Finite Group Databasebeta

← D3×C4⋊(C3⋊S4) #13180800

Subgroups — D3×C4⋊(C3⋊S4) #13180800

Order 1728.

1300 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1300 conjugacy classes of subgroups — too many to draw. Only its 102 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×D3×C3⋊S4C212yesmaximal, complemented
2C3×C4×D3×A4C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊(C3⋊S4))C211yesmaximal, characteristic, complemented
4C3×C4⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
5#11724202C212yesmaximal, complemented
6C2×C4⋊(C3⋊S4)—31nomaximal
7D3×C4⋊S4—33nomaximal
8D3×#930—31nomaximal
9C2×C3×D3×A4C2211yescharacteristic
10C2×C3⋊(C3⋊S4)C2212yescomplemented
11C2×C3×C3⋊S4C2212yescomplemented
12D3×C3⋊S4—24no—
13C32×C4×A4C2211yescharacteristic, complemented
14D3×C4⋊(C3⋊D3)—41nomaximal
15C3×Dic3×A4C2211yescharacteristic, complemented
16C22×C3⋊S4—32no—
17C2×D3×S4—36no—
18C22×C3×C4×D3D311yescharacteristic, complemented
19C2×C3×C4×A4—31no—
20C2×D3×#67—32no—
21C2×D3×D12—31no—
22C4×D3×A4D313yescomplemented
23C4⋊(C3⋊S4)—31no—
24C4⋊(C3⋊S4)—33no—
25C4⋊(C3⋊S4)—32no—
26C4⋊(C3⋊S4)—61no—
27C4⋊(C3⋊S4)—63no—
28C4⋊(C3⋊S4)D311yescharacteristic, direct product factor
29C3×C4⋊S4—33no—
30#64207—36no—
31#64139—31no—
32C3×#930—31no—
33#63978—32no—
34D3×(C2×D3)⋊C4—32no—
35#63909—31no—
36#63908—31no—
37#63906—32no—
38D3×C4⋊Dic3—31no—
39C2×C32×A4C2311yescharacteristic
40C2×D3×C3⋊D3—42no—
41C3×D3×A4D412yescomplemented
42C3⋊(C3⋊S4)—22no—
43C3×C3⋊S4—22no—
44C4⋊(C3⋊(C3⋊D3))—41no—
45C3×C4⋊(C3⋊D3)—41no—
46C32×C4×D3—41no—
47#6292—42no—
48C2×C4⋊S4—93no—
49C2×#930—91no—
50D3×#121—91no—
51C23×C3×D3C2×D311yescharacteristic
52C22×C3×A4—31no—
53C22×D32—32no—
54C2×D3×A4C2×D313yes—
55C2×C3⋊S4—32no—
56C2×C3⋊S4—36no—
57C2×C3⋊S4—64no—
58C2×C3⋊S4—62no—
59C2×C3⋊S4—66no—
60C2×C3⋊S4C2×D312yescomplemented
61C2×C3×S4—36no—
62D3×S4—612no—
63C22×C32×C4C2×D311yescharacteristic, complemented
64C2×#638—32no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—121no—
67C2×C3×#67—32no—
68C22×C3×Dic3C2×D311yescharacteristic, complemented
69C2×C3×D12—31no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—31no—
74C3×C4×A4—21no—
75C3×C4×A4—23no—
76C3×C4×A4—31no—
77C3×C4×A4C2×D313yescomplemented
78C3×C4×A4C2×D311yescharacteristic, complemented
79D3×#67—64no—
80D3×#67—64no—
81C2×#626—32no—
82C2×#626—32no—
83C2×#625—32no—
84C2×D3×Dic3—32no—
85D3×D12—61no—
86D3×D12—61no—
87D3×D12—61no—
88D3×D12—61no—
89D3×D12—123no—
90Dic3×A4C2×D313yescomplemented
91#3752—32no—
92C4⋊(C3⋊Dic3)—31no—
93C3×(C2×D3)⋊C4—32no—
94C3×C4⋊Dic3—31no—
95Dic3⋊Dic3—32no—
96#3722—32no—
97(C2×D3)⋊Dic3—32no—
98C2×C3⋊(C3⋊D3)—42no—
99C2×C3×C3⋊D3—42no—
100C2×C32×D3—41no—
101C32×A4C2×D411yescharacteristic, complemented
102D3×C3⋊D3—84no—
103C33×C4—41no—
104C32×Dic3—41no—
