Finite Group Databasebeta

← S4×Dic3⋊C4 #11921965

Subgroups — S4×Dic3⋊C4 #11921965

Order 1152.

864 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C2×C3×C4×S4C211yesmaximal, characteristic
2C2×Dic3×S4C211yesmaximal, characteristic
3C2×Dic3×S4C211yesmaximal, characteristic
4A4×Dic3⋊C4C211yesmaximal, characteristic, complemented
5#10593645C211yesmaximal, characteristic, complemented
6Dic3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
7#10593238C211yesmaximal, characteristic, complemented
8C4⋊C4×S4—31nomaximal
9D4×Dic3⋊C4—31nomaximal
10C22×C3×S4C2211yescharacteristic
11C2×C3×C4×A4C2211yescharacteristic
12C2×Dic3×A4C2211yescharacteristic
13C2×Dic3×A4C2211yescharacteristic
14C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
15C2×C3⋊SL(2,ℤ/4)C2211yescharacteristic
16C2×C3×SL(2,ℤ/4)C2211yescharacteristic
17C3×C4×S4—22no—
18Dic3×S4—22no—
19Dic3×S4C414yescomplemented
20D3×Dic3⋊C4—41nomaximal
21C2×C4×S4—31no—
22C2×C4×S4—31no—
23C2×C4×S4D311yescharacteristic
24C2×C3×C4×D4—31no—
25C2×D4×Dic3—31no—
26C2×D4×Dic3—31no—
27C22×Dic3⋊C4—31no—
28C22×Dic3⋊C4D311yescharacteristic, complemented
29A4×C4⋊C4—31no—
30C4⋊SL(2,ℤ/4)—31no—
31SL(2,ℤ/4)⋊C4—31no—
32SL(2,ℤ/4)⋊C4—31no—
33SL(2,ℤ/4)⋊C4—61no—
34#5210—31no—
35#5210—31no—
36#4975—31no—
37#4972—31no—
38#4946—31no—
39#4946—31no—
40#4942—31no—
41#4942—31no—
42C4×Dic3⋊C4—31no—
43#4928—31no—
44C22×C3×A4C2311yescharacteristic
45C2×C3×S4C2×C412yes—
46C2×C3×S4D412yes—
47C2×C3×S4Q812yes—
48C2×C3×C4×D3—41no—
49C3×C4×A4—21no—
50C2×D3×Dic3—41no—
51C2×D3×Dic3—41no—
52Dic3×A4—21no—
53Dic3×A4C2×C412yescomplemented
54C3⋊SL(2,ℤ/4)—21no—
55C3⋊SL(2,ℤ/4)C2×C412yescomplemented
56C3×SL(2,ℤ/4)—21no—
57#3750—41no—
58C3×Dic3⋊C4—41no—
59#3724—41no—
60Dic3⋊Dic3—41no—
61D4×C4⋊C4—91noSylow
62C22×S4C2×D311yescharacteristic
63C22×C3×D4—31no—
64C23×C3×C4—31no—
65C23×C3×C4C2×D311yescharacteristic
66C23×Dic3—31no—
67C23×Dic3—31no—
68C23×Dic3C2×D311yescharacteristic
69C23×Dic3C2×D311yescharacteristic
70C2×C4×A4—21no—
71C2×C4×A4—31no—
72C2×C4×A4—31no—
73C2×C4×A4C2×D311yescharacteristic
74C2×SL(2,ℤ/4)—31no—
75C2×SL(2,ℤ/4)—31no—
76C2×SL(2,ℤ/4)—61no—
77C2×SL(2,ℤ/4)—61no—
78C2×SL(2,ℤ/4)C2×D311yescharacteristic
79C4×S4—22no—
80C4×S4—34no—
81C4×S4—62no—
82C3×C4×D4—62no—
83C3×C4×D4—62no—
84C2×C3×C4⋊C4—31no—
85C2×C3×C22⋊C4—31no—
86C2×C3×C22⋊C4—31no—
87C2×C3×C42—31no—
88C2×#217—31no—
89C2×#217—31no—
90C2×#217—31no—
91C2×#217—31no—
92D4×Dic3—34no—
93D4×Dic3—34no—
94D4×Dic3—62no—
95D4×Dic3—62no—
96C2×C4⋊Dic3—31no—
