Finite Group Databasebeta

← C3×Dic6×S4 #13180678

Subgroups — C3×Dic6×S4 #13180678

Order 1728.

740 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 90 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×#64259C211yesmaximal, characteristic, complemented
3C32×C4×S4C211yesmaximal, characteristic
4C3×Dic3×S4C212yesmaximal
5C3×#64204C212yesmaximal, complemented
6C3×Q8×S4—31nomaximal
7Dic6×S4C311yesmaximal, characteristic, direct product factor
8C3×D4×Dic6—31nomaximal
9C2×C32×S4C2211yescharacteristic
10C32×C4×A4C2211yescharacteristic
11C3×D3×Dic6—41nomaximal
12C3×Dic3×A4C2212yes—
13C3×C3⋊SL(2,ℤ/4)C2212yes—
14C32×SL(2,ℤ/4)C2211yescharacteristic
15C22×C3×Dic6—31no—
16C22×C3×Dic6D311yescharacteristic, complemented
17C3×Q8×A4—31no—
18A4×Dic6—21no—
19A4×Dic6C2×C311yescharacteristic, complemented
20#64259C2×C311yescharacteristic, complemented
21C3×C4×S4—21no—
22C3×C4×S4—32no—
23C3×C4×S4D311yescharacteristic
24C3×C4×S4C2×C311yescharacteristic
25C3×A4⋊Q8—31no—
26C3×A4⋊Q8—32no—
27C3×A4⋊Q8—61no—
28Dic3×S4C2×C312yes—
29#64204C2×C312yescomplemented
30C32×C4×D4—31no—
31C3×D4×Dic3—32no—
32C3×#924—31no—
33C3×#924—31no—
34C3×#888—32no—
35C3×#878—32no—
36C3×#878—32no—
37C3×#869—31no—
38C3×C4×Dic6—31no—
39C2×C32×A4C2311yescharacteristic
40C32×S4Q812yescomplemented
41C3×#634—41no—
42C32×C4×D3—41no—
43C32×Dic6—41no—
44C3×#627—42no—
45C3×D3×Dic3—42no—
46Q8×S4—31no—
47C3×D4×Q8—91no—
48D4×Dic6—31no—
49C2×C3×S4—21no—
50C2×C3×S4C2×D311yescharacteristic
51C2×C3×S4C22×C311yescharacteristic
52C2×C32×D4—31no—
53C22×C32×C4—31no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C3×Dic3—32no—
56C22×C3×Dic3C2×D312yes—
57C3×D3×Q8—121no—
58C2×C3×Dic6—31no—
59C2×C3×Dic6—32no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—32no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64C2×C3×Dic6—62no—
65C2×C3×Dic6—61no—
66C3×C4×A4—21no—
67C3×C4×A4—21no—
68C3×C4×A4—21no—
69C3×C4×A4—32no—
70C3×C4×A4—41no—
71C3×C4×A4C2×D311yescharacteristic
72C3×C4×A4C22×C311yescharacteristic
73D3×Dic6—41no—
74Dic3×A4—22no—
75Dic3×A4C22×C312yes—
76C3⋊SL(2,ℤ/4)C22×C312yes—
77C3×SL(2,ℤ/4)—21no—
78C3×SL(2,ℤ/4)—32no—
79C3×SL(2,ℤ/4)—62no—
80C3×SL(2,ℤ/4)C2×D311yescharacteristic
81C3×SL(2,ℤ/4)C22×C311yescharacteristic
82C32×C4⋊C4—31no—
83C32×C22⋊C4—31no—
84C32×C22⋊C4—31no—
85C32×C42—31no—
86C3×#217—32no—
87C3×#217—32no—
88C3×C4⋊Dic3—32no—
89C3×C4⋊Dic3—32no—
90C3×C4⋊Dic3—31no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94C3×C4×Dic3—32no—
95C2×C32×D3—41no—
96C32×A4C2×Q811yescharacteristic, complemented
97C33×C4—41no—
98C3×C3⋊Dic3—42no—
99C32×Dic3—41no—
100C32×Dic3—42no—
101C22×C3×Q8—31no—
102C22×C3×Q8—91no—
103C22×Dic6—31no—
104C22×Dic6C3×D311yescharacteristic, complemented
105Q8×A4—31no—
106Q8×A4—61no—
107C4×S4—32no—
108C4×S4C3×D311yescharacteristic
109A4⋊Q8—31no—
110A4⋊Q8—32no—
111A4⋊Q8—61no—
112C3×C4⋊Q8—91no—
113C3×C4⋊Q8—92no—