105C22×S4—96no—
106C23×C3×C4—31no—
107C22×#67—92no—
108C2×D3×D4—91no—
109C2×D3×D4—92no—
110C22×D12—91no—
111C22×C4×D3C3⋊D311yescharacteristic, complemented
112C2×C4×A4—33no—
113C4⋊S4—33no—
114C4⋊S4—96no—
115C4⋊S4—93no—
116C4⋊S4—183no—
117C3×#121—91no—
118#939—92no—
119#938—91no—
120#930—31no—
121#930—91no—
122#930—92no—
123#930—91no—
124#930—92no—
125#930—91no—
126#930—91no—
127#930—181no—
128C2×(C2×D3)⋊C4—92no—
129C2×C4⋊Dic3—91no—
130#895—91no—
131D3×C4⋊C4—91no—
132#884—92no—
133D3×C22⋊C4—92no—
134C23×C32C22×D311yescharacteristic
135C22×C3⋊D3—32no—
136C22×C3⋊D3—122no—
137C22×C3×D3—31no—
138C22×C3×D3—31no—
139C22×C3×D3—32no—
140C22×C3×D3—32no—
141C22×C3×D3D1212yescomplemented
142C2×C3×A4—21no—
143C2×C3×A4—23no—
144C2×C3×A4—32no—
145C2×C3×A4C22×D313yes—
146C2×C3×A4C22×D311yescharacteristic
147C2×D32—64no—
148C2×D32—62no—
149C2×D32—64no—
150C2×D32—62no—
151C2×D32—126no—
152D3×A4D1216yescomplemented
153C3⋊S4—22no—
154C3⋊S4—62no—
155C3⋊S4—66no—
156C3⋊S4—122no—
157C3⋊S4—126no—
158C3×S4—66no—
159C2×C32×C4—31no—
160C2×C32×C4—31no—
161C2×C32×C4—121no—
162#638—62no—
163#638—62no—
164C2×C3⋊Dic3—32no—
165C4⋊(C3⋊D3)—41no—
166C4⋊(C3⋊D3)—61no—
167C4⋊(C3⋊D3)—61no—
168C4⋊(C3⋊D3)—121no—
169C4⋊(C3⋊D3)—122no—
170C4⋊(C3⋊D3)—123no—
171C4⋊(C3⋊D3)—241no—
172C4⋊(C3⋊D3)—243no—
173C3×#67—62no—
174C3×#67—62no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—32no—
178C3×D12—61no—
179C3×D12—61no—
180C3×D12—123no—
181C3×C4×D3—31no—
182C3×C4×D3—31no—
183C3×C4×D3—31no—
184C3×C4×D3—43no—
185C3×C4×D3—61no—
186C3×C4×D3S411yescharacteristic, complemented
187#626—62no—
188#626—64no—
189#626—62no—
190#626—126no—
191#625—64no—
192D3×Dic3—64no—
193C2×#121—271noSylow
194C2×C33—41no—
195C3⋊(C3⋊D3)—82no—
196C3×C3⋊D3—82no—
197C32×D3—42no—
198C24×C3—31no—
199C23×D3—92no—
200C23×D3—92no—
201C23×D3C2×C3⋊D311yescharacteristic
202C22×A4—33no—
203C2×S4—36no—
204C2×S4—96no—
205C2×S4—186no—
206C2×S4—1812no—
207C2×C3×D4—91no—
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×C4D3211yescharacteristic, complemented
216C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
217C2×#67—32no—
218C2×#67—92no—
219C2×#67—92no—
220C2×#67—92no—
221C2×#67—92no—
222C2×#67—92no—
223C2×#67—92no—
224C2×#67—182no—
225C2×#67—184no—
226C2×#67—184no—
227C22×Dic3—92no—
228C22×Dic3C2×C3⋊D311yescharacteristic, complemented
229D3×D4—181no—
230D3×D4—181no—
231D3×D4—181no—
232D3×D4—184no—
233D3×D4—184no—
234D3×D4—181no—
235C2×D12—31no—
236C2×D12—91no—
237C2×D12—91no—
238C2×D12—92no—
239C2×D12—92no—
240C2×D12—181no—
241C2×D12—181no—
242C2×D12—181no—
243C2×D12—181no—
244C2×D12—181no—
245C2×D12—363no—
246C2×C4×D3—31no—
247C2×C4×D3—31no—
248C2×C4×D3—31no—
249C2×C4×D3—31no—
250C2×C4×D3—92no—
251C4×A4—23no—
252C4×A4—33no—
253C4×A4D3213yescomplemented
254C3×C4⋊C4—91no—
255C3×C22⋊C4—92no—
