Finite Group Databasebeta

← C3×D4×C3⋊S4 #13181376

Subgroups — C3×D4×C3⋊S4 #13181376

Order 1728.

1314 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C22×C3×C3⋊S4C212yesmaximal, complemented
2C32×D4×A4C211yesmaximal, characteristic, complemented
3C3×#64269C212yesmaximal, complemented
4C3×C4⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
5C3×C4×C3⋊S4C211yesmaximal, characteristic, complemented
6D4×C3⋊S4C311yesmaximal, characteristic, direct product factor
7C3×D4×S4—33nomaximal
8C3×D4×#67—31nomaximal
9C22×C32×A4C2212yescomplemented
10C2×C3×C3⋊S4—24no—
11C2×C3×C3⋊S4C2211yescharacteristic
12C2×C3×C3⋊S4C2212yescomplemented
13C3×D4×C3⋊D3—41nomaximal
14C32×C4×A4C2211yescharacteristic, complemented
15C3×C3⋊SL(2,ℤ/4)C2211yescharacteristic, complemented
16C22×C3⋊S4C2×C312yescomplemented
17C22×C3×S4—36no—
18C22×C32×D4D311yescharacteristic, complemented
19C22×C3×#67—32no—
20C2×C3×D3×D4—31no—
21C3×D4×A4—21no—
22C3×D4×A4—23no—
23C3×D4×A4D313yescomplemented
24C3×D4×A4C2×C311yescharacteristic, complemented
25#64269C2×C312yescomplemented
26C4⋊(C3⋊S4)C2×C311yescharacteristic, complemented
27C4×C3⋊S4C2×C311yescharacteristic, complemented
28C3×GL(2,ℤ/4)—36no—
29C3×C4⋊S4—33no—
30C3×C4×S4—33no—
31C3×#953—32no—
32C3×#940—31no—
33C3×#939—32no—
34C3×#938—31no—
35C3×#937—32no—
36C3×D4×Dic3—31no—
37C3×#930—31no—
38C3×C4×#67—31no—
39C22×C3×C3⋊D3—42no—
40C2×C32×A4—22no—
41C2×C32×A4C2311yescharacteristic
42C3×C3⋊S4—22no—
43C3×C3⋊S4D412yesdirect product factor
44C33×D4—41no—
45C3×#638—42no—
46C3×C4⋊(C3⋊D3)—41no—
47C3×C4×C3⋊D3—41no—
48D4×S4—33no—
49C3×D42—91no—
50D4×#67—31no—
51C24×C32C2×D312yescomplemented
52C23×C3×D3—32no—
53C22×C3×A4—22no—
54C22×C3×A4—26no—
55C22×C3×A4C2×D316yescomplemented
56C22×C3×A4C22×C312yescomplemented
57C2×C3⋊S4—24no—
58C2×C3⋊S4C22×C311yescharacteristic
59C2×C3⋊S4C22×C312yescomplemented
60C2×C3×S4—33no—
61C2×C3×S4—36no—
62C2×C3×S4—612no—
63C2×C32×D4—31no—
64C2×C32×D4—32no—
65C2×C32×D4—31no—
66C22×C32×C4C2×D311yescharacteristic, complemented
67D4×C3⋊D3—41no—
68C2×C3×#67—31no—
69C2×C3×#67—32no—
70C2×C3×#67—32no—
71C2×C3×#67—32no—
72C2×C3×#67—32no—
73C2×C3×#67—32no—
74C2×C3×#67—64no—
75C2×C3×#67—64no—
76C22×C3×Dic3—32no—
77C3×D3×D4—62no—
78C3×D3×D4—61no—
79C3×D3×D4—61no—
80C3×D3×D4—123no—
81C2×C3×D12—31no—
82C2×C3×C4×D3—31no—
83C3×C4×A4—21no—
84C3×C4×A4—23no—
85C3×C4×A4C2×D313yescomplemented
86C3×C4×A4C22×C311yescharacteristic, complemented