97C2×C4⋊Dic3—31no—
98C2×Dic3⋊C4—31no—
99C2×Dic3⋊C4—31no—
100C2×Dic3⋊C4—31no—
101C2×Dic3⋊C4—31no—
102C2×Dic3⋊C4—31no—
103C2×Dic3⋊C4—31no—
104C2×Dic3⋊C4—31no—
105C2×Dic3⋊C4—31no—
106C2×Dic3⋊C4—31no—
107C2×Dic3⋊C4—31no—
108C2×Dic3⋊C4—61no—
109C2×Dic3⋊C4—61no—
110C2×Dic3⋊C4—61no—
111C2×Dic3⋊C4—61no—
112C2×C4×Dic3—31no—
113C2×C4×Dic3—31no—
114D3×C4⋊C4—121no—
115C3⋊#95—31no—
116C3⋊#95—31no—
117C3⋊#95—31no—
118C3⋊#95—31no—
119C3⋊#95—31no—
120C3⋊#95—31no—
121C22×C3×D3—41no—
122C2×C3×A4C22×C411yescharacteristic
123C2×C3×A4C2×D411yescharacteristic
124C2×C3×A4C2×Q811yescharacteristic
125C3×S4C4⋊C414yescomplemented
126C2×C32×C4—41no—
127C2×C3⋊Dic3—41no—
128C2×C3⋊Dic3—41no—
129C2×C3×Dic3—41no—
130C2×C3×Dic3—41no—
131C2×C3×Dic3—41no—
132C3×C4×D3—82no—
133D3×Dic3—44no—
134D3×Dic3—82no—
135C2×C4×D4—31no—
136C2×C4×D4—91no—
137C2×C4×D4—91no—
138C22×C4⋊C4—31no—
139C22×C4⋊C4—91no—
140#389—91no—
141#389—91no—
142#385—91no—
143#385—91no—
144#385—91no—
145#385—91no—
146#384—91no—
147#380—91no—
148#380—91no—
149C4×C4⋊C4—91no—
150C24×C3—31no—
151C24×C3C22×D311yescharacteristic
152C22×A4—21no—
153C22×A4C22×D311yescharacteristic
154C2×S4Dic612yes—
155C2×S4C4×D312yes—
156C2×S4#6712yes—
157C2×C3×D4—32no—
158C2×C3×D4—32no—
159C2×C3×D4—32no—
160C2×C3×D4—32no—
161C2×C3×D4—32no—
162C2×C3×D4—32no—
163C22×C3×C4—21no—
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C22×C3×C4—31no—
167C22×C3×C4—31no—
168C22×C3×C4—31no—
169C22×C3×C4—61no—
170C22×C3×C4—61no—
171C22×C3×C4—62no—
172C22×C3×C4—61no—
173C22×C3×C4—121no—
174C22×Dic3—21no—
175C22×Dic3—31no—
176C22×Dic3—31no—
177C22×Dic3—32no—
178C22×Dic3—31no—
179C22×Dic3—31no—
180C22×Dic3—34no—
181C22×Dic3—31no—
182C22×Dic3—31no—
183C22×Dic3—31no—
184C22×Dic3—31no—
185C22×Dic3—61no—
186C22×Dic3—62no—
187C22×Dic3—61no—
188C22×Dic3—61no—
189C22×Dic3—61no—
190C22×Dic3—62no—
191C22×Dic3—61no—
192C22×Dic3—121no—
193C22×Dic3C4×D312yescomplemented
194C2×C4×D3—41no—
195C2×C4×D3—121no—
196C2×C4×D3—121no—
197C4×A4—21no—
198C4×A4—32no—
199C4×A4—41no—
200C4×A4—61no—
201SL(2,ℤ/4)—21no—
202SL(2,ℤ/4)—32no—
203SL(2,ℤ/4)—62no—
204SL(2,ℤ/4)—61no—
205SL(2,ℤ/4)—121no—
206C3×C4⋊C4—62no—
207C3×C4⋊C4—121no—
208C3×C22⋊C4—62no—
209C3×C22⋊C4—61no—
210C3×C22⋊C4—61no—
211C3×C42—62no—
212#217—34no—
213#217—32no—
214#217—32no—
215#217—62no—
216#217—61no—
217#217—61no—
218C4⋊Dic3—34no—
219C4⋊Dic3—62no—
220C4⋊Dic3—121no—
221Dic3⋊C4—31no—
222Dic3⋊C4—31no—
223Dic3⋊C4—31no—
224Dic3⋊C4—61no—
225Dic3⋊C4—62no—
226Dic3⋊C4—61no—
227Dic3⋊C4—61no—
228Dic3⋊C4—61no—
