Finite Group Databasebeta

← C3×#10596425 #13180684

Subgroups — C3×#10596425 #13180684

Order 1728.

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

normal subgroups only

This group has 810 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×C4×C3⋊S4C211yesmaximal, characteristic, complemented
3C32×A4⋊Q8C211yesmaximal, characteristic, complemented
4C3×#64210C212yesmaximal, complemented
5C3×D3×SL(2,ℤ/4)C212yesmaximal, complemented
6C3×#5914—31nomaximal
7#10596425C311yesmaximal, characteristic, direct product factor
8C3×#5630—31nomaximal
9C2×C3×D3×A4C2212yes—
10C2×C3×C3⋊S4C2211yescharacteristic
11C32×C4×A4C2211yescharacteristic
12C3×#3796—41nomaximal
13C3×C3⋊SL(2,ℤ/4)C2211yescharacteristic
14C32×SL(2,ℤ/4)C2212yes—
15C2×C3×#239—31no—
16C22×C3×D12D311yescharacteristic, complemented
17C3×D4×A4—31no—
18A4×D12—21no—
19A4×D12C2×C311yescharacteristic, complemented
20C4×C3⋊S4C2×C311yescharacteristic, complemented
21C3×GL(2,ℤ/4)—32no—
22C2×C3×SL(2,ℤ/4)—32no—
23C3×C4×S4—31no—
24C3×C4×S4—61no—
25C3×A4⋊Q8—21no—
26C3×A4⋊Q8D311yescharacteristic, complemented
27C3×A4⋊Q8C2×C311yescharacteristic, complemented
28#64210C2×C312yescomplemented
29D3×SL(2,ℤ/4)C2×C312yescomplemented
30C32×C22⋊Q8—31no—
31C3×#947—31no—
32C3×C4×#67—31no—
33C3×#895—31no—
34C3×#895—32no—
35C3×#894—32no—
36C3×#893—31no—
37C3×#882—32no—
38C3×D3×C22⋊C4—32no—
39C2×C32×A4C2311yescharacteristic
40C3×D3×A4—22no—
41C3×C3⋊S4—21no—
42C3×C4×C3⋊D3—41no—
43C32×D12—41no—
44C32×Dic6—41no—
45C3×#626—42no—
46C3×D3×Dic3—42no—
47#5914—31no—
48C3×#546—91no—
49#5630—31no—
50C23×C3×D3C2×D312yes—
51C22×C3×A4—32no—
52C2×D3×A4—22no—
53C2×D3×A4C22×C312yes—
54C2×C3⋊S4C22×C311yescharacteristic
55C2×C3×S4—31no—
56C2×C3×S4—61no—
57C2×C32×Q8—31no—
58C22×C32×C4C2×D311yescharacteristic
59C2×C3×#67—31no—
60C3×#239—62no—
61C3×#239—61no—
62C3×#239—61no—
63C3×#237—121no—
64C2×C3×D12—31no—
65C2×C3×D12—32no—
66C2×C3×D12—31no—
67C2×C3×D12—32no—
68C2×C3×D12—32no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—32no—
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#3796—41no—
79C3⋊SL(2,ℤ/4)C22×C311yescharacteristic
80C3×SL(2,ℤ/4)—22no—
81C3×SL(2,ℤ/4)—31no—
82C3×SL(2,ℤ/4)—61no—
83C3×SL(2,ℤ/4)C2×D312yes—
84C3×SL(2,ℤ/4)C22×C312yes—
85C32×C4⋊C4—31no—
86C32×C4⋊C4—32no—
87C32×C22⋊C4—32no—
88C3×#217—31no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×(C2×D3)⋊C4—31no—
94C3×Dic3⋊C4—31no—
95C3×C4×Dic3—31no—
96C2×C3×C3⋊D3—41no—
97C2×C32×D3—42no—
98C32×A4#4111yescharacteristic, complemented
99C33×C4—41no—
100C3×C3⋊Dic3—41no—
101C32×Dic3—42no—
102C2×C3×#41—91no—
103C22×C3×D4—31no—
104C2×#239—31no—
105C22×D12C3×D311yescharacteristic, complemented
106D4×A4—31no—
107D4×A4—61no—
108GL(2,ℤ/4)—32no—
109C2×SL(2,ℤ/4)—32no—
110C4×S4—31no—
111C4×S4—61no—
112A4⋊Q8C3×D311yescharacteristic, complemented
113C3×#124—91no—
114C3×#123—92no—
115C3×C22⋊Q8—31no—
116C3×C22⋊Q8—31no—
117C3×C22⋊Q8—61no—
118C3×#121—91no—
119C3×#121—92no—
120C3×#120—92no—
121C3×C4×D4—91no—