87C3⋊SL(2,ℤ/4)C22×C311yescharacteristic, complemented
88C3×SL(2,ℤ/4)—33no—
89C3×#217—32no—
90C3×#217—32no—
91C3×#217—31no—
92C3×(C2×D3)⋊C4—31no—
93C3×(C2×D3)⋊C4—32no—
94C3×C4⋊Dic3—31no—
95C3×Dic3⋊C4—31no—
96C3×C4×Dic3—31no—
97C22×C33—42no—
98C2×C3×C3⋊D3—41no—
99C2×C3×C3⋊D3—42no—
100C2×C3×C3⋊D3—84no—
101C32×A4C2×D411yescharacteristic, complemented
102C33×C4—41no—
103C3×C3⋊Dic3—41no—
104C22×S4—36no—
105C22×C3×D4—21no—
106C22×C3×D4—91no—
107C22×C3×D4—92no—
108C22×C3×D4C3×D311yescharacteristic, complemented
109C22×C3×D4C3⋊D311yescharacteristic, complemented
110C22×#67—32no—
111C2×D3×D4—31no—
112D4×A4—23no—
113D4×A4C3×D313yescomplemented
114GL(2,ℤ/4)—36no—
115C4⋊S4—33no—
116C4×S4—33no—
117C3×#127—91no—
118C3×#121—91no—
119C3×#121—92no—
120C3×#121—91no—
121C3×#120—92no—
122C3×#120—92no—
123C3×C4×D4—91no—
124C3×C4×D4—91no—
125#953—32no—
126#940—31no—
127#939—32no—
128#938—31no—
129#937—32no—
130D4×Dic3—31no—
131#930—31no—
132C4×#67—31no—
133C23×C32—22no—
134C23×C32—32no—
135C23×C32—32no—
136C23×C32—62no—
137C23×C32C22×D311yescharacteristic
138C22×C3⋊D3—42no—
139C22×C3×D3—31no—
140C22×C3×D3—32no—
141C22×C3×D3—64no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—126no—
145C22×C3×D3—122no—
146C2×C3×A4—22no—
147C2×C3×A4—21no—
148C2×C3×A4—23no—
149C2×C3×A4—26no—
150C2×C3×A4—46no—
151C2×C3×A4—42no—
152C2×C3×A4C22×D313yes—
153C2×C3×A4C23×C311yescharacteristic
154C3⋊S4—22no—
155C3⋊S4C3×D412yesdirect product factor
156C3×S4—36no—
157C3×S4—66no—
158C32×D4—32no—
159C32×D4—31no—
160C32×D4—41no—
161C32×D4—43no—
162C32×D4—61no—
163C32×D4—83no—
164C32×D4—81no—
165C32×D4S411yescharacteristic, complemented
166C2×C32×C4—31no—
167C2×C32×C4—31no—
168#638—42no—
169C4⋊(C3⋊D3)—41no—
170C4×C3⋊D3—41no—
171C3×#67—32no—
172C3×#67—32no—
173C3×#67—64no—
174C3×#67—64no—
175C3×#67—62no—
176C3×#67—62no—
177C3×#67—62no—
178C3×#67—62no—
179C3×#67—126no—
180C3×#67—122no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—64no—
186C3×D12—61no—
187C3×D12—61no—
188C3×D12—123no—
189C3×C4×D3—61no—
190C3×C4×D3—61no—
191C3×C4×D3—123no—
192D42—91noSylow
193C2×C33—41no—
194C2×C33—82no—
195C3×C3⋊D3—42no—
196C3×C3⋊D3—82no—
197C24×C3—22no—
198C24×C3—92no—
199C24×C3C2×C3×D312yescomplemented
200C24×C3C2×C3⋊D312yescomplemented
201C23×D3—32no—
202C22×A4—26no—
203C22×A4C2×C3×D316yescomplemented
204C2×S4—33no—
205C2×S4—36no—
206C2×S4—612no—
207C2×C3×D4—31no—
208C2×C3×D4—31no—
209C2×C3×D4—32no—
210C2×C3×D4—31no—
211C2×C3×D4—32no—
