Finite Group Databasebeta

← C3×#10596420 #13180679

Subgroups — C3×#10596420 #13180679

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⋊S4C211yesmaximal, characteristic, complemented
3C32×C4⋊S4C211yesmaximal, characteristic, complemented
4C3×#64207C212yesmaximal, complemented
5C3×Dic3×S4C212yesmaximal, complemented
6C3×#5919—31nomaximal
7#10596420C311yesmaximal, characteristic, direct product factor
8C3×#5598—31nomaximal
9C2×C3×C3⋊S4C2211yescharacteristic
10C2×C32×S4C2212yes—
11C32×C4×A4C2211yescharacteristic
12C3×#3796—41nomaximal
13C3×Dic3×A4C2212yes—
14C3×C3⋊SL(2,ℤ/4)C2211yescharacteristic
15C2×C3×#237—31no—
16C22×C3×Dic6D311yescharacteristic, complemented
17C3×Q8×A4—31no—
18A4×Dic6—21no—
19A4×Dic6C2×C311yescharacteristic, complemented
20C4×C3⋊S4C2×C311yescharacteristic, complemented
21C3×C4⋊S4—21no—
22C3×C4⋊S4—32no—
23C3×C4⋊S4D311yescharacteristic, complemented
24C3×C4⋊S4C2×C311yescharacteristic, complemented
25C3×C4×S4—31no—
26C3×C4×S4—32no—
27C3×C4×S4—61no—
28#64207C2×C312yescomplemented
29Dic3×S4C2×C312yescomplemented
30C32×#121—31no—
31C3×#939—32no—
32C3×#936—31no—
33C3×D4×Dic3—32no—
34C3×C4×#67—31no—
35C3×#897—31no—
36C3×#887—31no—
37C3×#885—32no—
38C3×#878—32no—
39C2×C32×A4C2311yescharacteristic
40C3×C3⋊S4—21no—
41C32×S4—22no—
42C3×C4×C3⋊D3—41no—
43C32×D12—41no—
44C32×Dic6—41no—
45C3×#626—42no—
46C3×D3×Dic3—42no—
47#5919—31no—
48C3×#548—91no—
49#5598—31no—
50C2×C3⋊S4C22×C311yescharacteristic
51C2×C3×S4—22no—
52C2×C3×S4—31no—
53C2×C3×S4—61no—
54C2×C3×S4C2×D312yes—
55C2×C3×S4C22×C312yes—
56C2×C32×D4—31no—
57C2×C32×D4—32no—
58C22×C32×C4C2×D311yescharacteristic
59C2×C3×#67—32no—
60C2×C3×#67—31no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3C2×D312yes—
63C3×#239—121no—
64C3×#237—61no—
65C3×#237—62no—
66C3×#237—61no—
67C2×C3×C4×D3—31no—
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#3796—41no—
79Dic3×A4—22no—
80Dic3×A4C22×C312yes—
81C3⋊SL(2,ℤ/4)C22×C311yescharacteristic
82C3×SL(2,ℤ/4)—31no—
83C3×SL(2,ℤ/4)—61no—
84C32×C4⋊C4—31no—
85C32×C22⋊C4—32no—
86C3×#217—32no—
87C3×#217—32no—
88C3×#217—31no—
89C3×(C2×D3)⋊C4—31no—
90C3×(C2×D3)⋊C4—32no—
91C3×C4⋊Dic3—32no—
92C3×Dic3⋊C4—31no—
93C3×Dic3⋊C4—32no—
94C3×C4×Dic3—32no—
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×Q8—31no—
104C2×#237—31no—
105C22×Dic6C3×D311yescharacteristic, complemented
106Q8×A4—31no—
107Q8×A4—61no—
108C4⋊S4—32no—
109C4⋊S4C3×D311yescharacteristic, complemented
110C4×S4—32no—
111C4×S4—31no—
112C4×S4—61no—
113C3×#124—92no—
114C3×#124—91no—
115C3×C22⋊Q8—92no—
116C3×C22⋊Q8—91no—
117C3×#121—31no—
118C3×#121—31no—
119C3×#121—61no—
120C3×#121—92no—
121C3×C4×Q8—91no—
122C3×C4×D4—92no—
123C3×C4×D4—91no—
124#939—32no—
125#936—31no—
126D4×Dic3—32no—
127C4×#67—31no—
128#897—31no—
129#887—31no—
130#885—32no—
131#878—32no—