256#217—92no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—182no—
264C4⋊Dic3—31no—
265C4⋊Dic3—92no—
266C4⋊Dic3—91no—
267C4⋊Dic3—91no—
268C4⋊Dic3—181no—
269Dic3⋊C4—92no—
270C22×C32—31no—
271C22×C32—31no—
272C22×C32—121no—
273C22×C32C2×D1211yescharacteristic, complemented
274C2×C3⋊D3—42no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—62no—
277C2×C3⋊D3—62no—
278C2×C3⋊D3—122no—
279C2×C3⋊D3—126no—
280C2×C3⋊D3—242no—
281C2×C3⋊D3—246no—
282C2×C3⋊D3—244no—
283C2×C3×D3—31no—
284C2×C3×D3—32no—
285C2×C3×D3—32no—
286C2×C3×D3—32no—
287C2×C3×D3—43no—
288C2×C3×D3—61no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C2×C3×D3—126no—
293C2×C3×D3C2×S411yescharacteristic
294C3×A4—21no—
295C3×A4—23no—
296C3×A4C2×D1213yescomplemented
297C3×A4D3×D411yescharacteristic, complemented
298D32—124no—
299D32—124no—
300D32—2412no—
301C32×C4—31no—
302C32×C4—41no—
303C32×C4—43no—
304C32×C4—83no—
305C32×C4—81no—
306C32×C4—121no—
307C32×C4C2×S411yescharacteristic, complemented
308C3⋊Dic3—62no—
309C3×Dic3—31no—
310C3×Dic3—43no—
311C3×Dic3—62no—
312C3×Dic3C2×S411yescharacteristic, complemented
313C22×D4—271no—
314C22×D4—272no—
315C23×C4—31no—
316#121—91no—
317#121—272no—
318#121—271no—
319#121—272no—
320#121—271no—
321#121—271no—
322C2×C4⋊C4—271no—
323C2×C22⋊C4—272no—
324C33—41noSylow
325C23×C3—21no—
326C23×C3—32no—
327C23×C3—91no—
328C23×C3—91no—
329C23×C3—92no—
330C23×C3—92no—
331C23×C3C2×D3211yescharacteristic
332C23×C3C22×C3⋊D311yescharacteristic
333C22×D3—31no—
334C22×D3—31no—
335C22×D3—32no—
336C22×D3—32no—
337C22×D3—92no—
338C22×D3—92no—
339C22×D3—182no—
340C22×D3—182no—
341C22×D3—184no—
342C22×D3—184no—
343C22×D3—182no—
344C22×D3—182no—
345C22×D3—184no—
346C22×D3—184no—
347C22×D3—182no—
348C22×D3—366no—
349C22×D3C4⋊(C3⋊D3)12yescomplemented
350C2×A4—23no—
351C2×A4—36no—
352C2×A4C2×D3213yes—
353S4—66no—
354S4—186no—
355S4—366no—
356C3×D4—181no—
357C3×D4—182no—
358C3×D4—182no—
359C3×D4—181no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—91no—
368C2×C3×C4—91no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—91no—
372C2×C3×C4—92no—
373C2×C3×C4—123no—
374C2×C3×C4—181no—
375#67—62no—
376#67—62no—
377#67—182no—
378#67—184no—
379#67—184no—
380#67—184no—
381#67—182no—
382#67—182no—
383#67—182no—
384#67—182no—
385#67—182no—
386#67—362no—
387#67—362no—
388C2×Dic3—31no—
389C2×Dic3—31no—
390C2×Dic3—32no—
391C2×Dic3—92no—
392C2×Dic3—92no—
393C2×Dic3—182no—
394C2×Dic3—184no—
395D12—61no—
396D12—61no—
397D12—123no—
398D12—182no—
399D12—181no—
400D12—181no—
401D12—182no—
402D12—184no—
403D12—181no—
404D12—181no—
405D12—361no—
406D12—363no—
407D12—361no—
408D12—366no—
409D12—723no—
410C4×D3—31no—
411C4×D3—31no—
412C4×D3—31no—
413C4×D3—61no—
414C4×D3—184no—
415C4×D3C3⋊S411yescharacteristic, complemented
