Finite Group Databasebeta

← C3×#10596427 #13180686

Subgroups — C3×#10596427 #13180686

Order 1728.

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

normal subgroups only

This group has 808 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×C4×D3×A4C211yesmaximal, characteristic, complemented
2C3×#64259C211yesmaximal, characteristic, complemented
3C32×C4⋊S4C211yesmaximal, characteristic, complemented
4C3×#64209C212yesmaximal, complemented
5C3×Dic3×S4C212yesmaximal, complemented
6C3×#5912—31nomaximal
7#10596427C311yesmaximal, characteristic, direct product factor
8C3×#5606—31nomaximal
9C2×C3×D3×A4C2211yescharacteristic
10C2×C32×S4C2212yes—
11C32×C4×A4C2211yescharacteristic
12C3×#3795—41nomaximal
13C3×Dic3×A4C2211yescharacteristic
14C3×C3⋊SL(2,ℤ/4)C2212yes—
15C2×C3×#237—31no—
16C22×C3×C4×D3D311yescharacteristic, complemented
17C2×C3×C4×A4—31no—
18C4×D3×A4—21no—
19C4×D3×A4C2×C311yescharacteristic, complemented
20#64259C2×C311yescharacteristic, complemented
21C3×GL(2,ℤ/4)—32no—
22C3×C4⋊S4—21no—
23C3×C4⋊S4D311yescharacteristic, complemented
24C3×C4⋊S4C2×C311yescharacteristic, complemented
25C3×C4×S4—32no—
26C3×A4⋊Q8—31no—
27C3×A4⋊Q8—61no—
28#64209C2×C312yescomplemented
29Dic3×S4C2×C312yescomplemented
30C32×#121—31no—
31C3×#938—31no—
32C3×#937—32no—
33C3×D4×Dic3—32no—
34C3×#924—31no—
35C3×#897—31no—
36C3×#892—31no—
37C3×#883—32no—
38C3×#67⋊C4—32no—
39C2×C32×A4C2311yescharacteristic
40C3×D3×A4—21no—
41C32×S4—22no—
42C3×#634—41no—
43C32×D12—41no—
44C32×C4×D3—41no—
45C3×#625—42no—
46C3×D3×Dic3—42no—
47#5912—31no—
48C3×#525—91no—
49#5606—31no—
50C23×C3×D3C2×D311yescharacteristic
51C22×C3×A4—31no—
52C2×D3×A4—21no—
53C2×D3×A4C22×C311yescharacteristic
54C2×C3×S4—22no—
55C2×C3×S4C2×D312yes—
56C2×C3×S4C22×C312yes—
57C2×C32×D4—31no—
58C2×C32×D4—32no—
59C22×C32×C4C2×D311yescharacteristic
60C2×C3×#67—32no—
61C2×C3×#67—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3C2×D311yescharacteristic
64C3×#237—61no—
65C3×#237—61no—
66C3×#237—61no—
67C3×#237—61no—
68C3×#235—121no—
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—31no—
73C2×C3×Dic6—31no—
74C3×C4×A4—21no—
75C3×C4×A4—21no—
76C3×C4×A4—21no—
77C3×C4×A4—31no—
78C3×C4×A4—41no—
79C3×C4×A4C2×D311yescharacteristic
80C3×C4×A4C22×C311yescharacteristic
81#3795—41no—
82Dic3×A4—21no—
83Dic3×A4C22×C311yescharacteristic
84C3⋊SL(2,ℤ/4)C22×C312yes—
85C3×SL(2,ℤ/4)—32no—
86C3×SL(2,ℤ/4)—62no—
87C32×C4⋊C4—31no—
88C32×C22⋊C4—32no—
89C3×#217—32no—
90C3×#217—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—31no—
95C3×Dic3⋊C4—32no—
96C3×C4×Dic3—32no—
97C2×C32×D3—42no—
98C2×C32×D3—41no—
99C32×A4#4111yescharacteristic, complemented
100C33×C4—41no—
101C3×C3⋊Dic3—42no—
102C32×Dic3—41no—
103C2×C3×#41—91no—
104C23×C3×C4—31no—
105C2×#237—31no—
106C22×C4×D3C3×D311yescharacteristic, complemented
107C2×C4×A4—31no—
108C2×C4×A4—61no—
109GL(2,ℤ/4)—32no—
110C4⋊S4C3×D311yescharacteristic, complemented
111C4×S4—32no—
112A4⋊Q8—31no—
113A4⋊Q8—61no—
114C3×#123—92no—
115C3×C22⋊Q8—91no—
116C3×C22⋊Q8—91no—
117C3×#121—31no—
118C3×#121—31no—
119C3×#121—61no—