212C2×C3×D4—31no—
213C2×C3×D4—61no—
214C2×C3×D4—62no—
215C2×C3×D4—61no—
216C2×C3×D4—91no—
217C2×C3×D4—91no—
218C2×C3×D4—92no—
219C2×C3×D4—92no—
220C2×C3×D4—92no—
221C2×C3×D4—92no—
222C2×C3×D4—92no—
223C2×C3×D4—182no—
224C2×C3×D4—181no—
225C2×C3×D4—184no—
226C2×C3×D4—184no—
227C2×C3×D4—181no—
228C22×C3×C4—21no—
229C22×C3×C4—91no—
230C22×C3×C4—92no—
231C22×C3×C4C2×C3×D311yescharacteristic, complemented
232C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
233C2×#67—31no—
234C2×#67—32no—
235C2×#67—32no—
236C2×#67—32no—
237C2×#67—32no—
238C2×#67—32no—
239C2×#67—64no—
240C2×#67—64no—
241C22×Dic3—32no—
242D3×D4—62no—
243D3×D4—61no—
244D3×D4—61no—
245D3×D4—123no—
246C2×D12—31no—
247C2×C4×D3—31no—
248C4×A4—23no—
249C4×A4C2×C3×D313yescomplemented
250SL(2,ℤ/4)—33no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—91no—
253C3×C22⋊C4—91no—
254C3×C22⋊C4—92no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4—92no—
257C3×C22⋊C4—91no—
258C3×C42—91no—
259#217—32no—
260#217—32no—
261#217—31no—
262(C2×D3)⋊C4—31no—
263(C2×D3)⋊C4—32no—
264C4⋊Dic3—31no—
265Dic3⋊C4—31no—
266C4×Dic3—31no—
267C22×C32—31no—
268C22×C32—31no—
269C22×C32—32no—
270C22×C32—42no—
271C22×C32—46no—
272C22×C32—62no—
273C22×C32—62no—
274C22×C32—62no—
275C22×C32—62no—
276C22×C32—86no—
277C22×C32—82no—
278C22×C32D3×D411yescharacteristic, complemented
279C22×C32C2×S412yescomplemented
280C2×C3⋊D3—41no—
281C2×C3⋊D3—42no—
282C2×C3⋊D3—84no—
283C2×C3×D3—32no—
284C2×C3×D3—61no—
285C2×C3×D3—61no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—62no—
289C2×C3×D3—123no—
290C2×C3×D3—126no—
291C2×C3×D3—124no—
292C2×C3×D3—124no—
293C2×C3×D3—2412no—
294C3×A4—21no—
295C3×A4—23no—
296C3×A4D3×D413yescomplemented
297C3×A4C2×C3×D411yescharacteristic, complemented
298C32×C4—31no—
299C32×C4—41no—
300C32×C4—43no—
301C32×C4—83no—
302C32×C4—81no—
303C32×C4C2×S411yescharacteristic, complemented
304C3⋊Dic3—41no—
305C3×Dic3—32no—
306C3×Dic3—62no—
307C3×Dic3—61no—
308C3×Dic3—61no—
309C3×Dic3—123no—
310C22×D4—91no—
311C22×D4—92no—
312C22×D4C3×C3⋊D311yescharacteristic, complemented
313#127—91no—
314#121—91no—
315#121—92no—
316#121—91no—
317#120—92no—
318#120—92no—
319C4×D4—91no—
320C4×D4—91no—
321C33—41noSylow
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—21no—
325C23×C3—32no—
326C23×C3—32no—
327C23×C3—32no—
328C23×C3—32no—
329C23×C3—42no—
330C23×C3—62no—
331C23×C3—62no—
332C23×C3—62no—
333C23×C3—62no—
334C23×C3—91no—
