Finite Group Databasebeta

← C3×#10596424 #13180683

Subgroups — C3×#10596424 #13180683

Order 1728.

816 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C3×A4×D12C211yesmaximal, characteristic, complemented
2C3×#64259C211yesmaximal, characteristic, complemented
3C32×C4×S4C211yesmaximal, characteristic, complemented
4C3×#64209C212yesmaximal, complemented
5C3×D3×SL(2,ℤ/4)C212yesmaximal, complemented
6C3×#5914—31nomaximal
7#10596424C311yesmaximal, characteristic, direct product factor
8C3×#5550—31nomaximal
9C2×C3×D3×A4C2212yes—
10C2×C32×S4C2211yescharacteristic
11C32×C4×A4C2211yescharacteristic
12C3×#3795—41nomaximal
13C3×C3⋊SL(2,ℤ/4)C2212yes—
14C32×SL(2,ℤ/4)C2211yescharacteristic
15C2×C3×#235—31no—
16C22×C3×D12D311yescharacteristic, complemented
17C3×D4×A4—31no—
18A4×D12—21no—
19A4×D12C2×C311yescharacteristic, complemented
20#64259C2×C311yescharacteristic, complemented
21C3×GL(2,ℤ/4)—32no—
22C2×C3×SL(2,ℤ/4)—32no—
23C3×C4×S4—21no—
24C3×C4×S4D311yescharacteristic, complemented
25C3×C4×S4C2×C311yescharacteristic, complemented
26C3×A4⋊Q8—31no—
27C3×A4⋊Q8—61no—
28#64209C2×C312yescomplemented
29D3×SL(2,ℤ/4)C2×C312yescomplemented
30C32×C4×D4—31no—
31C3×#937—32no—
32C3×#930—31no—
33C3×#924—31no—
34C3×#894—32no—
35C3×#884—32no—
36C3×D3×C22⋊C4—32no—
37C3×#875—31no—
38C3×C4×D12—31no—
39C2×C32×A4C2311yescharacteristic
40C3×D3×A4—22no—
41C32×S4—21no—
42C3×#634—41no—
43C32×D12—41no—
44C32×C4×D3—41no—
45C3×#625—42no—
46C3×D3×Dic3—42no—
47#5914—31no—
48C3×#546—91no—
49#5550—31no—
50C23×C3×D3C2×D312yes—
51C22×C3×A4—32no—
52C2×D3×A4—22no—
53C2×D3×A4C22×C312yes—
54C2×C3×S4—21no—
55C2×C3×S4C2×D311yescharacteristic
56C2×C3×S4C22×C311yescharacteristic
57C2×C32×D4—31no—
58C22×C32×C4—31no—
59C22×C32×C4C2×D311yescharacteristic
60C2×C3×#67—32no—
61C2×C3×#67—32no—
62C3×#237—121no—
63C3×#235—62no—
64C3×#235—61no—
65C3×#235—61no—
66C2×C3×D12—31no—
67C2×C3×D12—32no—
68C2×C3×D12—31no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—32no—
71C2×C3×Dic6—31no—
72C3×C4×A4—21no—
73C3×C4×A4—21no—
74C3×C4×A4—21no—
75C3×C4×A4—41no—
76C3×C4×A4C2×D311yescharacteristic
77C3×C4×A4C22×C311yescharacteristic
78#3795—41no—
79C3⋊SL(2,ℤ/4)C22×C312yes—
80C3×SL(2,ℤ/4)—21no—
81C3×SL(2,ℤ/4)—32no—
82C3×SL(2,ℤ/4)—62no—
83C3×SL(2,ℤ/4)C2×D311yescharacteristic
84C3×SL(2,ℤ/4)C22×C311yescharacteristic
85C32×C4⋊C4—31no—
86C32×C22⋊C4—31no—
87C32×C22⋊C4—31no—
88C32×C42—31no—
89C3×#217—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—32no—
94C3×C4⋊Dic3—31no—
95C3×Dic3⋊C4—32no—
96C2×C32×D3—41no—
97C2×C32×D3—42no—
98C32×A4#4111yescharacteristic, complemented
99C33×C4—41no—
100C3×C3⋊Dic3—42no—
101C32×Dic3—41no—
102C2×C3×#41—91no—
103C22×C3×D4—31no—
104C2×#235—31no—
105C22×D12C3×D311yescharacteristic, complemented
106D4×A4—31no—
107D4×A4—61no—
108GL(2,ℤ/4)—32no—
109C2×SL(2,ℤ/4)—32no—
110C4×S4C3×D311yescharacteristic, complemented
111A4⋊Q8—31no—
112A4⋊Q8—61no—
113C3×#124—91no—
114C3×#123—92no—
115C3×C22⋊Q8—91no—
116C3×#121—92no—
117C3×#121—91no—
118C3×#120—92no—
119C3×C4×D4—31no—
120C3×C4×D4—31no—
121C3×C4×D4—61no—
122C3×C4×D4—91no—