229Dic3⋊C4—61no—
230Dic3⋊C4—61no—
231Dic3⋊C4—61no—
232Dic3⋊C4—121no—
233Dic3⋊C4—121no—
234Dic3⋊C4—121no—
235Dic3⋊C4—121no—
236Dic3⋊C4—241no—
237Dic3⋊C4S411yescharacteristic, direct product factor
238C4×Dic3—34no—
239C4×Dic3—62no—
240C22×C32—41no—
241C2×C3×D3—42no—
242C2×C3×D3—42no—
243C2×C3×D3—42no—
244C3×A4C2×C4⋊C411yescharacteristic, complemented
245C32×C4—81no—
246C3⋊Dic3—42no—
247C3⋊Dic3—81no—
248C3×Dic3—42no—
249C3×Dic3—81no—
250C3×Dic3—81no—
251C22×D4—31no—
252C23×C4—31no—
253C23×C4—31no—
254C23×C4—31no—
255C23×C4—91no—
256C23×C4—91no—
257C23×C4D3211yescharacteristic
258C4×D4—62no—
259C4×D4—62no—
260C4×D4—94no—
261C4×D4—94no—
262C4×D4—182no—
263C4×D4—182no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—91no—
266C2×C4⋊C4—91no—
267C2×C4⋊C4—91no—
268C2×C4⋊C4—91no—
269C2×C4⋊C4—91no—
270C2×C4⋊C4—91no—
271C2×C4⋊C4—91no—
272C2×C4⋊C4—91no—
273C2×C4⋊C4—91no—
274C2×C4⋊C4—91no—
275C2×C4⋊C4—91no—
276C2×C4⋊C4—91no—
277C2×C4⋊C4—181no—
278C2×C4⋊C4—181no—
279C2×C4⋊C4—181no—
280C2×C4⋊C4—181no—
281C2×C22⋊C4—31no—
282C2×C22⋊C4—31no—
283C2×C22⋊C4—91no—
284C2×C22⋊C4—91no—
285C2×C22⋊C4—91no—
286C2×C22⋊C4—91no—
287C2×C42—31no—
288C2×C42—91no—
289C2×C42—91no—
290#95—91no—
291#95—91no—
292#95—91no—
293#95—91no—
294#95—91no—
295#95—91no—
296C23×C3—31no—
297C23×C3—31no—
298C23×C3—31no—
299C23×C3—31no—
300C23×C3—32no—
301C23×C3—32no—
302C23×C3—32no—
303C23×C3—61no—
304C23×C3—61no—
305C23×C3—61no—
306C23×C3—61no—
307C23×C3C2×C4×D311yescharacteristic
308C23×C3D3×D411yescharacteristic
309C23×C3D3×Q811yescharacteristic
310C22×D3—41no—
311C2×A4—21no—
312C2×A4—21no—
313C2×A4—21no—
314C2×A4C2×Dic611yescharacteristic
315C2×A4C2×C4×D311yescharacteristic
316C2×A4C2×#6711yescharacteristic
317S4Dic3⋊C414yesdirect product factor
318C3×D4—34no—
319C3×D4—34no—
320C3×D4—34no—
321C3×D4—34no—
322C2×C3×C4—32no—
323C2×C3×C4—32no—
324C2×C3×C4—32no—
325C2×C3×C4—31no—
326C2×C3×C4—41no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—81no—
338C2×C3×C4—121no—
339C2×C3×C4—121no—
340C2×C3×C4—122no—
341C2×C3×C4—122no—
342C2×C3×C4C2×S411yescharacteristic
343C2×Dic3—31no—
344C2×Dic3—32no—
345C2×Dic3—32no—
346C2×Dic3—32no—
347C2×Dic3—32no—
348C2×Dic3—32no—
349C2×Dic3—31no—
350C2×Dic3—41no—
351C2×Dic3—61no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—64no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—64no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—121no—
369C2×Dic3—121no—
370C2×Dic3—122no—
371C2×Dic3—122no—
372C2×Dic3—241no—
373C2×Dic3—241no—