114C3×C22⋊Q8—92no—
115C3×C22⋊Q8—92no—
116C3×C22⋊Q8—91no—
117C3×C22⋊Q8—91no—
118C3×C4×Q8—91no—
119C3×C4×D4—31no—
120C3×C4×D4—31no—
121C3×C4×D4—61no—
122C3×C4×D4—92no—
123D4×Dic3—32no—
124#924—31no—
125#924—31no—
126#888—32no—
127#878—32no—
128#878—32no—
129#869—31no—
130C4×Dic6—31no—
131C23×C32—31no—
132C23×C32C22×D311yescharacteristic
133C2×C3×A4—21no—
134C2×C3×A4—21no—
135C2×C3×A4—21no—
136C2×C3×A4—41no—
137C2×C3×A4C22×D311yescharacteristic
138C2×C3×A4C23×C311yescharacteristic
139C3×S4—22no—
140C3×S4Dic612yesdirect product factor
141C3×S4C3×Q812yescomplemented
142C32×Q8—121no—
143C32×D4—32no—
144C32×D4—32no—
145C2×C32×C4—31no—
146C2×C32×C4—31no—
147C2×C32×C4—31no—
148C2×C32×C4—31no—
149C2×C32×C4—31no—
150C2×C32×C4—61no—
151C2×C32×C4—61no—
152#634—41no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—32no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—62no—
159C3×C4×D3—41no—
160C3×C4×D3—41no—
161C3×C4×D3—81no—
162C3×C4×D3—122no—
163C3×Dic6—31no—
164C3×Dic6—32no—
165C3×Dic6—41no—
166C3×Dic6—61no—
167C3×Dic6—62no—
168C3×Dic6—61no—
169C3×Dic6—61no—
170C3×Dic6—62no—
171C3×Dic6—81no—
172C3×Dic6—122no—
173C3×Dic6—121no—
174C3×Dic6—241no—
175C3×Dic6S411yescharacteristic, direct product factor
176#627—42no—
177D3×Dic3—42no—
178D4×Q8—91noSylow
179C2×C33—41no—
180C32×D3—42no—
181C2×S4C2×C3×D311yescharacteristic
182C2×C3×Q8—91no—
183C2×C3×Q8—91no—
184C2×C3×Q8—92no—
185C2×C3×Q8—91no—
186C2×C3×Q8—92no—
187C2×C3×Q8—181no—
188C2×C3×Q8—182no—
189C2×C3×Q8—181no—
190C2×C3×D4—31no—
191C2×C3×D4—31no—
192C2×C3×D4—61no—
193C22×C3×C4—21no—
194C22×C3×C4—31no—
195C22×C3×C4—31no—
196C22×C3×C4—32no—
197C22×C3×C4—61no—
198C22×C3×C4—92no—
199C22×C3×C4D3211yescharacteristic
200C22×C3×C4C2×C3×D311yescharacteristic
201C22×Dic3—32no—
202C22×Dic3C2×C3×D312yes—
203D3×Q8—121no—
204C2×Dic6—31no—
205C2×Dic6—31no—
206C2×Dic6—32no—
207C2×Dic6—31no—
208C2×Dic6—32no—
209C2×Dic6—61no—
210C2×Dic6—62no—
211C2×Dic6—61no—
212C4×A4—21no—
213C4×A4—21no—
214C4×A4—32no—
215C4×A4—41no—
216C4×A4—62no—
217C4×A4C2×C3×D311yescharacteristic
218SL(2,ℤ/4)—32no—
219SL(2,ℤ/4)—62no—
220SL(2,ℤ/4)C2×C3×D311yescharacteristic
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—92no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—92no—
230C3×C22⋊C4—31no—
231C3×C22⋊C4—31no—
232C3×C22⋊C4—31no—
233C3×C22⋊C4—31no—
234C3×C22⋊C4—61no—
235C3×C22⋊C4—61no—
236C3×C22⋊C4—92no—
237C3×C22⋊C4—92no—
238C3×C42—31no—
239C3×C42—31no—
240C3×C42—61no—
241C3×C42—92no—
242#217—32no—
243#217—32no—
244C4⋊Dic3—31no—
245C4⋊Dic3—32no—
246C4⋊Dic3—32no—
247Dic3⋊C4—32no—
248Dic3⋊C4—32no—
249Dic3⋊C4—32no—
250C4×Dic3—32no—
251C22×C32—31no—
252C22×C32—31no—
253C22×C32—32no—
254C22×C32—61no—
255C22×C32—61no—
256C22×C32D3×Q811yescharacteristic, complemented
257C2×C3×D3—41no—
258C2×C3×D3—41no—
259C2×C3×D3—81no—
260C3×A4—21no—
261C3×A4—21no—
262C3×A4—21no—
263C3×A4—41no—