123C2×C3×C22⋊C4—92no—
124#937—32no—
125#930—31no—
126#924—31no—
127#894—32no—
128#884—32no—
129D3×C22⋊C4—32no—
130#875—31no—
131C4×D12—31no—
132C23×C32—31no—
133C23×C32C22×D311yescharacteristic
134C22×C3×D3—22no—
135C22×C3×D3—32no—
136C22×C3×D3—32no—
137C22×C3×D3—62no—
138C2×C3×A4—21no—
139C2×C3×A4—21no—
140C2×C3×A4—21no—
141C2×C3×A4—41no—
142C2×C3×A4—62no—
143C2×C3×A4C22×D311yescharacteristic
144C2×C3×A4C23×C311yescharacteristic
145D3×A4—22no—
146D3×A4—42no—
147C3×S4—22no—
148C3×S4—21no—
149C3×S4—21no—
150C32×D4—61no—
151C32×D4—61no—
152C32×D4—121no—
153C2×C32×C4—31no—
154C2×C32×C4—31no—
155C2×C32×C4—31no—
156C2×C32×C4—31no—
157C2×C32×C4—31no—
158C2×C32×C4—61no—
159C2×C32×C4—61no—
160#634—41no—
161C3×#67—62no—
162C3×#67—62no—
163C3×#67—122no—
164C3×#67—122no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—122no—
168C3×D12—32no—
169C3×D12—31no—
170C3×D12—41no—
171C3×D12—61no—
172C3×D12—81no—
173C3×D12S411yescharacteristic, complemented
174C3×C4×D3—41no—
175C3×C4×D3—41no—
176C3×C4×D3—62no—
177C3×C4×D3—62no—
178C3×C4×D3—81no—
179C3×C4×D3—122no—
180C3×Dic6—61no—
181C3×Dic6—61no—
182C3×Dic6—121no—
183C3×Dic6—241no—
184#625—42no—
185D3×Dic3—42no—
186#546—91noSylow
187C2×C33—41no—
188C32×D3—82no—
189C32×D3—81no—
190C24×C3—32no—
191C23×D3C2×C3×D312yes—
192C22×A4—32no—
193C22×A4—62no—
194C2×S4C2×C3×D311yescharacteristic
195C3×#41—181no—
196C3×#41—182no—
197C3×#41—181no—
198C2×C3×Q8—91no—
199C2×C3×D4—31no—
200C2×C3×D4—31no—
201C2×C3×D4—61no—
202C2×C3×D4—91no—
203C2×C3×D4—92no—
204C2×C3×D4—91no—
205C2×C3×D4—92no—
206C2×C3×D4—92no—
207C22×C3×C4—21no—
208C22×C3×C4—31no—
209C22×C3×C4—31no—
210C22×C3×C4—61no—
211C22×C3×C4—92no—
212C22×C3×C4—92no—
213C22×C3×C4D3211yescharacteristic
214C22×C3×C4C2×C3×D311yescharacteristic
215C2×#67—32no—
216C2×#67—32no—
217#237—121no—
218#235—62no—
219#235—61no—
220#235—61no—
221C2×D12—31no—
222C2×D12—31no—
223C2×D12—32no—
224C2×C4×D3—32no—
225C2×C4×D3—32no—
226C2×Dic6—31no—
227C4×A4—21no—
228C4×A4—21no—
229C4×A4—41no—
230C4×A4C2×C3×D311yescharacteristic
231SL(2,ℤ/4)—32no—
232SL(2,ℤ/4)—62no—
233SL(2,ℤ/4)C2×C3×D311yescharacteristic
234C3×C4⋊C4—31no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—61no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—91no—
239C3×C22⋊C4—31no—
240C3×C22⋊C4—31no—
241C3×C22⋊C4—31no—
242C3×C22⋊C4—31no—
243C3×C22⋊C4—61no—
244C3×C22⋊C4—61no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C22⋊C4—92no—
250C3×C42—31no—
251C3×C42—31no—
252C3×C42—61no—
253#217—32no—
254(C2×D3)⋊C4—32no—
255(C2×D3)⋊C4—32no—
256(C2×D3)⋊C4—32no—
257(C2×D3)⋊C4—32no—
258C4⋊Dic3—31no—
259Dic3⋊C4—32no—
260C22×C32—31no—
261C22×C32—31no—
262C22×C32—61no—
263C22×C32—61no—
264C22×C32—61no—
265C22×C32—122no—
266C22×C32#23711yescharacteristic, complemented
267C2×C3×D3—32no—
268C2×C3×D3—41no—
269C2×C3×D3—41no—
270C2×C3×D3—42no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—81no—
276C2×C3×D3—82no—
277C2×C3×D3C2×S412yes—
278C3×A4—21no—
279C3×A4—21no—
280C3×A4—21no—
281C3×A4—41no—
282C3×A4#23511yescharacteristic, complemented