122C3×C4×D4—91no—
123C2×C3×C22⋊C4—92no—
124#947—31no—
125C4×#67—31no—
126#895—31no—
127#895—32no—
128#894—32no—
129#893—31no—
130#882—32no—
131D3×C22⋊C4—32no—
132C23×C32C22×D311yescharacteristic
133C22×C3×D3—22no—
134C22×C3×D3—32no—
135C22×C3×D3—32no—
136C22×C3×D3—31no—
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—21no—
148C3×S4—62no—
149C3×S4—61no—
150C32×Q8—61no—
151C32×Q8—61no—
152C32×D4—121no—
153C2×C32×C4—31no—
154C2×C32×C4—31no—
155C2×C32×C4—32no—
156C2×C32×C4—32no—
157C4×C3⋊D3—41no—
158C3×#67—61no—
159C3×#67—61no—
160C3×#67—122no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—122no—
165C3×D12—32no—
166C3×D12—31no—
167C3×D12—41no—
168C3×D12—61no—
169C3×D12—62no—
170C3×D12—62no—
171C3×D12—81no—
172C3×D12—122no—
173C3×D12S411yescharacteristic, complemented
174C3×C4×D3—61no—
175C3×C4×D3—62no—
176C3×C4×D3—62no—
177C3×C4×D3—61no—
178C3×C4×D3—121no—
179C3×C4×D3—122no—
180C3×C4×D3—241no—
181C3×Dic6—41no—
182C3×Dic6—41no—
183C3×Dic6—81no—
184#626—42no—
185D3×Dic3—42no—
186#546—91noSylow
187C2×C33—41no—
188C3×C3⋊D3—81no—
189C32×D3—82no—
190C24×C3—32no—
191C23×D3C2×C3×D312yes—
192C22×A4—32no—
193C22×A4—62no—
194C2×S4—31no—
195C2×S4—61no—
196C3×#41—181no—
197C3×#41—182no—
198C3×#41—181no—
199C2×C3×Q8—31no—
200C2×C3×Q8—31no—
201C2×C3×Q8—61no—
202C2×C3×D4—91no—
203C2×C3×D4—92no—
204C2×C3×D4—91no—
205C2×C3×D4—92no—
206C2×C3×D4—92no—
207C2×C3×D4—91no—
208C22×C3×C4—21no—
209C22×C3×C4—92no—
210C22×C3×C4—91no—
211C22×C3×C4—92no—
212C22×C3×C4D3211yescharacteristic
213C22×C3×C4C2×C3×D311yescharacteristic
214C2×#67—31no—
215#239—62no—
216#239—61no—
217#239—61no—
218#237—121no—
219C2×D12—31no—
220C2×D12—32no—
221C2×D12—31no—
222C2×D12—32no—
223C2×D12—32no—
224C2×C4×D3—32no—
225C2×C4×D3—31no—
226C2×C4×D3—32no—
227C4×A4—21no—
228C4×A4—21no—
229C4×A4—41no—
230C4×A4C2×C3×D311yescharacteristic
231SL(2,ℤ/4)—31no—
232SL(2,ℤ/4)—61no—
233SL(2,ℤ/4)C2×C3×D312yes—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—32no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—32no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—62no—
240C3×C4⋊C4—91no—
241C3×C22⋊C4—32no—
242C3×C22⋊C4—32no—
243C3×C22⋊C4—62no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—91no—
248C3×C22⋊C4—92no—
249C3×C22⋊C4—91no—
250C3×C42—91no—
251#217—31no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—32no—
255(C2×D3)⋊C4—32no—
256(C2×D3)⋊C4—31no—
257Dic3⋊C4—31no—
258C4×Dic3—31no—
259C22×C32—31no—
260C22×C32—31no—
261C22×C32—122no—
262C22×C32#23711yescharacteristic, complemented
263C2×C3⋊D3—41no—
264C2×C3×D3—32no—
265C2×C3×D3—42no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—82no—
274C2×C3×D3—121no—
275C2×C3×D3—241no—
276C2×C3×D3C2×S412yes—
277C3×A4—21no—
278C3×A4—21no—
279C3×A4—21no—
280C3×A4—41no—
281C3×A4#23911yescharacteristic, complemented
282C3×A4C3×#4111yescharacteristic, complemented
283C32×C4—31no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4—62no—
287C32×C4—62no—
288C32×C4—81no—