416C2×C32—31no—
417C2×C32—31no—
418C2×C32—41no—
419C2×C32—43no—
420C2×C32—83no—
421C2×C32—81no—
422C2×C32—122no—
423C2×C32C22×S411yescharacteristic
424C3⋊D3—82no—
425C3⋊D3—122no—
426C3⋊D3—242no—
427C3⋊D3—246no—
428C3⋊D3—482no—
429C3⋊D3—486no—
430C3×D3—32no—
431C3×D3—46no—
432C3×D3—122no—
433C3×D3—246no—
434C3×D3C4⋊S412yescomplemented
435C24—31no—
436C24—272no—
437C2×D4—91no—
438C2×D4—92no—
439C2×D4—271no—
440C2×D4—272no—
441C2×D4—272no—
442C2×D4—272no—
443C2×D4—272no—
444C2×D4—541no—
445C2×D4—541no—
446C2×D4—541no—
447C2×D4—541no—
448C2×D4—544no—
449C2×D4—544no—
450C22×C4—31no—
451C22×C4—91no—
452C22×C4—91no—
453C22×C4—91no—
454C22×C4—91no—
455C22×C4—272no—
456C22×C4D3×C3⋊D311yescharacteristic, complemented
457C4⋊C4—91no—
458C4⋊C4—271no—
459C4⋊C4—272no—
460C22⋊C4—92no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4—272no—
464C22×C3—21no—
465C22×C3—31no—
466C22×C3—31no—
467C22×C3—31no—
468C22×C3—31no—
469C22×C3—31no—
470C22×C3—61no—
471C22×C3—61no—
472C22×C3—92no—
473C22×C3—92no—
474C22×C3—92no—
475C22×C3—91no—
476C22×C3—123no—
477C22×C3—181no—
478C22×C3—182no—
479C22×C3—182no—
480C22×C3—182no—
481C22×C3D3×D1211yescharacteristic, complemented
482C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic, complemented
483C2×D3—31no—
484C2×D3—32no—
485C2×D3—32no—
486C2×D3—32no—
487C2×D3—61no—
488C2×D3—62no—
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—126no—
492C2×D3—182no—
493C2×D3—182no—
494C2×D3—182no—
495C2×D3—182no—
496C2×D3—182no—
497C2×D3—182no—
498C2×D3—362no—
499C2×D3—366no—
500C2×D3—362no—
501C2×D3—362no—
502C2×D3—364no—
503C2×D3—364no—
504C2×D3—364no—
505C2×D3—364no—
506C2×D3—726no—
507C2×D3—7212no—
508C2×D3C2×C3⋊S411yescharacteristic
509A4—23no—
510A4D3×D1213yescomplemented
511C3×C4—21no—
512C3×C4—31no—
513C3×C4—31no—
514C3×C4—31no—
515C3×C4—43no—
516C3×C4—61no—
517C3×C4—83no—
518C3×C4—91no—
519C3×C4—123no—
520C3×C4—182no—
521C3×C4D3×S411yescharacteristic, complemented
522C3×C4C2×C3⋊S411yescharacteristic, complemented
523Dic3—31no—
524Dic3—62no—
525Dic3—182no—
526Dic3—182no—
527Dic3—362no—
528Dic3C2×C3⋊S411yescharacteristic, complemented
529C32—41no—
530C32—43no—
531C32—83no—
532C32—81no—
533C32C2×C4⋊S411yescharacteristic, complemented
534C23—32no—
535C23—91no—
536C23—91no—
537C23—92no—
538C23—92no—
539C23—272no—
540C23—542no—
541C23—544no—
542C23—542no—
543C23—544no—
544C23C2×D3×C3⋊D311yescharacteristic
545D4—181no—
546D4—182no—
547D4—182no—
548D4—181no—
549D4—542no—
550D4—541no—
551D4—542no—
552D4—544no—
553D4—542no—
554D4—544no—
555D4—542no—
556D4—541no—
557C2×C4—31no—
558C2×C4—31no—
559C2×C4—31no—
560C2×C4—91no—
561C2×C4—91no—
562C2×C4—91no—
563C2×C4—91no—
564C2×C4—91no—
565C2×C4—92no—
566C2×C4—181no—
567C2×C4—272no—
568C2×C4—544no—
569C2×C3—21no—
570C2×C3—31no—