374C2×Dic3C2×S411yescharacteristic
375C2×Dic3C2×S411yescharacteristic
376C4×D3—82no—
377C4×D3—124no—
378C4×D3—242no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—41no—
382C3×D3—44no—
383C24—31no—
384C24C2×D3211yescharacteristic
385C2×D4—32no—
386C2×D4—32no—
387C2×D4—32no—
388C2×D4—32no—
389C2×D4—32no—
390C2×D4—32no—
391C22×C4—21no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—31no—
397C22×C4—32no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—62no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—91no—
404C22×C4—91no—
405C22×C4—91no—
406C22×C4—92no—
407C22×C4—91no—
408C22×C4—91no—
409C22×C4—94no—
410C22×C4—91no—
411C22×C4—91no—
412C22×C4—91no—
413C22×C4—121no—
414C22×C4—181no—
415C22×C4—181no—
416C22×C4—181no—
417C22×C4—182no—
418C22×C4—181no—
419C22×C4—181no—
420C22×C4—182no—
421C22×C4—361no—
422C4⋊C4—31no—
423C4⋊C4—62no—
424C4⋊C4—91no—
425C4⋊C4—91no—
426C4⋊C4—94no—
427C4⋊C4—91no—
428C4⋊C4—181no—
429C4⋊C4—181no—
430C4⋊C4—181no—
431C4⋊C4—181no—
432C4⋊C4—182no—
433C4⋊C4—181no—
434C4⋊C4—181no—
435C4⋊C4—182no—
436C4⋊C4—181no—
437C4⋊C4—361no—
438C4⋊C4—361no—
439C22⋊C4—62no—
440C22⋊C4—61no—
441C22⋊C4—61no—
442C22⋊C4—94no—
443C22⋊C4—92no—
444C22⋊C4—92no—
445C22⋊C4—182no—
446C22⋊C4—181no—
447C22⋊C4—181no—
448C42—62no—
449C42—94no—
450C42—182no—
451C22×C3—31no—
452C22×C3—31no—
453C22×C3—31no—
454C22×C3—31no—
455C22×C3—31no—
456C22×C3—31no—
457C22×C3—31no—
458C22×C3—31no—
459C22×C3—31no—
460C22×C3—34no—
461C22×C3—41no—
462C22×C3—61no—
463C22×C3—62no—
464C22×C3—62no—
465C22×C3—62no—
466C22×C3—62no—
467C22×C3—62no—
468C22×C3—62no—
469C22×C3—81no—
470C22×C3D3×C4⋊C411yescharacteristic, complemented
471C22×C3C22×S411yescharacteristic
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—42no—
475A4—21no—
476A4C2×Dic3⋊C411yescharacteristic, complemented
477C3×C4—21no—
478C3×C4—34no—
479C3×C4—61no—
480C3×C4—81no—
481C3×C4—122no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—161no—
485C3×C4—241no—
486Dic3—21no—
487Dic3—32no—
488Dic3—62no—
489Dic3—62no—
490Dic3—61no—
491Dic3—81no—
492Dic3—122no—
493Dic3—121no—
494Dic3—121no—
495Dic3—242no—
496Dic3—241no—
497Dic3—481no—
498Dic3C4×S412yescomplemented
499C32—41noSylow
500C23—31no—
501C23—31no—
502C23—31no—
503C23—31no—
504C23—32no—
505C23—32no—
506C23—32no—
507C23—61no—
508C23—61no—
509C23—61no—
510C23—61no—
511C23D3×Dic611yescharacteristic
512C23C4×D3211yescharacteristic
513C23D3×#6711yescharacteristic
514D4—34no—
515D4—34no—
516D4—34no—
517D4—34no—