289C32×C4—81no—
290C32×C4—81no—
291C32×C4—161no—
292C32×C4C2×S411yescharacteristic
293C3⋊Dic3—41no—
294C3×Dic3—42no—
295C3×Dic3—42no—
296C3×Dic3—61no—
297C3×Dic3—61no—
298C3×Dic3—61no—
299C3×Dic3—82no—
300C3×Dic3—121no—
301C3×Dic3—241no—
302C2×#41—91no—
303C22×D4—31no—
304#124—91no—
305#123—92no—
306C22⋊Q8—31no—
307#121—91no—
308#121—92no—
309#120—92no—
310C4×D4—91no—
311C4×D4—91no—
312C2×C22⋊C4—92no—
313C33—41noSylow
314C23×C3—21no—
315C23×C3—62no—
316C23×C3—92no—
317C23×C3—92no—
318C23×C3—91no—
319C23×C3—182no—
320C23×C3C2×D3211yescharacteristic
321C23×C3C22×C3×D311yescharacteristic
322C22×D3—22no—
323C22×D3—32no—
324C22×D3—32no—
325C22×D3—31no—
326C22×D3—62no—
327C2×A4—21no—
328C2×A4—21no—
329C2×A4—41no—
330C2×A4—62no—
331C2×A4—122no—
332C2×A4C22×C3×D311yescharacteristic
333S4—62no—
334S4—61no—
335C3×Q8—61no—
336C3×Q8—61no—
337C3×Q8—61no—
338C3×Q8—61no—
339C3×Q8—121no—
340C3×Q8—121no—
341C3×D4—31no—
342C3×D4—92no—
343C3×D4—91no—
344C3×D4—121no—
345C3×D4—181no—
346C3×D4—182no—
347C3×D4—182no—
348C3×D4—181no—
349C3×D4—181no—
350C3×D4—241no—
351C3×D4—362no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—32no—
357C2×C3×C4—32no—
358C2×C3×C4—32no—
359C2×C3×C4—32no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—91no—
367C2×C3×C4—181no—
368C2×C3×C4—182no—
369C2×C3×C4—182no—
370C2×C3×C4—181no—
371C2×C3×C4—362no—
372#67—61no—
373#67—61no—
374#67—122no—
375C2×Dic3—31no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—122no—
379D12—32no—
380D12—31no—
381D12—61no—
382D12—62no—
383D12—62no—
384D12—122no—
385D12C3×S411yescharacteristic, complemented
386C4×D3—61no—
387C4×D3—62no—
388C4×D3—62no—
389C4×D3—61no—
390C4×D3—121no—
391C4×D3—122no—
392C4×D3—241no—
393Dic6—41no—
394C2×C32—31no—
395C2×C32—31no—
396C2×C32—41no—
397C2×C32—41no—
398C2×C32—81no—
399C2×C32—81no—
400C2×C32—81no—
401C2×C32—161no—
402C2×C32—242no—
403C2×C32C22×S411yescharacteristic
404C3⋊D3—81no—
405C3×D3—22no—
406C3×D3—62no—
407C3×D3—82no—
408C3×D3—121no—
409C3×D3—162no—
410C3×D3—241no—
411C3×D3—242no—
412C24—32no—
413#41—181no—
414#41—182no—
415#41—181no—
416C2×Q8—31no—
417C2×D4—91no—
418C2×D4—92no—
419C2×D4—91no—
420C2×D4—92no—
421C2×D4—92no—
422C2×D4—91no—
423C22×C4—92no—
424C22×C4—91no—
425C22×C4—92no—
426C22×C4C3×D3211yescharacteristic
427C4⋊C4—31no—
428C4⋊C4—32no—
429C4⋊C4—91no—
430C22⋊C4—32no—
431C22⋊C4—92no—
432C22⋊C4—92no—
433C22⋊C4—92no—
434C22⋊C4—91no—
435C22⋊C4—92no—
436C22⋊C4—91no—
437C42—91no—
438C22×C3—21no—
439C22×C3—31no—
440C22×C3—31no—
441C22×C3—31no—
442C22×C3—31no—
443C22×C3—32no—
444C22×C3—61no—
445C22×C3—61no—
446C22×C3—92no—
447C22×C3—122no—
448C22×C3—182no—
449C22×C3—182no—
450C22×C3—182no—
451C22×C3—182no—
452C22×C3—181no—
453C22×C3—181no—
454C22×C3—181no—
455C22×C3—242no—
456C22×C3#379611yescharacteristic, complemented
457C22×C3C3×#23711yescharacteristic, complemented
458C2×D3—32no—
459C2×D3—62no—
460C2×D3—62no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—61no—