132C23×C32—32no—
133C23×C32C22×D311yescharacteristic
134C22×C3×D3—31no—
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—21no—
142C3×S4—22no—
143C3×S4—22no—
144C3×S4—42no—
145C3×S4—62no—
146C3×S4—61no—
147C32×Q8—121no—
148C32×D4—61no—
149C32×D4—62no—
150C32×D4—62no—
151C32×D4—61no—
152C2×C32×C4—31no—
153C2×C32×C4—31no—
154C2×C32×C4—32no—
155C4×C3⋊D3—41no—
156C3×#67—62no—
157C3×#67—61no—
158C3×#67—61no—
159C3×#67—62no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—62no—
167C3×D12—41no—
168C3×D12—41no—
169C3×D12—81no—
170C3×D12—122no—
171C3×C4×D3—61no—
172C3×C4×D3—61no—
173C3×C4×D3—121no—
174C3×C4×D3—122no—
175C3×C4×D3—241no—
176C3×Dic6—31no—
177C3×Dic6—32no—
178C3×Dic6—41no—
179C3×Dic6—61no—
180C3×Dic6—81no—
181C3×Dic6S411yescharacteristic, complemented
182#626—42no—
183D3×Dic3—42no—
184#548—91noSylow
185C2×C33—41no—
186C3×C3⋊D3—81no—
187C32×D3—82no—
188C2×S4—31no—
189C2×S4—61no—
190C2×S4C2×C3×D312yes—
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—32no—
200C2×C3×D4—32no—
201C2×C3×D4—62no—
202C2×C3×D4—61no—
203C2×C3×D4—92no—
204C2×C3×D4—91no—
205C22×C3×C4—21no—
206C22×C3×C4—32no—
207C22×C3×C4—91no—
208C22×C3×C4—92no—
209C22×C3×C4D3211yescharacteristic
210C22×C3×C4C2×C3×D311yescharacteristic
211C2×#67—32no—
212C2×#67—31no—
213C22×Dic3—32no—
214C22×Dic3C2×C3×D312yes—
215#239—121no—
216#237—61no—
217#237—62no—
218#237—61no—
219C2×C4×D3—31no—
220C2×Dic6—31no—
221C2×Dic6—32no—
222C2×Dic6—31no—
223C4×A4—21no—
224C4×A4—21no—
225C4×A4—32no—
226C4×A4—41no—
227C4×A4—62no—
228C4×A4C2×C3×D311yescharacteristic
229SL(2,ℤ/4)—31no—
230SL(2,ℤ/4)—61no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C22⋊C4—32no—
238C3×C22⋊C4—32no—
239C3×C22⋊C4—62no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—92no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—91no—
245C3×C42—91no—
246C3×C42—92no—
247#217—32no—
248#217—32no—
249#217—31no—
250(C2×D3)⋊C4—31no—
251(C2×D3)⋊C4—32no—
252C4⋊Dic3—32no—
253Dic3⋊C4—31no—
254Dic3⋊C4—32no—
255C4×Dic3—32no—
256C4×Dic3—31no—
257C22×C32—31no—
258C22×C32—31no—
259C22×C32—62no—
260C22×C32—62no—
261C22×C32—62no—
262C22×C32#23911yescharacteristic, complemented
263C2×C3⋊D3—41no—
264C2×C3×D3—42no—
265C2×C3×D3—42no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—82no—
270C2×C3×D3—121no—
271C2×C3×D3—241no—
272C3×A4—21no—
273C3×A4—21no—
274C3×A4—21no—
275C3×A4—41no—
276C3×A4#23711yescharacteristic, complemented
277C3×A4C3×#4111yescharacteristic, complemented
278C32×C4—31no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—62no—
282C32×C4—81no—
283C32×C4—81no—
284C32×C4—81no—
285C32×C4—122no—
286C32×C4—161no—
287C32×C4C2×S411yescharacteristic
288C3⋊Dic3—41no—
289C3×Dic3—32no—
290C3×Dic3—42no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3—61no—
294C3×Dic3—82no—
