Finite Group Databasebeta

← C3×#10596421 #13180680

Subgroups — C3×#10596421 #13180680

Order 1728.

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

normal subgroups only

This group has 776 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×Dic6C211yesmaximal, characteristic, complemented
2C3×C4⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
3C32×C4×S4C211yesmaximal, characteristic, complemented
4C3×#64207C212yesmaximal, complemented
5C3×(C3⋊S4)⋊C4C212yesmaximal, complemented
6C3×#5919—31nomaximal
7#10596421C311yesmaximal, characteristic, direct product factor
8C3×#5552—31nomaximal
9C2×C3×C3⋊S4C2212yes—
10C2×C32×S4C2211yescharacteristic
11C32×C4×A4C2211yescharacteristic
12C3×#3799—41nomaximal
13C3×Dic3×A4C2212yes—
14C32×SL(2,ℤ/4)C2211yescharacteristic
15C2×C3×#235—31no—
16C22×C3×Dic6D311yescharacteristic, complemented
17C3×Q8×A4—31no—
18A4×Dic6—21no—
19A4×Dic6C2×C311yescharacteristic, complemented
20C4⋊(C3⋊S4)C2×C311yescharacteristic, complemented
21C3×C4⋊S4—31no—
22C3×C4⋊S4—32no—
23C3×C4⋊S4—61no—
24C3×C4×S4—21no—
25C3×C4×S4—32no—
26C3×C4×S4D311yescharacteristic, complemented
27C3×C4×S4C2×C311yescharacteristic, complemented
28#64207C2×C312yescomplemented
29(C3⋊S4)⋊C4C2×C312yescomplemented
30C32×C4×D4—31no—
31C3×#939—32no—
32C3×#930—31no—
33C3×#924—31no—
34C3×#896—32no—
35C3×#885—32no—
36C3×#67⋊C4—32no—
37C3×#875—31no—
38C3×C4×Dic6—31no—
39C2×C32×A4C2311yescharacteristic
40C3×C3⋊S4—22no—
41C32×S4—21no—
42C3×C4⋊(C3⋊D3)—41no—
43C32×C4×D3—41no—
44C32×Dic6—41no—
45C3×#626—42no—
46C3×#624—42no—
47#5919—31no—
48C3×#548—91no—
49#5552—31no—
50C2×C3⋊S4C22×C312yes—
51C2×C3×S4—21no—
52C2×C3×S4—32no—
53C2×C3×S4—62no—
54C2×C3×S4C2×D311yescharacteristic
55C2×C3×S4C22×C311yescharacteristic
56C2×C32×D4—31no—
57C22×C32×C4—31no—
58C22×C32×C4C2×D311yescharacteristic
59C2×C3×#67—32no—
60C2×C3×#67—32no—
61C22×C3×Dic3C2×D312yes—
62C3×#239—121no—
63C3×#235—61no—
64C3×#235—62no—
65C3×#235—61no—
66C2×C3×D12—31no—
67C2×C3×C4×D3—32no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—32no—
70C2×C3×Dic6—31no—
71C3×C4×A4—21no—
72C3×C4×A4—21no—
73C3×C4×A4—21no—
74C3×C4×A4—32no—
75C3×C4×A4—41no—
76C3×C4×A4C2×D311yescharacteristic
77C3×C4×A4C22×C311yescharacteristic
78#3799—41no—
79Dic3×A4—22no—
80Dic3×A4C22×C312yes—
81C3×SL(2,ℤ/4)—21no—
82C3×SL(2,ℤ/4)C2×D311yescharacteristic
83C3×SL(2,ℤ/4)C22×C311yescharacteristic
84C32×C4⋊C4—31no—
85C32×C22⋊C4—31no—
86C32×C22⋊C4—31no—
87C32×C42—31no—
88C3×#217—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×C4⋊Dic3—31no—
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4—32no—
95C3×C4×Dic3—32no—
96C2×C3×C3⋊D3—42no—
97C2×C32×D3—41no—
98C32×A4#4111yescharacteristic, complemented
99C33×C4—41no—
100C32×Dic3—41no—
101C32×Dic3—42no—
102C2×C3×#41—91no—
103C22×C3×Q8—31no—
104C2×#235—31no—
105C22×Dic6C3×D311yescharacteristic, complemented
106Q8×A4—31no—
107Q8×A4—61no—
108C4⋊S4—32no—
109C4⋊S4—31no—
110C4⋊S4—61no—
111C4×S4—32no—
112C4×S4C3×D311yescharacteristic, complemented