335C23×C3—92no—
336C23×C3—122no—
337C23×C3—182no—
338C23×C3—184no—
339C23×C3—182no—
340C23×C3—362no—
341C23×C3C22×C3×D311yescharacteristic
342C23×C3C22×C3⋊D311yescharacteristic
343C22×D3—31no—
344C22×D3—32no—
345C22×D3—64no—
346C22×D3—62no—
347C22×D3—62no—
348C22×D3—126no—
349C22×D3—122no—
350C2×A4—26no—
351C2×A4—23no—
352C2×A4—46no—
353C2×A4C22×C3×D313yes—
354S4—36no—
355S4—66no—
356C3×D4—21no—
357C3×D4—32no—
358C3×D4—32no—
359C3×D4—31no—
360C3×D4—31no—
361C3×D4—43no—
362C3×D4—61no—
363C3×D4—62no—
364C3×D4—61no—
365C3×D4—61no—
366C3×D4—62no—
367C3×D4—83no—
368C3×D4—92no—
369C3×D4—92no—
370C3×D4—181no—
371C3×D4—184no—
372C3×D4—182no—
373C3×D4—182no—
374C3×D4—184no—
375C3×D4—182no—
376C3×D4—182no—
377C3×D4—181no—
378C3×D4—362no—
379C3×D4C3×S411yescharacteristic, complemented
380C3×D4C3⋊S411yescharacteristic, direct product factor
381C2×C3×C4—31no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—91no—
388C2×C3×C4—91no—
389C2×C3×C4—92no—
390C2×C3×C4—91no—
391C2×C3×C4—181no—
392C2×C3×C4—184no—
393C2×C3×C4—181no—
394#67—32no—
395#67—32no—
396#67—64no—
397#67—64no—
398#67—62no—
399#67—62no—
400#67—62no—
401#67—62no—
402#67—126no—
403#67—122no—
404C2×Dic3—31no—
405C2×Dic3—31no—
406C2×Dic3—31no—
407C2×Dic3—32no—
408C2×Dic3—64no—
409D12—61no—
410D12—61no—
411D12—123no—
412C4×D3—61no—
413C4×D3—61no—
414C4×D3—123no—
415C2×C32—22no—
416C2×C32—31no—
417C2×C32—31no—
418C2×C32—41no—
419C2×C32—43no—
420C2×C32—62no—
421C2×C32—83no—
422C2×C32—82no—
423C2×C32—86no—
424C2×C32—81no—
425C2×C32—162no—
426C2×C32—166no—
427C2×C32C22×S411yescharacteristic
428C3⋊D3—42no—
429C3⋊D3—82no—
430C3×D3—62no—
431C3×D3—126no—
432C3×D3—122no—
433C3×D3—246no—
434C24—92no—
435C24C2×C3×C3⋊D312yescomplemented
436C2×D4—31no—
437C2×D4—32no—
438C2×D4—31no—
439C2×D4—91no—
440C2×D4—91no—
441C2×D4—92no—
442C2×D4—92no—
443C2×D4—92no—
444C2×D4—92no—
445C2×D4—92no—
446C2×D4—182no—
447C2×D4—181no—
448C2×D4—184no—
449C2×D4—184no—
450C2×D4—181no—
451C22×C4—91no—
452C22×C4—92no—
453C22×C4C2×C3×C3⋊D311yescharacteristic, complemented
454C4⋊C4—91no—
455C4⋊C4—91no—
456C22⋊C4—91no—
457C22⋊C4—92no—
458C22⋊C4—92no—
459C22⋊C4—92no—
460C22⋊C4—91no—
461C42—91no—
462C22×C3—22no—
463C22×C3—21no—
464C22×C3—31no—
465C22×C3—31no—
466C22×C3—31no—
467C22×C3—31no—
468C22×C3—32no—
469C22×C3—32no—
470C22×C3—46no—
471C22×C3—61no—
472C22×C3—62no—
473C22×C3—61no—
474C22×C3—62no—
475C22×C3—62no—
476C22×C3—62no—
477C22×C3—62no—
478C22×C3—62no—