295C3×Dic3—121no—
296C3×Dic3—241no—
297C3×Dic3C2×S412yes—
298C2×#41—91no—
299C22×Q8—31no—
300#124—92no—
301#124—91no—
302C22⋊Q8—92no—
303C22⋊Q8—91no—
304#121—31no—
305#121—92no—
306C4×Q8—91no—
307C4×D4—92no—
308C4×D4—91no—
309C33—41noSylow
310C23×C3—21no—
311C23×C3—32no—
312C23×C3—32no—
313C23×C3—62no—
314C23×C3—91no—
315C23×C3C2×D3211yescharacteristic
316C23×C3C22×C3×D311yescharacteristic
317C22×D3—31no—
318C2×A4—21no—
319C2×A4—21no—
320C2×A4—41no—
321C2×A4C22×C3×D311yescharacteristic
322S4—22no—
323S4—62no—
324S4—61no—
325C3×Q8—31no—
326C3×Q8—92no—
327C3×Q8—91no—
328C3×Q8—121no—
329C3×Q8—181no—
330C3×Q8—241no—
331C3×D4—61no—
332C3×D4—61no—
333C3×D4—62no—
334C3×D4—62no—
335C3×D4—62no—
336C3×D4—62no—
337C3×D4—61no—
338C3×D4—61no—
339C3×D4—121no—
340C3×D4—121no—
341C3×D4—122no—
342C3×D4—122no—
343C3×D4—182no—
344C3×D4—182no—
345C3×D4—181no—
346C3×D4—181no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—32no—
352C2×C3×C4—32no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—62no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—182no—
362C2×C3×C4—181no—
363C2×C3×C4—182no—
364C2×C3×C4—181no—
365#67—62no—
366#67—61no—
367#67—61no—
368#67—62no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—31no—
372C2×Dic3—31no—
373C2×Dic3—31no—
374C2×Dic3—62no—
375C2×Dic3—62no—
376D12—41no—
377D12—122no—
378C4×D3—61no—
379C4×D3—61no—
380C4×D3—121no—
381C4×D3—122no—
382C4×D3—241no—
383Dic6—31no—
384Dic6—32no—
385Dic6—61no—
386Dic6C3×S411yescharacteristic, complemented
387C2×C32—31no—
388C2×C32—31no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32—81no—
392C2×C32—81no—
393C2×C32—81no—
394C2×C32—122no—
395C2×C32—161no—
396C2×C32C22×S411yescharacteristic
397C3⋊D3—81no—
398C3×D3—82no—
399C3×D3—82no—
400C3×D3—121no—
401C3×D3—162no—
402C3×D3—241no—
403C3×D3—242no—
404#41—181no—
405#41—182no—
406#41—181no—
407C2×Q8—91no—
408C2×Q8—91no—
409C2×Q8—92no—
410C2×D4—31no—
411C2×D4—32no—
412C2×D4—92no—
413C2×D4—91no—
414C22×C4—32no—
415C22×C4—91no—
416C22×C4—92no—
417C22×C4C3×D3211yescharacteristic
418C4⋊C4—31no—
419C4⋊C4—91no—
420C4⋊C4—92no—
421C4⋊C4—92no—
422C22⋊C4—32no—
423C22⋊C4—91no—
424C22⋊C4—92no—
425C22⋊C4—92no—
426C22⋊C4—92no—
427C22⋊C4—91no—
428C42—91no—
429C42—92no—
430C22×C3—21no—
431C22×C3—31no—
432C22×C3—31no—
433C22×C3—31no—
434C22×C3—31no—
435C22×C3—61no—
436C22×C3—61no—
437C22×C3—62no—
438C22×C3—62no—
439C22×C3—62no—
440C22×C3—62no—
441C22×C3—62no—
442C22×C3—62no—
443C22×C3—122no—
444C22×C3—122no—
445C22×C3—122no—
446C22×C3—181no—
447C22×C3—181no—
448C22×C3—181no—
449C22×C3#379611yescharacteristic, complemented
450C22×C3C3×#23911yescharacteristic, complemented
451C2×D3—42no—
452C2×D3—61no—
453C2×D3—61no—
454C2×D3—61no—
455C2×D3—121no—
456C2×D3—241no—
457A4—21no—
458A4—21no—
459A4—41no—