113C3×#124—91no—
114C3×#124—92no—
115C3×C22⋊Q8—92no—
116C3×C22⋊Q8—91no—
117C3×#121—92no—
118C3×#121—91no—
119C3×C4×Q8—91no—
120C3×C4×D4—31no—
121C3×C4×D4—31no—
122C3×C4×D4—61no—
123C3×C4×D4—92no—
124#939—32no—
125#930—31no—
126#924—31no—
127#896—32no—
128#885—32no—
129#67⋊C4—32no—
130#875—31no—
131C4×Dic6—31no—
132C23×C32—31no—
133C23×C32C22×D311yescharacteristic
134C22×C3×D3—32no—
135C2×C3×A4—21no—
136C2×C3×A4—21no—
137C2×C3×A4—21no—
138C2×C3×A4—41no—
139C2×C3×A4C22×D311yescharacteristic
140C2×C3×A4C23×C311yescharacteristic
141C3⋊S4—22no—
142C3×S4—22no—
143C3×S4—21no—
144C3×S4—21no—
145C3×S4—62no—
146C3×S4—122no—
147C32×Q8—121no—
148C32×D4—61no—
149C32×D4—61no—
150C2×C32×C4—31no—
151C2×C32×C4—31no—
152C2×C32×C4—31no—
153C2×C32×C4—31no—
154C2×C32×C4—31no—
155C2×C32×C4—61no—
156C2×C32×C4—61no—
157C4⋊(C3⋊D3)—41no—
158C3×#67—62no—
159C3×#67—62no—
160C3×#67—62no—
161C3×#67—62no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C3×D12—61no—
166C3×D12—61no—
167C3×D12—121no—
168C3×D12—122no—
169C3×D12—241no—
170C3×C4×D3—41no—
171C3×C4×D3—41no—
172C3×C4×D3—62no—
173C3×C4×D3—62no—
174C3×C4×D3—81no—
175C3×C4×D3—122no—
176C3×Dic6—31no—
177C3×Dic6—32no—
178C3×Dic6—41no—
179C3×Dic6—61no—
180C3×Dic6—81no—
181C3×Dic6S411yescharacteristic, complemented
182#626—42no—
183#624—42no—
184#548—91noSylow
185C2×C33—41no—
186C3×C3⋊D3—82no—
187C32×D3—81no—
188C2×S4—32no—
189C2×S4—62no—
190C2×S4C2×C3×D311yescharacteristic
191C3×#41—181no—
192C3×#41—182no—
193C3×#41—181no—
194C2×C3×Q8—91no—
195C2×C3×Q8—91no—
196C2×C3×Q8—92no—
197C2×C3×D4—31no—
198C2×C3×D4—31no—
199C2×C3×D4—61no—
200C2×C3×D4—91no—
201C2×C3×D4—92no—
202C2×C3×D4—92no—
203C22×C3×C4—21no—
204C22×C3×C4—31no—
205C22×C3×C4—31no—
206C22×C3×C4—32no—
207C22×C3×C4—61no—
208C22×C3×C4—92no—
209C22×C3×C4D3211yescharacteristic
210C22×C3×C4C2×C3×D311yescharacteristic
211C2×#67—32no—
212C2×#67—32no—
213C22×Dic3C2×C3×D312yes—
214#239—121no—
215#235—61no—
216#235—62no—
217#235—61no—
218C2×D12—31no—
219C2×C4×D3—32no—
220C2×Dic6—31no—
221C2×Dic6—31no—
222C2×Dic6—32no—
223C4×A4—21no—
224C4×A4—21no—
225C4×A4—32no—
226C4×A4—41no—
227C4×A4—62no—
228C4×A4C2×C3×D311yescharacteristic
229SL(2,ℤ/4)C2×C3×D311yescharacteristic
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—91no—
236C3×C22⋊C4—31no—
237C3×C22⋊C4—31no—
238C3×C22⋊C4—31no—
239C3×C22⋊C4—31no—
240C3×C22⋊C4—61no—
241C3×C22⋊C4—61no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C42—31no—
247C3×C42—31no—
248C3×C42—61no—
249C3×C42—92no—
250#217—32no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254C4⋊Dic3—31no—
255Dic3⋊C4—32no—
256Dic3⋊C4—32no—
257C4×Dic3—32no—
258C22×C32—31no—
259C22×C32—31no—
260C22×C32—61no—
261C22×C32—61no—
262C22×C32—61no—
263C22×C32#23911yescharacteristic, complemented
264C2×C3⋊D3—42no—
265C2×C3×D3—41no—
266C2×C3×D3—41no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—81no—
271C2×C3×D3—122no—
272C2×C3×D3—242no—
273C3×A4—21no—
274C3×A4—21no—