283C3×A4C3×#4111yescharacteristic, complemented
284C32×C4—31no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—61no—
288C32×C4—61no—
289C32×C4—61no—
290C32×C4—81no—
291C32×C4—81no—
292C32×C4—81no—
293C32×C4—161no—
294C32×C4C2×S411yescharacteristic
295C3⋊Dic3—42no—
296C3×Dic3—41no—
297C3×Dic3—41no—
298C3×Dic3—62no—
299C3×Dic3—62no—
300C3×Dic3—81no—
301C3×Dic3—122no—
302C3×Dic3—242no—
303C2×#41—91no—
304C22×D4—31no—
305#124—91no—
306#123—92no—
307C22⋊Q8—91no—
308#121—92no—
309#121—91no—
310#120—92no—
311C4×D4—31no—
312C4×D4—91no—
313C2×C22⋊C4—92no—
314C33—41noSylow
315C23×C3—21no—
316C23×C3—31no—
317C23×C3—31no—
318C23×C3—61no—
319C23×C3—62no—
320C23×C3—92no—
321C23×C3—92no—
322C23×C3—182no—
323C23×C3C2×D3211yescharacteristic
324C23×C3C22×C3×D311yescharacteristic
325C22×D3—22no—
326C22×D3—32no—
327C22×D3—32no—
328C22×D3—62no—
329C2×A4—21no—
330C2×A4—21no—
331C2×A4—41no—
332C2×A4—62no—
333C2×A4—122no—
334C2×A4C22×C3×D311yescharacteristic
335S4—21no—
336C3×Q8—181no—
337C3×Q8—181no—
338C3×D4—31no—
339C3×D4—62no—
340C3×D4—61no—
341C3×D4—61no—
342C3×D4—62no—
343C3×D4—61no—
344C3×D4—61no—
345C3×D4—92no—
346C3×D4—91no—
347C3×D4—121no—
348C3×D4—181no—
349C3×D4—182no—
350C3×D4—182no—
351C3×D4—241no—
352C3×D4—362no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—92no—
373C2×C3×C4—92no—
374C2×C3×C4—121no—
375C2×C3×C4—121no—
376C2×C3×C4—182no—
377C2×C3×C4—182no—
378C2×C3×C4—362no—
379#67—62no—
380#67—62no—
381#67—122no—
382#67—122no—
383C2×Dic3—32no—
384C2×Dic3—32no—
385C2×Dic3—122no—
386D12—32no—
387D12—31no—
388D12—61no—
389D12C3×S411yescharacteristic, complemented
390C4×D3—41no—
391C4×D3—62no—
392C4×D3—62no—
393C4×D3—122no—
394Dic6—61no—
395Dic6—61no—
396Dic6—121no—
397Dic6—241no—
398C2×C32—31no—
399C2×C32—31no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—81no—
403C2×C32—81no—
404C2×C32—81no—
405C2×C32—121no—
406C2×C32—161no—
407C2×C32—242no—
408C2×C32C22×S411yescharacteristic
409C3×D3—22no—
410C3×D3—62no—
411C3×D3—82no—
412C3×D3—81no—
413C3×D3—81no—
414C3×D3—82no—
415C3×D3—162no—
416C24—32no—
417#41—181no—
418#41—182no—
419#41—181no—
420C2×Q8—91no—
421C2×D4—31no—
422C2×D4—91no—
423C2×D4—92no—
424C2×D4—91no—
425C2×D4—92no—
426C2×D4—92no—
427C22×C4—31no—
428C22×C4—92no—
429C22×C4—92no—
430C22×C4C3×D3211yescharacteristic
431C4⋊C4—31no—
432C4⋊C4—91no—
433C4⋊C4—92no—
434C22⋊C4—31no—
435C22⋊C4—31no—
436C22⋊C4—92no—
437C22⋊C4—92no—
438C22⋊C4—92no—
439C22⋊C4—92no—
440C22⋊C4—92no—
441C42—31no—
442C22×C3—21no—
443C22×C3—31no—
444C22×C3—31no—
445C22×C3—31no—
446C22×C3—31no—
447C22×C3—32no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—61no—
454C22×C3—62no—
455C22×C3—61no—
456C22×C3—61no—
457C22×C3—92no—
458C22×C3—122no—
459C22×C3—121no—
460C22×C3—121no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3—182no—
464C22×C3—182no—
465C22×C3—242no—
466C22×C3#379511yescharacteristic, complemented
467C22×C3C3×#23711yescharacteristic, complemented
468C2×D3—32no—
469C2×D3—41no—
470C2×D3—62no—
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—62no—
474C2×D3C2×C3×S412yes—
475A4—21no—
476A4—21no—