460A4C3×#23711yescharacteristic, complemented
461C3×C4—21no—
462C3×C4—31no—
463C3×C4—32no—
464C3×C4—31no—
465C3×C4—41no—
466C3×C4—61no—
467C3×C4—62no—
468C3×C4—62no—
469C3×C4—81no—
470C3×C4—81no—
471C3×C4—92no—
472C3×C4—122no—
473C3×C4—122no—
474C3×C4—161no—
475C3×C4—181no—
476C3×C4—181no—
477C3×C4—181no—
478C3×C4—242no—
479C3×C4D3×S411yescharacteristic
480C3×C4C2×C3×S411yescharacteristic
481Dic3—32no—
482Dic3—61no—
483Dic3—61no—
484Dic3—61no—
485Dic3—121no—
486Dic3—241no—
487Dic3C2×C3×S412yes—
488C32—41no—
489C32—41no—
490C32—81no—
491C32—81no—
492C32—81no—
493C32—161no—
494C32#591911yescharacteristic, complemented
495C23—32no—
496C23—91no—
497C23C2×C3×D3211yescharacteristic
498Q8—31no—
499Q8—92no—
500Q8—91no—
501Q8—181no—
502D4—61no—
503D4—62no—
504D4—62no—
505D4—61no—
506D4—182no—
507D4—182no—
508D4—181no—
509D4—181no—
510C2×C4—31no—
511C2×C4—31no—
512C2×C4—32no—
513C2×C4—92no—
514C2×C4—92no—
515C2×C4—91no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—182no—
519C2×C4—181no—
520C2×C4—182no—
521C2×C4—181no—
522C2×C3—21no—
523C2×C3—31no—
524C2×C3—31no—
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—41no—
528C2×C3—61no—
529C2×C3—61no—
530C2×C3—81no—
531C2×C3—81no—
532C2×C3—122no—
533C2×C3—122no—
534C2×C3—161no—
535C2×C3—242no—
536C2×C3—361no—
537C2×C3C2×D3×S411yescharacteristic
538C2×C3C22×C3×S411yescharacteristic
539D3—82no—
540D3—121no—
541D3—241no—
542D3—242no—
543C22—31no—
544C22—31no—
545C22—62no—
546C22—62no—
547C22—62no—
548C22—181no—
549C22—181no—
550C22—181no—
551C22C3×#379611yescharacteristic, complemented
552C4—31no—
553C4—32no—
554C4—62no—
555C4—92no—
556C4—181no—
557C4—181no—
558C4—181no—
559C4C3×D3×S411yescharacteristic
560C3—21no—
561C3—41no—
562C3—81no—
563C3—81no—
564C3—161no—
565C3#1059642011yescharacteristic, direct product factor
566C3C3×#591911yescharacteristic, complemented
567C2—31no—
568C2—31no—
569C2—122no—
570C2—361no—
571C2C2×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⋊S4C2semidirect
3. C32×C4⋊S4C2semidirect
4. C3×#64207C2semidirect
5. C3×Dic3×S4C2semidirect
7. #10596420C3direct (N × H)
16. C22×C3×Dic6D3semidirect
19. A4×Dic6C2×C3semidirect
20. C4×C3⋊S4C2×C3semidirect
23. C3×C4⋊S4D3semidirect
24. C3×C4⋊S4C2×C3semidirect
28. #64207C2×C3semidirect
29. Dic3×S4C2×C3semidirect
98. C32×A4#41semidirect
105. C22×Dic6C3×D3semidirect
109. C4⋊S4C3×D3semidirect
181. C3×Dic6S4semidirect
262. C22×C32#239semidirect
276. C3×A4#237semidirect
277. C3×A4C3×#41semidirect
386. Dic6C3×S4semidirect
449. C22×C3#3796semidirect
450. C22×C3C3×#239semidirect
460. A4C3×#237semidirect
494. C32#5919semidirect
551. C22C3×#3796semidirect
565. C3#10596420direct (N × H)
566. 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⋊S42
C32×C4⋊S42
C3×#642072
C3×Dic3×S42
C3×#59193
#105964203
C3×#55983
C3×#37964