275C3×A4—21no—
276C3×A4—41no—
277C3×A4#23511yescharacteristic, complemented
278C3×A4C3×#4111yescharacteristic, complemented
279C32×C4—31no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4—61no—
283C32×C4—61no—
284C32×C4—61no—
285C32×C4—81no—
286C32×C4—81no—
287C32×C4—81no—
288C32×C4—122no—
289C32×C4—161no—
290C32×C4C2×S411yescharacteristic
291C3×Dic3—32no—
292C3×Dic3—41no—
293C3×Dic3—41no—
294C3×Dic3—42no—
295C3×Dic3—62no—
296C3×Dic3—81no—
297C3×Dic3—82no—
298C3×Dic3C2×S412yes—
299C2×#41—91no—
300C22×Q8—31no—
301#124—91no—
302#124—92no—
303C22⋊Q8—92no—
304C22⋊Q8—91no—
305#121—92no—
306#121—91no—
307C4×Q8—91no—
308C4×D4—31no—
309C4×D4—92no—
310C33—41noSylow
311C23×C3—21no—
312C23×C3—31no—
313C23×C3—31no—
314C23×C3—61no—
315C23×C3—92no—
316C23×C3C2×D3211yescharacteristic
317C23×C3C22×C3×D311yescharacteristic
318C22×D3—32no—
319C2×A4—21no—
320C2×A4—21no—
321C2×A4—41no—
322C2×A4C22×C3×D311yescharacteristic
323S4—21no—
324S4—62no—
325S4—122no—
326C3×Q8—31no—
327C3×Q8—92no—
328C3×Q8—91no—
329C3×Q8—121no—
330C3×Q8—181no—
331C3×Q8—241no—
332C3×D4—62no—
333C3×D4—61no—
334C3×D4—61no—
335C3×D4—62no—
336C3×D4—61no—
337C3×D4—61no—
338C3×D4—182no—
339C3×D4—181no—
340C3×D4—182no—
341C3×D4—182no—
342C3×D4—182no—
343C3×D4—181no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—92no—
364C2×C3×C4—92no—
365C2×C3×C4—92no—
366C2×C3×C4—121no—
367C2×C3×C4—121no—
368C2×C3×C4—182no—
369C2×C3×C4—182no—
370#67—62no—
371#67—62no—
372#67—62no—
373#67—62no—
374C2×Dic3—32no—
375C2×Dic3—32no—
376C2×Dic3—32no—
377D12—61no—
378D12—61no—
379D12—121no—
380D12—122no—
381D12—241no—
382C4×D3—41no—
383C4×D3—62no—
384C4×D3—62no—
385C4×D3—122no—
386Dic6—31no—
387Dic6—32no—
388Dic6—61no—
389Dic6C3×S411yescharacteristic, complemented
390C2×C32—31no—
391C2×C32—31no—
392C2×C32—41no—
393C2×C32—41no—
394C2×C32—81no—
395C2×C32—81no—
396C2×C32—81no—
397C2×C32—121no—
398C2×C32—161no—
399C2×C32C22×S411yescharacteristic
400C3⋊D3—82no—
401C3×D3—81no—
402C3×D3—81no—
403C3×D3—82no—
404C3×D3—122no—
405C3×D3—242no—
406C3×D3—482no—
407#41—181no—
408#41—182no—
409#41—181no—
410C2×Q8—91no—
411C2×Q8—91no—
412C2×Q8—92no—
413C2×D4—31no—
414C2×D4—91no—
415C2×D4—92no—
416C2×D4—92no—
417C22×C4—31no—
418C22×C4—32no—
419C22×C4—92no—
420C22×C4C3×D3211yescharacteristic
421C4⋊C4—31no—
422C4⋊C4—92no—
423C4⋊C4—92no—
424C4⋊C4—91no—
425C22⋊C4—31no—
426C22⋊C4—31no—
427C22⋊C4—92no—
428C22⋊C4—92no—
429C22⋊C4—92no—
430C22⋊C4—92no—
431C42—31no—
432C42—92no—
433C22×C3—21no—
434C22×C3—31no—
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—61no—
439C22×C3—61no—
440C22×C3—61no—
441C22×C3—61no—
442C22×C3—61no—
443C22×C3—61no—
444C22×C3—62no—
445C22×C3—61no—
446C22×C3—61no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3—182no—
450C22×C3—182no—
451C22×C3—182no—
452C22×C3#379911yescharacteristic, complemented
453C22×C3C3×#23911yescharacteristic, complemented
454C2×D3—41no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—62no—
458C2×D3—122no—