477A4—41no—
478A4C3×#23511yescharacteristic, complemented
479C3×C4—21no—
480C3×C4—31no—
481C3×C4—31no—
482C3×C4—41no—
483C3×C4—61no—
484C3×C4—61no—
485C3×C4—61no—
486C3×C4—61no—
487C3×C4—61no—
488C3×C4—62no—
489C3×C4—61no—
490C3×C4—61no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4—121no—
494C3×C4—121no—
495C3×C4—161no—
496C3×C4—182no—
497C3×C4—182no—
498C3×C4D3×S411yescharacteristic
499C3×C4C2×C3×S411yescharacteristic
500Dic3—41no—
501Dic3—62no—
502Dic3—62no—
503Dic3—122no—
504Dic3—242no—
505C32—41no—
506C32—41no—
507C32—81no—
508C32—81no—
509C32—81no—
510C32—161no—
511C32#591411yescharacteristic, complemented
512C23—31no—
513C23—62no—
514C23—92no—
515C23—92no—
516C23—182no—
517C23C2×C3×D3211yescharacteristic
518Q8—181no—
519Q8—181no—
520D4—31no—
521D4—61no—
522D4—61no—
523D4—92no—
524D4—91no—
525D4—181no—
526D4—182no—
527D4—182no—
528D4—362no—
529C2×C4—31no—
530C2×C4—31no—
531C2×C4—31no—
532C2×C4—31no—
533C2×C4—31no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—92no—
537C2×C4—92no—
538C2×C4—182no—
539C2×C4—182no—
540C2×C4—362no—
541C2×C3—21no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—31no—
546C2×C3—41no—
547C2×C3—61no—
548C2×C3—61no—
549C2×C3—62no—
550C2×C3—81no—
551C2×C3—81no—
552C2×C3—122no—
553C2×C3—121no—
554C2×C3—121no—
555C2×C3—161no—
556C2×C3—182no—
557C2×C3—242no—
558C2×C3—482no—
559C2×C3C2×D3×S411yescharacteristic
560C2×C3C22×C3×S411yescharacteristic
561D3—22no—
562D3—62no—
563D3—81no—
564C22—31no—
565C22—31no—
566C22—32no—
567C22—61no—
568C22—61no—
569C22—61no—
570C22—92no—
571C22—182no—
572C22—182no—
573C22—182no—
574C22—182no—
575C22C3×#379511yescharacteristic, complemented
576C4—31no—
577C4—61no—
578C4—61no—
579C4—61no—
580C4—182no—
581C4—182no—
582C4C3×D3×S411yescharacteristic
583C3—21no—
584C3—41no—
585C3—81no—
586C3—81no—
587C3—161no—
588C3#1059642411yescharacteristic, direct product factor
589C3C3×#591411yescharacteristic, complemented
590C2—31no—
591C2—31no—
592C2—62no—
593C2—121no—
594C2—182no—
595C2C2×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×A4C3×#41
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. C3×A4×D12C2semidirect
2. C3×#64259C2semidirect
3. C32×C4×S4C2semidirect
4. C3×#64209C2semidirect
5. C3×D3×SL(2,ℤ/4)C2semidirect
7. #10596424C3direct (N × H)
16. C22×C3×D12D3semidirect
19. A4×D12C2×C3semidirect
20. #64259C2×C3semidirect
24. C3×C4×S4D3semidirect
25. C3×C4×S4C2×C3semidirect
28. #64209C2×C3semidirect
29. D3×SL(2,ℤ/4)C2×C3semidirect
98. C32×A4#41semidirect
105. C22×D12C3×D3semidirect
110. C4×S4C3×D3semidirect
173. C3×D12S4semidirect
266. C22×C32#237semidirect
282. C3×A4#235semidirect
283. C3×A4C3×#41semidirect
389. D12C3×S4semidirect
466. C22×C3#3795semidirect
467. C22×C3C3×#237semidirect
478. A4C3×#235semidirect
511. C32#5914semidirect
575. C22C3×#3795semidirect
588. C3#10596424direct (N × H)
589. C3C3×#5914semidirect

Maximal subgroups

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

subgroupindex
C3×A4×D122
C3×#642592
C32×C4×S42
C3×#642092
C3×D3×SL(2,ℤ/4)2
C3×#59143
#105964243
C3×#55503
C3×#37954