518C2×C4—32no—
519C2×C4—32no—
520C2×C4—32no—
521C2×C4—31no—
522C2×C4—31no—
523C2×C4—31no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—92no—
535C2×C4—92no—
536C2×C4—91no—
537C2×C4—92no—
538C2×C4—92no—
539C2×C4—92no—
540C2×C4—91no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—184no—
544C2×C4—181no—
545C2×C4—181no—
546C2×C4—181no—
547C2×C4—184no—
548C2×C4—181no—
549C2×C4—181no—
550C2×C4—181no—
551C2×C4—181no—
552C2×C4—181no—
553C2×C4—181no—
554C2×C4—181no—
555C2×C4—181no—
556C2×C4—181no—
557C2×C4—181no—
558C2×C4—181no—
559C2×C4—181no—
560C2×C4—362no—
561C2×C4—362no—
562C2×C4D3×S411yescharacteristic
563C2×C3—31no—
564C2×C3—31no—
565C2×C3—31no—
566C2×C3—31no—
567C2×C3—41no—
568C2×C3—41no—
569C2×C3—41no—
570C2×C3—64no—
571C2×C3—81no—
572C2×C3—81no—
573C2×C3—81no—
574C2×C3C2×C4×S411yescharacteristic
575C2×C3D4×S411yescharacteristic
576C2×C3Q8×S411yescharacteristic
577D3—44no—
578C22—31no—
579C22—31no—
580C22—31no—
581C22—31no—
582C22—31no—
583C22—31no—
584C22—31no—
585C22—31no—
586C22—31no—
587C22—34no—
588C22—61no—
589C22—62no—
590C22—62no—
591C22—62no—
592C22—62no—
593C22—62no—
594C22—62no—
595C22D3×Dic3⋊C411yescharacteristic, complemented
596C22C2×D3×S411yescharacteristic
597C4—21no—
598C4—34no—
599C4—32no—
600C4—61no—
601C4—61no—
602C4—92no—
603C4—121no—
604C4—121no—
605C4—182no—
606C4—182no—
607C4—181no—
608C4—361no—
609C4—361no—
610C3—41no—
611C3—81no—
612C3C4⋊C4×S411yescharacteristic, complemented
613C2—31no—
614C2—31no—
615C2—31no—
616C2—31no—
617C2—64no—
618C2Dic6×S411yescharacteristic
619C2C4×D3×S411yescharacteristic
620C2#67×S411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutatorC2×C3×A4C22×C4
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC22×C3×S4C22
cosocleC22×C3×A4C23
hypercenterC2×C4D3×S4
hypocenterC3×A4C2×C4⋊C4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. A4×Dic3⋊C4C2semidirect
5. #10593645C2semidirect
6. Dic3⋊SL(2,ℤ/4)C2semidirect
7. #10593238C2semidirect
19. Dic3×S4C4semidirect
28. C22×Dic3⋊C4D3semidirect
53. Dic3×A4C2×C4semidirect
55. C3⋊SL(2,ℤ/4)C2×C4semidirect
125. C3×S4C4⋊C4semidirect
193. C22×Dic3C4×D3semidirect
237. Dic3⋊C4S4direct (N × H)
244. C3×A4C2×C4⋊C4semidirect
317. S4Dic3⋊C4direct (N × H)
470. C22×C3D3×C4⋊C4semidirect
476. A4C2×Dic3⋊C4semidirect
498. Dic3C4×S4semidirect
595. C22D3×Dic3⋊C4semidirect
612. C3C4⋊C4×S4semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×S42
C2×Dic3×S42
A4×Dic3⋊C42
#105936452
Dic3⋊SL(2,ℤ/4)2
#105932382
C4⋊C4×S43
D4×Dic3⋊C43
D3×Dic3⋊C44