459C2×D3—242no—
460A4—21no—
461A4—21no—
462A4—41no—
463A4C3×#23511yescharacteristic, complemented
464C3×C4—21no—
465C3×C4—31no—
466C3×C4—32no—
467C3×C4—31no—
468C3×C4—41no—
469C3×C4—61no—
470C3×C4—61no—
471C3×C4—61no—
472C3×C4—61no—
473C3×C4—61no—
474C3×C4—62no—
475C3×C4—61no—
476C3×C4—61no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4—92no—
480C3×C4—122no—
481C3×C4—121no—
482C3×C4—121no—
483C3×C4—161no—
484C3×C4—182no—
485C3×C4—242no—
486C3×C4D3×S411yescharacteristic
487C3×C4C2×C3×S411yescharacteristic
488Dic3—32no—
489Dic3—41no—
490Dic3—62no—
491Dic3C2×C3×S412yes—
492C32—41no—
493C32—41no—
494C32—81no—
495C32—81no—
496C32—81no—
497C32—161no—
498C32#591911yescharacteristic, complemented
499C23—31no—
500C23—92no—
501C23C2×C3×D3211yescharacteristic
502Q8—31no—
503Q8—92no—
504Q8—91no—
505Q8—181no—
506D4—61no—
507D4—61no—
508D4—182no—
509D4—181no—
510D4—182no—
511D4—182no—
512D4—182no—
513D4—181no—
514C2×C4—31no—
515C2×C4—31no—
516C2×C4—31no—
517C2×C4—31no—
518C2×C4—31no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—92no—
522C2×C4—92no—
523C2×C4—92no—
524C2×C4—182no—
525C2×C4—182no—
526C2×C3—21no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—41no—
532C2×C3—61no—
533C2×C3—61no—
534C2×C3—81no—
535C2×C3—81no—
536C2×C3—122no—
537C2×C3—121no—
538C2×C3—121no—
539C2×C3—161no—
540C2×C3—362no—
541C2×C3C2×D3×S411yescharacteristic
542C2×C3C22×C3×S411yescharacteristic
543D3—81no—
544D3—122no—
545D3—242no—
546D3—482no—
547C22—31no—
548C22—31no—
549C22—61no—
550C22—61no—
551C22—61no—
552C22—182no—
553C22—182no—
554C22—182no—
555C22C3×#379911yescharacteristic, complemented
556C4—31no—
557C4—32no—
558C4—61no—
559C4—61no—
560C4—61no—
561C4—92no—
562C4—182no—
563C4C3×D3×S411yescharacteristic
564C3—21no—
565C3—41no—
566C3—81no—
567C3—81no—
568C3—161no—
569C3#1059642111yescharacteristic, direct product factor
570C3C3×#591911yescharacteristic, complemented
571C2—31no—
572C2—31no—
573C2—121no—
574C2—362no—
575C2C2×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×Dic6C2semidirect
2. C3×C4⋊(C3⋊S4)C2semidirect
3. C32×C4×S4C2semidirect
4. C3×#64207C2semidirect
5. C3×(C3⋊S4)⋊C4C2semidirect
7. #10596421C3direct (N × H)
16. C22×C3×Dic6D3semidirect
19. A4×Dic6C2×C3semidirect
20. C4⋊(C3⋊S4)C2×C3semidirect
26. C3×C4×S4D3semidirect
27. C3×C4×S4C2×C3semidirect
28. #64207C2×C3semidirect
29. (C3⋊S4)⋊C4C2×C3semidirect
98. C32×A4#41semidirect
105. C22×Dic6C3×D3semidirect
112. C4×S4C3×D3semidirect
181. C3×Dic6S4semidirect
263. C22×C32#239semidirect
277. C3×A4#235semidirect
278. C3×A4C3×#41semidirect
389. Dic6C3×S4semidirect
452. C22×C3#3799semidirect
453. C22×C3C3×#239semidirect
463. A4C3×#235semidirect
498. C32#5919semidirect
555. C22C3×#3799semidirect
569. C3#10596421direct (N × H)
570. C3C3×#5919semidirect

Maximal subgroups

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

subgroupindex
C3×A4×Dic62
C3×C4⋊(C3⋊S4)2
C32×C4×S42
C3×#642072
C3×(C3⋊S4)⋊C42
C3×#59193
#105964213
C3×#55523
C3×#37994