264C3×A4C2×Dic611yescharacteristic, complemented
265C3×A4C2×C3×Q811yescharacteristic, complemented
266C32×C4—32no—
267C32×C4—31no—
268C32×C4—41no—
269C32×C4—41no—
270C32×C4—61no—
271C32×C4—61no—
272C32×C4—81no—
273C32×C4—81no—
274C32×C4—81no—
275C32×C4—122no—
276C32×C4—161no—
277C32×C4C2×S411yescharacteristic
278C3⋊Dic3—42no—
279C3×Dic3—32no—
280C3×Dic3—41no—
281C3×Dic3—41no—
282C3×Dic3—42no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—81no—
286C3×Dic3—82no—
287C3×Dic3—122no—
288C3×Dic3—242no—
289C3×Dic3C2×S412yes—
290C22×Q8—31no—
291C22×Q8—91no—
292C4⋊Q8—91no—
293C4⋊Q8—92no—
294C22⋊Q8—92no—
295C22⋊Q8—92no—
296C22⋊Q8—91no—
297C22⋊Q8—91no—
298C4×Q8—91no—
299C4×D4—31no—
300C4×D4—92no—
301C33—41noSylow
302C23×C3—21no—
303C23×C3—31no—
304C23×C3—31no—
305C23×C3—61no—
306C23×C3C2×D3211yescharacteristic
307C23×C3C22×C3×D311yescharacteristic
308C2×A4—21no—
309C2×A4—21no—
310C2×A4—41no—
311C2×A4C22×C3×D311yescharacteristic
312S4C3×Dic612yesdirect product factor
313C3×Q8—31no—
314C3×Q8—92no—
315C3×Q8—91no—
316C3×Q8—121no—
317C3×Q8—182no—
318C3×Q8—181no—
319C3×Q8—181no—
320C3×Q8—182no—
321C3×Q8—181no—
322C3×Q8—241no—
323C3×D4—32no—
324C3×D4—32no—
325C3×D4—32no—
326C3×D4—32no—
327C3×D4—62no—
328C3×D4—62no—
329C2×C3×C4—31no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—92no—
349C2×C3×C4—92no—
350C2×C3×C4—92no—
351C2×C3×C4—92no—
352C2×C3×C4—121no—
353C2×C3×C4—121no—
354C2×C3×C4—182no—
355C2×C3×C4—182no—
356C2×Dic3—32no—
357C2×Dic3—32no—
358C2×Dic3—32no—
359C2×Dic3—32no—
360C2×Dic3—62no—
361C2×Dic3—62no—
362C4×D3—41no—
363C4×D3—122no—
364Dic6—31no—
365Dic6—32no—
366Dic6—61no—
367Dic6—62no—
368Dic6—61no—
369Dic6—61no—
370Dic6—62no—
371Dic6—122no—
372Dic6—121no—
373Dic6—241no—
374Dic6C3×S411yescharacteristic, direct product factor
375C2×C32—31no—
376C2×C32—31no—
377C2×C32—41no—
378C2×C32—41no—
379C2×C32—62no—
380C2×C32—81no—
381C2×C32—81no—
382C2×C32—81no—
383C2×C32—161no—
384C2×C32C22×S411yescharacteristic
385C3×D3—42no—
386C3×D3—42no—
387C3×D3—82no—
388C2×Q8—91no—
389C2×Q8—91no—
390C2×Q8—92no—
391C2×Q8—91no—
392C2×Q8—92no—
393C2×Q8—181no—
394C2×Q8—182no—
395C2×Q8—181no—
396C2×D4—31no—
397C22×C4—31no—
398C22×C4—32no—
399C22×C4—92no—
400C22×C4C3×D3211yescharacteristic
401C4⋊C4—31no—
402C4⋊C4—92no—
403C4⋊C4—92no—
404C4⋊C4—91no—
405C4⋊C4—92no—
406C4⋊C4—92no—
407C4⋊C4—92no—
408C22⋊C4—31no—
409C22⋊C4—31no—
410C22⋊C4—92no—
411C22⋊C4—92no—
412C42—31no—
413C42—92no—
414C22×C3—21no—
415C22×C3—31no—
416C22×C3—31no—
417C22×C3—31no—
418C22×C3—31no—
419C22×C3—32no—
420C22×C3—32no—
421C22×C3—61no—
422C22×C3—61no—
423C22×C3—62no—
424C22×C3—61no—
425C22×C3—61no—
426C22×C3—61no—
427C22×C3—61no—
428C22×C3—121no—
429C22×C3—121no—
430C22×C3D3×Dic611yescharacteristic, complemented
431C22×C3C3×D3×Q811yescharacteristic, complemented
432C2×D3—41no—
433A4—21no—
434A4—21no—
435A4—41no—
436A4C2×C3×Dic611yescharacteristic, complemented