479C22×C3—62no—
480C22×C3—62no—
481C22×C3—62no—
482C22×C3—86no—
483C22×C3—92no—
484C22×C3—122no—
485C22×C3—122no—
486C22×C3—122no—
487C22×C3—122no—
488C22×C3—181no—
489C22×C3—182no—
490C22×C3—182no—
491C22×C3—181no—
492C22×C3—182no—
493C22×C3—364no—
494C22×C3—364no—
495C22×C3C3×D3×D411yescharacteristic, complemented
496C22×C3D4×C3⋊D311yescharacteristic, complemented
497C22×C3C2×C3×S412yescomplemented
498C22×C3C2×C3⋊S412yescomplemented
499C2×D3—32no—
500C2×D3—61no—
501C2×D3—61no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—62no—
505C2×D3—123no—
506C2×D3—126no—
507C2×D3—124no—
508C2×D3—124no—
509C2×D3—2412no—
510A4—23no—
511A4C3×D3×D413yescomplemented
512C3×C4—21no—
513C3×C4—31no—
514C3×C4—31no—
515C3×C4—43no—
516C3×C4—61no—
517C3×C4—83no—
518C3×C4—92no—
519C3×C4—182no—
520C3×C4—181no—
521C3×C4—181no—
522C3×C4C2×C3×S411yescharacteristic, complemented
523C3×C4C2×C3⋊S411yescharacteristic, complemented
524Dic3—32no—
525Dic3—62no—
526Dic3—61no—
527Dic3—61no—
528Dic3—123no—
529C32—41no—
530C32—43no—
531C32—83no—
532C32—81no—
533C32D4×S411yescharacteristic, complemented
534C23—22no—
535C23—32no—
536C23—32no—
537C23—62no—
538C23—91no—
539C23—92no—
540C23—182no—
541C23—184no—
542C23—182no—
543C23—362no—
544C23C22×C3×C3⋊D311yescharacteristic
545D4—32no—
546D4—31no—
547D4—61no—
548D4—92no—
549D4—92no—
550D4—181no—
551D4—184no—
552D4—182no—
553D4—182no—
554D4—184no—
555D4—182no—
556D4—182no—
557D4—181no—
558D4—362no—
559D4C3×C3⋊S411yescharacteristic, direct product factor
560C2×C4—31no—
561C2×C4—31no—
562C2×C4—91no—
563C2×C4—91no—
564C2×C4—92no—
565C2×C4—91no—
566C2×C4—181no—
567C2×C4—184no—
568C2×C4—181no—
569C2×C3—22no—
570C2×C3—22no—
571C2×C3—21no—
572C2×C3—31no—
573C2×C3—31no—
574C2×C3—31no—
575C2×C3—31no—
576C2×C3—42no—
577C2×C3—43no—
578C2×C3—61no—
579C2×C3—61no—
580C2×C3—62no—
581C2×C3—62no—
582C2×C3—86no—
583C2×C3—83no—
584C2×C3—122no—
585C2×C3—166no—
586C2×C3—182no—
587C2×C3—362no—
588C2×C3C22×C3×S411yescharacteristic
589C2×C3C22×C3⋊S411yescharacteristic
590D3—62no—
591D3—126no—
592D3—122no—
593D3—246no—
594C22—31no—
595C22—31no—
596C22—32no—
597C22—62no—
598C22—62no—
599C22—62no—
600C22—62no—
601C22—92no—
602C22—181no—
603C22—182no—
604C22—182no—
605C22—181no—
606C22—182no—
607C22—364no—
608C22—364no—
609C22C3×D4×C3⋊D311yescharacteristic, complemented
610C22C2×C3×C3⋊S412yescomplemented
611C4—31no—
612C4—92no—
613C4—182no—
614C4—181no—
615C4—181no—