120C3×#121—91no—
121C3×#120—92no—
122C3×C4×D4—92no—
123C3×C4×D4—92no—
124C3×#117—91no—
125#938—31no—
126#937—32no—
127D4×Dic3—32no—
128#924—31no—
129#897—31no—
130#892—31no—
131#883—32no—
132#67⋊C4—32no—
133C23×C32—32no—
134C23×C32C22×D311yescharacteristic
135C22×C3×D3—21no—
136C22×C3×D3—31no—
137C22×C3×D3—31no—
138C22×C3×D3—61no—
139C2×C3×A4—21no—
140C2×C3×A4—21no—
141C2×C3×A4—21no—
142C2×C3×A4—41no—
143C2×C3×A4—61no—
144C2×C3×A4C22×D311yescharacteristic
145C2×C3×A4C23×C311yescharacteristic
146D3×A4—22no—
147D3×A4—21no—
148C3×S4—22no—
149C3×S4—22no—
150C3×S4—42no—
151C32×D4—61no—
152C32×D4—62no—
153C32×D4—62no—
154C32×D4—61no—
155C2×C32×C4—31no—
156C2×C32×C4—31no—
157C2×C32×C4—32no—
158C2×C32×C4—121no—
159#634—41no—
160C3×#67—62no—
161C3×#67—64no—
162C3×#67—62no—
163C3×#67—122no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C3×D12—41no—
171C3×D12—41no—
172C3×D12—81no—
173C3×C4×D3—31no—
174C3×C4×D3—31no—
175C3×C4×D3—31no—
176C3×C4×D3—41no—
177C3×C4×D3—61no—
178C3×C4×D3—81no—
179C3×C4×D3—122no—
180C3×C4×D3S411yescharacteristic, complemented
181C3×Dic6—61no—
182C3×Dic6—61no—
183C3×Dic6—121no—
184C3×Dic6—241no—
185#625—42no—
186D3×Dic3—42no—
187#525—91noSylow
188C2×C33—41no—
189C32×D3—81no—
190C32×D3—82no—
191C24×C3—31no—
192C23×D3C2×C3×D311yescharacteristic
193C22×A4—31no—
194C22×A4—61no—
195C2×S4C2×C3×D312yes—
196C3×#41—181no—
197C3×#41—181no—
198C3×#41—181no—
199C3×#41—181no—
200C2×C3×Q8—91no—
201C2×C3×D4—31no—
202C2×C3×D4—31no—
203C2×C3×D4—32no—
204C2×C3×D4—32no—
205C2×C3×D4—62no—
206C2×C3×D4—61no—
207C2×C3×D4—92no—
208C2×C3×D4—92no—
209C22×C3×C4—21no—
210C22×C3×C4—31no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—92no—
216C22×C3×C4D3211yescharacteristic
217C22×C3×C4C2×C3×D311yescharacteristic
218C2×#67—32no—
219C2×#67—32no—
220C22×Dic3—32no—
221C22×Dic3C2×C3×D311yescharacteristic
222#237—61no—
223#237—61no—
224#237—61no—
225#237—61no—
226#235—121no—
227C2×C4×D3—31no—
228C2×C4×D3—31no—
229C2×C4×D3—31no—
230C2×C4×D3—31no—
231C2×Dic6—31no—
232C4×A4—21no—
233C4×A4—21no—
234C4×A4—31no—
235C4×A4—41no—
236C4×A4—61no—
237C4×A4C2×C3×D311yescharacteristic
238SL(2,ℤ/4)—32no—
239SL(2,ℤ/4)—62no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—92no—
246C3×C22⋊C4—32no—
247C3×C22⋊C4—32no—
248C3×C22⋊C4—62no—
249C3×C22⋊C4—92no—
250C3×C22⋊C4—92no—
251C3×C22⋊C4—92no—
252C3×C22⋊C4—92no—
253C3×C42—92no—
254#217—32no—
255#217—32no—
256(C2×D3)⋊C4—32no—
257(C2×D3)⋊C4—32no—
258C4⋊Dic3—32no—
259C4⋊Dic3—31no—
260Dic3⋊C4—32no—
261C4×Dic3—32no—
262C22×C32—31no—
263C22×C32—31no—
264C22×C32—62no—
265C22×C32—62no—
266C22×C32—62no—
267C22×C32—121no—
268C22×C32#23511yescharacteristic, complemented
269C2×C3×D3—32no—
270C2×C3×D3—31no—
271C2×C3×D3—42no—
272C2×C3×D3—42no—
273C2×C3×D3—41no—
274C2×C3×D3—61no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C2×C3×D3—82no—
278C2×C3×D3—81no—
279C2×C3×D3C2×S411yescharacteristic
280C3×A4—21no—
281C3×A4—21no—
282C3×A4—21no—
283C3×A4—41no—
284C3×A4#23711yescharacteristic, complemented
285C3×A4C3×#4111yescharacteristic, complemented