464C2×D3—61no—
465C2×D3—61no—
466C2×D3—121no—
467C2×D3—241no—
468C2×D3C2×C3×S412yes—
469A4—21no—
470A4—21no—
471A4—41no—
472A4C3×#23911yescharacteristic, complemented
473C3×C4—21no—
474C3×C4—31no—
475C3×C4—31no—
476C3×C4—41no—
477C3×C4—61no—
478C3×C4—62no—
479C3×C4—62no—
480C3×C4—62no—
481C3×C4—62no—
482C3×C4—81no—
483C3×C4—81no—
484C3×C4—122no—
485C3×C4—122no—
486C3×C4—161no—
487C3×C4—181no—
488C3×C4—181no—
489C3×C4—181no—
490C3×C4D3×S411yescharacteristic
491C3×C4C2×C3×S411yescharacteristic
492Dic3—42no—
493Dic3—61no—
494Dic3—61no—
495Dic3—61no—
496Dic3—121no—
497Dic3—241no—
498C32—41no—
499C32—41no—
500C32—81no—
501C32—81no—
502C32—81no—
503C32—161no—
504C32#591411yescharacteristic, complemented
505C23—62no—
506C23—92no—
507C23—92no—
508C23—91no—
509C23—182no—
510C23C2×C3×D3211yescharacteristic
511Q8—61no—
512Q8—61no—
513D4—31no—
514D4—92no—
515D4—91no—
516D4—181no—
517D4—182no—
518D4—182no—
519D4—181no—
520D4—181no—
521D4—362no—
522C2×C4—31no—
523C2×C4—31no—
524C2×C4—32no—
525C2×C4—32no—
526C2×C4—91no—
527C2×C4—91no—
528C2×C4—91no—
529C2×C4—181no—
530C2×C4—182no—
531C2×C4—182no—
532C2×C4—181no—
533C2×C4—362no—
534C2×C3—21no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—41no—
540C2×C3—61no—
541C2×C3—61no—
542C2×C3—62no—
543C2×C3—81no—
544C2×C3—81no—
545C2×C3—161no—
546C2×C3—182no—
547C2×C3—242no—
548C2×C3—361no—
549C2×C3—482no—
550C2×C3C2×D3×S411yescharacteristic
551C2×C3C22×C3×S411yescharacteristic
552D3—22no—
553D3—62no—
554D3—121no—
555D3—241no—
556D3—242no—
557C22—31no—
558C22—31no—
559C22—32no—
560C22—92no—
561C22—182no—
562C22—182no—
563C22—182no—
564C22—182no—
565C22—181no—
566C22—181no—
567C22—181no—
568C22C3×#379611yescharacteristic, complemented
569C4—31no—
570C4—62no—
571C4—62no—
572C4—181no—
573C4—181no—
574C4—181no—
575C4C3×D3×S411yescharacteristic
576C3—21no—
577C3—41no—
578C3—81no—
579C3—81no—
580C3—161no—
581C3#1059642511yescharacteristic, direct product factor
582C3C3×#591411yescharacteristic, complemented
583C2—31no—
584C2—31no—
585C2—62no—
586C2—182no—
587C2—361no—
588C2C2×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×C4×C3⋊S4C2semidirect
3. C32×A4⋊Q8C2semidirect
4. C3×#64210C2semidirect
5. C3×D3×SL(2,ℤ/4)C2semidirect
7. #10596425C3direct (N × H)
16. C22×C3×D12D3semidirect
19. A4×D12C2×C3semidirect
20. C4×C3⋊S4C2×C3semidirect
26. C3×A4⋊Q8D3semidirect
27. C3×A4⋊Q8C2×C3semidirect
28. #64210C2×C3semidirect
29. D3×SL(2,ℤ/4)C2×C3semidirect
98. C32×A4#41semidirect
105. C22×D12C3×D3semidirect
112. A4⋊Q8C3×D3semidirect
173. C3×D12S4semidirect
262. C22×C32#237semidirect
281. C3×A4#239semidirect
282. C3×A4C3×#41semidirect
385. D12C3×S4semidirect
456. C22×C3#3796semidirect
457. C22×C3C3×#237semidirect
472. A4C3×#239semidirect
504. C32#5914semidirect
568. C22C3×#3796semidirect
581. C3#10596425direct (N × H)
582. 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×C4×C3⋊S42
C32×A4⋊Q82
C3×#642102
C3×D3×SL(2,ℤ/4)2
C3×#59143
#105964253
C3×#56303
C3×#37964