182no—
615C2×C3—182no—
616C2×C3—242no—
617C2×C3—242no—
618C2×C3—482no—
619C2×C3C2×D3×S411yescharacteristic
620C2×C3C22×C3×S411yescharacteristic
621D3—22no—
622D3—62no—
623D3—62no—
624D3—82no—
625D3—122no—
626D3—242no—
627C22—31no—
628C22—31no—
629C22—32no—
630C22—62no—
631C22—62no—
632C22—62no—
633C22—92no—
634C22—92no—
635C22—182no—
636C22—182no—
637C22—182no—
638C22—182no—
639C22—181no—
640C22—181no—
641C22—364no—
642C22—364no—
643C22C3×#380211yescharacteristic, complemented
644C4—31no—
645C4—62no—
646C4—92no—
647C4—181no—
648C4—181no—
649C4C3×D3×S411yescharacteristic, complemented
650C3—21no—
651C3—41no—
652C3—81no—
653C3—81no—
654C3—161no—
655C3#1059643211yescharacteristic, direct product factor
656C3C3×D4×S411yescharacteristic, complemented
657C2—31no—
658C2—31no—
659C2—62no—
660C2—122no—
661C2—182no—
662C2—182no—
663C2C2×C3×D3×S411yescharacteristic
subgroupG/H
centerC2×C3C2×D3×S4
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2C2×C3×D3×S4
FittingC22×C32×C4C2×D3
socleC23×C32C22×D3
hypercenterC3×C4D3×S4
hypocenterC3×A4C2×C3×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×S4C2semidirect
2. C3×A4×D12C2semidirect
3. C3×C4×C3⋊S4C2semidirect
4. C32×C4⋊S4C2semidirect
5. C3×#64209C2semidirect
7. #10596432C3direct (N × H)
9. C2×C3×D3×A4C22semidirect
11. C2×C32×S4C22semidirect
13. C32×C4×A4C22semidirect
15. C3×C3⋊SL(2,ℤ/4)C22semidirect
17. C2×D3×S4C2×C3semidirect
20. C22×C3×D12D3semidirect
23. A4×D12C2×C3semidirect
24. C4×C3⋊S4C2×C3semidirect
27. C3×C4⋊S4D3semidirect
28. C3×C4⋊S4C2×C3semidirect
31. #64209C2×C3semidirect
43. C3×C3⋊S4D4semidirect
53. C23×C3×D3C2×D3semidirect
56. C2×D3×A4C22×C3semidirect
62. C2×C3×S4C2×D3semidirect
63. C2×C3×S4C22×C3semidirect
67. C22×C32×C4C2×D3semidirect
88. C3×C4×A4C2×D3semidirect
89. C3×C4×A4C22×C3semidirect
91. C3⋊SL(2,ℤ/4)C22×C3semidirect
105. C32×A4C2×D4semidirect
115. C22×D12C3×D3semidirect
119. C4⋊S4C3×D3semidirect
161. C3⋊S4C3×D4semidirect
195. C3×D12S4semidirect
210. C23×D3C2×C3×D3semidirect
216. C2×S4C2×C3×D3semidirect
239. C22×C3×C4D32semidirect
240. C22×C3×C4C2×C3×D3semidirect
260. C4×A4C2×C3×D3semidirect
287. C22×C32D3×D4semidirect
307. C2×C3×D3C2×S4semidirect
312. C3×A4D3×D4semidirect
313. C3×A4C2×C3×D4semidirect
323. C32×C4C2×S4semidirect
431. D12C3×S4semidirect
478. C22×C4C3×D32semidirect
516. C22×C3#3802semidirect
517. C22×C3C3×D3×D4semidirect
532. C2×D3C2×C3×S4semidirect
536. A4C3×D3×D4semidirect
551. C3×C4D3×S4semidirect
552. C3×C4C2×C3×S4semidirect
564. C32D4×S4semidirect
643. C22C3×#3802semidirect
649. C4C3×D3×S4semidirect
655. C3#10596432direct (N × H)
656. C3C3×D4×S4semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×S42
C3×A4×D122
C3×C4×C3⋊S42
C32×C4⋊S42
C3×#642092
C3×D4×S43
#105964323
C3×#56093
C3×#38024