616C4C2×C3×C3⋊S411yescharacteristic, complemented
617C3—21no—
618C3—43no—
619C3—83no—
620C3C3×D4×S411yescharacteristic, complemented
621C3D4×C3⋊S411yescharacteristic, direct product factor
622C2—22no—
623C2—31no—
624C2—31no—
625C2—62no—
626C2—182no—
627C2—362no—
628C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC2×C3C22×C3⋊S4
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC22×C32×D4D3
socleC23×C32C22×D3
hypercenterC3×D4C3⋊S4
hypocenterC3×A4C2×C3×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C3⋊S4C2semidirect
2. C32×D4×A4C2semidirect
3. C3×#64269C2semidirect
4. C3×C4⋊(C3⋊S4)C2semidirect
5. C3×C4×C3⋊S4C2semidirect
6. D4×C3⋊S4C3direct (N × H)
9. C22×C32×A4C22semidirect
12. C2×C3×C3⋊S4C22semidirect
14. C32×C4×A4C22semidirect
15. C3×C3⋊SL(2,ℤ/4)C22semidirect
16. C22×C3⋊S4C2×C3semidirect
18. C22×C32×D4D3semidirect
23. C3×D4×A4D3semidirect
24. C3×D4×A4C2×C3semidirect
25. #64269C2×C3semidirect
26. C4⋊(C3⋊S4)C2×C3semidirect
27. C4×C3⋊S4C2×C3semidirect
43. C3×C3⋊S4D4direct (N × H)
51. C24×C32C2×D3semidirect
55. C22×C3×A4C2×D3semidirect
56. C22×C3×A4C22×C3semidirect
59. C2×C3⋊S4C22×C3semidirect
66. C22×C32×C4C2×D3semidirect
85. C3×C4×A4C2×D3semidirect
86. C3×C4×A4C22×C3semidirect
87. C3⋊SL(2,ℤ/4)C22×C3semidirect
101. C32×A4C2×D4semidirect
108. C22×C3×D4C3×D3semidirect
109. C22×C3×D4C3⋊D3semidirect
113. D4×A4C3×D3semidirect
155. C3⋊S4C3×D4direct (N × H)
165. C32×D4S4semidirect
199. C24×C3C2×C3×D3semidirect
200. C24×C3C2×C3⋊D3semidirect
203. C22×A4C2×C3×D3semidirect
231. C22×C3×C4C2×C3×D3semidirect
232. C22×C3×C4C2×C3⋊D3semidirect
249. C4×A4C2×C3×D3semidirect
278. C22×C32D3×D4semidirect
279. C22×C32C2×S4semidirect
296. C3×A4D3×D4semidirect
297. C3×A4C2×C3×D4semidirect
303. C32×C4C2×S4semidirect
312. C22×D4C3×C3⋊D3semidirect
379. C3×D4C3×S4semidirect
380. C3×D4C3⋊S4direct (N × H)
435. C24C2×C3×C3⋊D3semidirect
453. C22×C4C2×C3×C3⋊D3semidirect
495. C22×C3C3×D3×D4semidirect
496. C22×C3D4×C3⋊D3semidirect
497. C22×C3C2×C3×S4semidirect
498. C22×C3C2×C3⋊S4semidirect
511. A4C3×D3×D4semidirect
522. C3×C4C2×C3×S4semidirect
523. C3×C4C2×C3⋊S4semidirect
533. C32D4×S4semidirect
559. D4C3×C3⋊S4direct (N × H)
609. C22C3×D4×C3⋊D3semidirect
610. C22C2×C3×C3⋊S4semidirect
616. C4C2×C3×C3⋊S4semidirect
620. C3C3×D4×S4semidirect
621. C3D4×C3⋊S4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊S42
C32×D4×A42
C3×#642692
C3×C4⋊(C3⋊S4)2
C3×C4×C3⋊S42
D4×C3⋊S43
C3×D4×S43
C3×D4×#673
C3×D4×C3⋊D34