286C32×C4—31no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—62no—
290C32×C4—81no—
291C32×C4—81no—
292C32×C4—81no—
293C32×C4—121no—
294C32×C4—161no—
295C32×C4C2×S411yescharacteristic
296C3⋊Dic3—42no—
297C3×Dic3—31no—
298C3×Dic3—41no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C3×Dic3—81no—
302C3×Dic3—122no—
303C3×Dic3—242no—
304C3×Dic3C2×S411yescharacteristic
305C2×#41—91no—
306C23×C4—31no—
307#123—92no—
308C22⋊Q8—91no—
309C22⋊Q8—91no—
310#121—31no—
311#121—91no—
312#120—92no—
313C4×D4—92no—
314C4×D4—92no—
315#117—91no—
316C33—41noSylow
317C23×C3—21no—
318C23×C3—32no—
319C23×C3—32no—
320C23×C3—62no—
321C23×C3—61no—
322C23×C3—91no—
323C23×C3—91no—
324C23×C3—181no—
325C23×C3C2×D3211yescharacteristic
326C23×C3C22×C3×D311yescharacteristic
327C22×D3—21no—
328C22×D3—31no—
329C22×D3—31no—
330C22×D3—61no—
331C2×A4—21no—
332C2×A4—21no—
333C2×A4—41no—
334C2×A4—62no—
335C2×A4—61no—
336C2×A4C22×C3×D311yescharacteristic
337S4—22no—
338C3×Q8—181no—
339C3×Q8—181no—
340C3×D4—61no—
341C3×D4—61no—
342C3×D4—62no—
343C3×D4—62no—
344C3×D4—62no—
345C3×D4—62no—
346C3×D4—61no—
347C3×D4—61no—
348C3×D4—121no—
349C3×D4—121no—
350C3×D4—122no—
351C3×D4—122no—
352C3×D4—182no—
353C3×D4—184no—
354C3×D4—182no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—32no—
361C2×C3×C4—32no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—62no—
365C2×C3×C4—91no—
366C2×C3×C4—91no—
367C2×C3×C4—91no—
368C2×C3×C4—91no—
369C2×C3×C4—91no—
370C2×C3×C4—92no—
371C2×C3×C4—92no—
372C2×C3×C4—121no—
373C2×C3×C4—181no—
374C2×C3×C4—182no—
375C2×C3×C4—182no—
376C2×C3×C4—241no—
377#67—62no—
378#67—64no—
379#67—62no—
380#67—122no—
381C2×Dic3—31no—
382C2×Dic3—31no—
383C2×Dic3—32no—
384C2×Dic3—32no—
385C2×Dic3—62no—
386C2×Dic3—62no—
387D12—41no—
388C4×D3—31no—
389C4×D3—31no—
390C4×D3—31no—
391C4×D3—61no—
392C4×D3—122no—
393C4×D3C3×S411yescharacteristic, complemented
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—122no—
406C2×C32—161no—
407C2×C32—241no—
408C2×C32C22×S411yescharacteristic
409C3×D3—21no—
410C3×D3—61no—
411C3×D3—81no—
412C3×D3—82no—
413C3×D3—82no—
414C3×D3—82no—
415C3×D3—162no—
416C24—31no—
417#41—181no—
418#41—181no—
419#41—181no—
420#41—181no—
421C2×Q8—91no—
422C2×D4—31no—
423C2×D4—32no—
424C2×D4—92no—
425C2×D4—92no—
426C22×C4—31no—
427C22×C4—91no—
428C22×C4—91no—
429C22×C4—91no—
430C22×C4—91no—
431C22×C4—92no—
432C22×C4C3×D3211yescharacteristic
433C4⋊C4—31no—
434C4⋊C4—91no—
435C4⋊C4—92no—
436C4⋊C4—92no—
437C22⋊C4—32no—
438C22⋊C4—92no—
439C22⋊C4—92no—
440C22⋊C4—92no—
441C22⋊C4—92no—
442C42—92no—
443C22×C3—21no—
444C22×C3—31no—
445C22×C3—31no—
446C22×C3—31no—
447C22×C3—31no—
448C22×C3—31no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—62no—
452C22×C3—62no—
453C22×C3—62no—
454C22×C3—62no—
455C22×C3—62no—
456C22×C3—62no—
457C22×C3—92no—
458C22×C3—91no—
459C22×C3—121no—
460C22×C3—122no—
461C22×C3—122no—
462C22×C3—122no—
463C22×C3—181no—
464C22×C3—181no—
465C22×C3—181no—
466C22×C3—241no—
467C22×C3#379511yescharacteristic, complemented
468C22×C3C3×#23511yescharacteristic, complemented
469C2×D3—32no—