437C3×C4—21no—
438C3×C4—32no—
439C3×C4—32no—
440C3×C4—31no—
441C3×C4—32no—
442C3×C4—31no—
443C3×C4—41no—
444C3×C4—61no—
445C3×C4—62no—
446C3×C4—61no—
447C3×C4—61no—
448C3×C4—61no—
449C3×C4—61no—
450C3×C4—81no—
451C3×C4—81no—
452C3×C4—92no—
453C3×C4—122no—
454C3×C4—121no—
455C3×C4—121no—
456C3×C4—161no—
457C3×C4—182no—
458C3×C4—182no—
459C3×C4—242no—
460C3×C4D3×S411yescharacteristic
461C3×C4C2×C3×S411yescharacteristic
462Dic3—32no—
463Dic3—41no—
464Dic3—62no—
465Dic3—62no—
466Dic3—122no—
467Dic3—242no—
468Dic3C2×C3×S412yes—
469C32—41no—
470C32—41no—
471C32—81no—
472C32—81no—
473C32—81no—
474C32—161no—
475C32Q8×S411yescharacteristic, complemented
476C23—31no—
477C23C2×C3×D3211yescharacteristic
478Q8—31no—
479Q8—92no—
480Q8—91no—
481Q8—182no—
482Q8—181no—
483Q8—181no—
484Q8—182no—
485Q8—181no—
486D4—32no—
487D4—32no—
488C2×C4—31no—
489C2×C4—31no—
490C2×C4—31no—
491C2×C4—31no—
492C2×C4—31no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—92no—
496C2×C4—92no—
497C2×C4—92no—
498C2×C4—92no—
499C2×C4—182no—
500C2×C4—182no—
501C2×C3—21no—
502C2×C3—31no—
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—41no—
507C2×C3—61no—
508C2×C3—61no—
509C2×C3—62no—
510C2×C3—62no—
511C2×C3—81no—
512C2×C3—81no—
513C2×C3—122no—
514C2×C3—161no—
515C2×C3C2×D3×S411yescharacteristic
516C2×C3C22×C3×S411yescharacteristic
517D3—42no—
518C22—31no—
519C22—31no—
520C22—32no—
521C22—61no—
522C22—61no—
523C22C3×D3×Dic611yescharacteristic, complemented
524C4—32no—
525C4—31no—
526C4—32no—
527C4—61no—
528C4—61no—
529C4—92no—
530C4—182no—
531C4—182no—
532C4C3×D3×S411yescharacteristic
533C3—21no—
534C3—41no—
535C3—81no—
536C3—81no—
537C3—161no—
538C3Dic6×S411yescharacteristic, direct product factor
539C3C3×Q8×S411yescharacteristic, complemented
540C2—31no—
541C2—31no—
542C2—62no—
543C2C2×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
prime radicalC2×C32×S4C22
hypercenterC3×C4D3×S4
hypocenterC3×A4C2×C3×Q8
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×#64259C2semidirect
5. C3×#64204C2semidirect
7. Dic6×S4C3direct (N × H)
16. C22×C3×Dic6D3semidirect
19. A4×Dic6C2×C3semidirect
20. #64259C2×C3semidirect
29. #64204C2×C3semidirect
40. C32×S4Q8semidirect
96. C32×A4C2×Q8semidirect
104. C22×Dic6C3×D3semidirect
140. C3×S4Dic6direct (N × H)
141. C3×S4C3×Q8semidirect
175. C3×Dic6S4direct (N × H)
256. C22×C32D3×Q8semidirect
264. C3×A4C2×Dic6semidirect
265. C3×A4C2×C3×Q8semidirect
312. S4C3×Dic6direct (N × H)
374. Dic6C3×S4direct (N × H)
430. C22×C3D3×Dic6semidirect
431. C22×C3C3×D3×Q8semidirect
436. A4C2×C3×Dic6semidirect
475. C32Q8×S4semidirect
523. C22C3×D3×Dic6semidirect
538. C3Dic6×S4direct (N × H)
539. C3C3×Q8×S4semidirect

Maximal subgroups

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

subgroupindex
C3×A4×Dic62
C3×#642592
C32×C4×S42
C3×Dic3×S42
C3×#642042
C3×Q8×S43
Dic6×S43
C3×D4×Dic63
C3×D3×Dic64