571C2×C3—31no—
572C2×C3—32no—
573C2×C3—31no—
574C2×C3—31no—
575C2×C3—43no—
576C2×C3—61no—
577C2×C3—61no—
578C2×C3—83no—
579C2×C3—92no—
580C2×C3—126no—
581C2×C3—362no—
582C2×C3C2×D3×S411yescharacteristic
583C2×C3C22×C3⋊S411yescharacteristic
584D3—32no—
585D3—122no—
586D3—246no—
587D3—362no—
588D3—362no—
589D3—722no—
590D3—726no—
591D3—1446no—
592D3C4⋊(C3⋊S4)12yesdirect product factor
593C22—31no—
594C22—31no—
595C22—31no—
596C22—92no—
597C22—92no—
598C22—92no—
599C22—91no—
600C22—181no—
601C22—182no—
602C22—182no—
603C22—182no—
604C22—542no—
605C22—542no—
606C22—542no—
607C22—1084no—
608C22—1084no—
609C22D3×C4⋊(C3⋊D3)11yescharacteristic, complemented
610C4—31no—
611C4—31no—
612C4—91no—
613C4—182no—
614C4—542no—
615C4D3×C3⋊S411yescharacteristic, complemented
616C3—21no—
617C3—43no—
618C3—83no—
619C3D3×C4⋊S411yescharacteristic, complemented
620C3C2×C4⋊(C3⋊S4)11yescharacteristic, complemented
621C2—31no—
622C2—31no—
623C2—32no—
624C2—92no—
625C2—362no—
626C2—1082no—
627C2C2×D3×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×C3⋊S4
commutator, cosocleC2×C32×A4C23
FittingC22×C32×C4C2×D3
socleC23×C32C22×D3
hypercenterC4D3×C3⋊S4
hypocenterC32×A4C2×D4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×C3⋊S4C2semidirect
2. C3×C4×D3×A4C2semidirect
3. C4⋊(C3⋊(C3⋊S4))C2semidirect
4. C3×C4⋊(C3⋊S4)C2semidirect
5. #11724202C2semidirect
10. C2×C3⋊(C3⋊S4)C22semidirect
11. C2×C3×C3⋊S4C22semidirect
13. C32×C4×A4C22semidirect
15. C3×Dic3×A4C22semidirect
18. C22×C3×C4×D3D3semidirect
22. C4×D3×A4D3semidirect
28. C4⋊(C3⋊S4)D3direct (N × H)
41. C3×D3×A4D4semidirect
60. C2×C3⋊S4C2×D3semidirect
63. C22×C32×C4C2×D3semidirect
68. C22×C3×Dic3C2×D3semidirect
77. C3×C4×A4C2×D3semidirect
78. C3×C4×A4C2×D3semidirect
90. Dic3×A4C2×D3semidirect
101. C32×A4C2×D4semidirect
111. C22×C4×D3C3⋊D3semidirect
141. C22×C3×D3D12semidirect
152. D3×A4D12semidirect
186. C3×C4×D3S4semidirect
215. C22×C3×C4D32semidirect
216. C22×C3×C4C2×C3⋊D3semidirect
228. C22×Dic3C2×C3⋊D3semidirect
253. C4×A4D32semidirect
273. C22×C32C2×D12semidirect
296. C3×A4C2×D12semidirect
297. C3×A4D3×D4semidirect
307. C32×C4C2×S4semidirect
312. C3×Dic3C2×S4semidirect
349. C22×D3C4⋊(C3⋊D3)semidirect
415. C4×D3C3⋊S4semidirect
434. C3×D3C4⋊S4semidirect
456. C22×C4D3×C3⋊D3semidirect
481. C22×C3D3×D12semidirect
482. C22×C3C2×C4⋊(C3⋊D3)semidirect
510. A4D3×D12semidirect
521. C3×C4D3×S4semidirect
522. C3×C4C2×C3⋊S4semidirect
528. Dic3C2×C3⋊S4semidirect
533. C32C2×C4⋊S4semidirect
592. D3C4⋊(C3⋊S4)direct (N × H)
609. C22D3×C4⋊(C3⋊D3)semidirect
615. C4D3×C3⋊S4semidirect
619. C3D3×C4⋊S4semidirect
620. C3C2×C4⋊(C3⋊S4)semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C3⋊S42
C3×C4×D3×A42
C4⋊(C3⋊(C3⋊S4))2
C3×C4⋊(C3⋊S4)2
#117242022
C2×C4⋊(C3⋊S4)3
D3×C4⋊S43
D3×#9303
D3×C4⋊(C3⋊D3)4