470C2×D3—31no—
471C2×D3—42no—
472C2×D3—61no—
473C2×D3—61no—
474C2×D3—61no—
475C2×D3C2×C3×S411yescharacteristic
476A4—21no—
477A4—21no—
478A4—41no—
479A4C3×#23711yescharacteristic, complemented
480C3×C4—21no—
481C3×C4—31no—
482C3×C4—31no—
483C3×C4—31no—
484C3×C4—41no—
485C3×C4—61no—
486C3×C4—62no—
487C3×C4—62no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—91no—
491C3×C4—121no—
492C3×C4—122no—
493C3×C4—161no—
494C3×C4—182no—
495C3×C4—182no—
496C3×C4—241no—
497C3×C4D3×S411yescharacteristic
498C3×C4C2×C3×S411yescharacteristic
499Dic3—31no—
500Dic3—62no—
501Dic3—62no—
502Dic3—122no—
503Dic3—242no—
504Dic3C2×C3×S411yescharacteristic
505C32—41no—
506C32—41no—
507C32—81no—
508C32—81no—
509C32—81no—
510C32—161no—
511C32#591211yescharacteristic, complemented
512C23—32no—
513C23—61no—
514C23—91no—
515C23—91no—
516C23—181no—
517C23C2×C3×D3211yescharacteristic
518Q8—181no—
519Q8—181no—
520D4—61no—
521D4—62no—
522D4—62no—
523D4—61no—
524D4—182no—
525D4—182no—
526D4—184no—
527C2×C4—31no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—32no—
531C2×C4—91no—
532C2×C4—91no—
533C2×C4—91no—
534C2×C4—91no—
535C2×C4—91no—
536C2×C4—92no—
537C2×C4—92no—
538C2×C4—181no—
539C2×C4—182no—
540C2×C4—182no—
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—61no—
550C2×C3—81no—
551C2×C3—81no—
552C2×C3—122no—
553C2×C3—122no—
554C2×C3—161no—
555C2×C3—181no—
556C2×C3—241no—
557C2×C3—242no—
558C2×C3—242no—
559C2×C3C2×D3×S411yescharacteristic
560C2×C3C22×C3×S411yescharacteristic
561D3—21no—
562D3—61no—
563D3—82no—
564C22—31no—
565C22—31no—
566C22—31no—
567C22—62no—
568C22—62no—
569C22—62no—
570C22—92no—
571C22—91no—
572C22—181no—
573C22—181no—
574C22—181no—
575C22C3×#379511yescharacteristic, complemented
576C4—31no—
577C4—31no—
578C4—62no—
579C4—91no—
580C4—182no—
581C4—182no—
582C4C3×D3×S411yescharacteristic
583C3—21no—
584C3—41no—
585C3—81no—
586C3—81no—
587C3—161no—
588C3#1059642711yescharacteristic, direct product factor
589C3C3×#591211yescharacteristic, complemented
590C2—31no—
591C2—31no—
592C2—61no—
593C2—122no—
594C2—181no—
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×C4×D3×A4C2semidirect
2. C3×#64259C2semidirect
3. C32×C4⋊S4C2semidirect
4. C3×#64209C2semidirect
5. C3×Dic3×S4C2semidirect
7. #10596427C3direct (N × H)
16. C22×C3×C4×D3D3semidirect
19. C4×D3×A4C2×C3semidirect
20. #64259C2×C3semidirect
23. C3×C4⋊S4D3semidirect
24. C3×C4⋊S4C2×C3semidirect
28. #64209C2×C3semidirect
29. Dic3×S4C2×C3semidirect
99. C32×A4#41semidirect
106. C22×C4×D3C3×D3semidirect
110. C4⋊S4C3×D3semidirect
180. C3×C4×D3S4semidirect
268. C22×C32#235semidirect
284. C3×A4#237semidirect
285. C3×A4C3×#41semidirect
393. C4×D3C3×S4semidirect
467. C22×C3#3795semidirect
468. C22×C3C3×#235semidirect
479. A4C3×#237semidirect
511. C32#5912semidirect
575. C22C3×#3795semidirect
588. C3#10596427direct (N × H)
589. C3C3×#5912semidirect

Maximal subgroups

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

subgroupindex
C3×C4×D3×A42
C3×#642592
C32×C4⋊S42
C3×#642092
C3×Dic3×S42
C3×#59123
#105964273
C3×#56063
C3×#37954