Finite Group Databasebeta

← D4×#6295 #13178652

Subgroups — D4×#6295 #13178652

Order 1728.

2332 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C212yesmaximal, complemented
2C22×#6295C212yesmaximal, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#11723727C214yesmaximal, complemented
5#11723717C212yesmaximal, complemented
6#11723697C211yesmaximal, characteristic, complemented
7#11723696C212yesmaximal, complemented
8C4×#6295C211yesmaximal, characteristic, complemented
9D4×#626—32nomaximal
10D4×#625—31nomaximal
11C23×C3×C3⋊D3C2214yescomplemented
12C2×C33×D4C2211yescharacteristic
13C2×C3×#638C2214yescomplemented
14C22×C3×C3⋊Dic3C2212yescomplemented
15C3×D4×C3⋊D3—28no—
16C2×C3×C4⋊(C3⋊D3)C2212yescomplemented
17C2×C3×C4×C3⋊D3C2212yescomplemented
18C2×#6295—28no—
19C2×#6295C2211yescharacteristic
20C2×#6295C2214yescomplemented
21C2×#6295C2212yescomplemented
22C2×#6295C2212yescomplemented
23C3×#3757C2212yescomplemented
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
25C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
26#88815C2211yescharacteristic, complemented
27#88814C2212yescomplemented
28#88814C2214yescomplemented
29C2×D4×C3⋊D3D312yescomplemented
30C2×C3×D3×D4—32no—
31C2×C3×D3×D4—32no—
32C2×C3×D3×D4—62no—
33C22×#626—34no—
34C22×#625—32no—
35D4×C3⋊Dic3D311yescharacteristic, complemented
36C3×D4×Dic3—32no—
37C3×D4×Dic3—61no—
38#63981—34no—
39#63980—32no—
40#63978—34no—
41#63977—34no—
42#63976—34no—
43#63916—32no—
44#63914—31no—
45#63911—32no—
46#63909—32no—
47#63908—32no—
48C4×#626—32no—
49C4×#625—31no—
50C23×C33C2312yes—
51C22×C3×C3⋊D3—28no—
52C22×C3×C3⋊D3—28no—
53C22×C3×C3⋊D3—44no—
54C22×C3×C3⋊D3C2314yes—
55C22×C3×C3⋊D3C2312yes—
56C33×D4D414yescomplemented
57C2×C33×C4C2311yescharacteristic
58C3×#638—28no—
59C2×C3×C3⋊Dic3—24no—
60C2×C3×C3⋊Dic3C2311yescharacteristic
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2312yes—
63C2×C3×C3⋊Dic3C2311yescharacteristic
64C3×C4⋊(C3⋊D3)—24no—
65C3×C4×C3⋊D3—24no—
66#6295—28no—
67#6295—24no—
68#6295—42no—
69#6295D414yesdirect product factor
70D4×#67—92no—
71D4×D12—91no—
72C23×C3⋊D3C2×D314yescomplemented
73C23×C3×D3—34no—
74C23×C3×D3—34no—
75C23×C3×D3—64no—
76C2×C32×D4—21no—
77C2×C32×D4—22no—
78C2×C32×D4—41no—
79C2×C32×D4C2×D311yescharacteristic
80C2×C32×D4C2×D312yes—
81C2×#638C2×D314yescomplemented
82C22×C3⋊Dic3C2×D312yescomplemented
83D4×C3⋊D3—28no—
84C2×C4⋊(C3⋊D3)C2×D312yescomplemented
85C2×C4×C3⋊D3C2×D312yescomplemented
86C2×C3×#67—34no—
87C2×C3×#67—34no—
88C2×C3×#67—64no—
89C22×C3×Dic3—34no—
90C22×C3×Dic3—62no—
91C3×D3×D4—616no—
92C3×D3×D4—68no—
93C3×D3×D4—68no—
94C2×C3×D12—32no—
95C2×C3×D12—32no—
96C2×C3×D12—62no—
97C2×C3×C4×D3—32no—
98C2×C3×C4×D3—32no—
99C2×C3×C4×D3—62no—
100C2×#626—32no—
101C2×#626—34no—
102C2×#626—34no—
103C2×#626—34no—
104C2×#626—34no—
105C2×#626—616no—
106C2×#625—31no—
107C2×#625—34no—
108C2×#625—32no—
109C2×#625—32no—
110C2×#625—68no—
111#3757C2×D312yescomplemented
112C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
113C4×C3⋊Dic3C2×D311yescharacteristic, complemented
114C3×#217—34no—
115C3×#217—62no—
116C3×C4⋊Dic3—32no—
117C3×C4⋊Dic3—61no—
118C3×C4×Dic3—32no—
119C3×C4×Dic3—61no—
120#3724—31no—
121Dic3⋊Dic3—32no—
122#3722—32no—
123#3722—34no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—32no—
126(C2×D3)⋊Dic3—34no—
127(C2×D3)⋊Dic3—34no—
128C22×C33—22no—
129C22×C33—22no—
130C22×C33C2×D414yescomplemented
131C22×C33C2411yescharacteristic
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—22no—
134C2×C3×C3⋊D3—24no—
135C2×C3×C3⋊D3—24no—
136C2×C3×C3⋊D3—24no—
137C2×C3×C3⋊D3—416no—
138C2×C3×C3⋊D3C2×D414yescomplemented
139C33×C4C2×D412yescomplemented
140C3×C3⋊Dic3—22no—
141C3×C3⋊Dic3—22no—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3C2×D412yescomplemented
144C22×C3×D4—92no—
145C22×#67—94no—
146C2×D3×D4—32no—
147C2×D3×D4—32no—
148C2×D3×D4—62no—
149C22×D12—92no—
150C3×C4×D4—91no—
151#953—94no—
152#940—92no—
153#939—94no—
154#938—92no—
155#937—94no—
156D4×Dic3—32no—
157D4×Dic3—61no—
158#930—92no—
159#930—92no—
160C4×#67—92no—
161#895—92no—
162#882—94no—
163#874—91no—
164C4×D12—91no—
165C23×C32—24no—
166C23×C32—22no—
167C23×C32—42no—
168C23×C32C22×D312yes—
169C23×C32C22×D314yes—
170C22×C3⋊D3—28no—
171C22×C3⋊D3—28no—
172C22×C3⋊D3—44no—
173C22×C3⋊D3C22×D314yes—
174C22×C3⋊D3C22×D312yes—
175C22×C3×D3—32no—
176C22×C3×D3—32no—
177C22×C3×D3—34no—
178C22×C3×D3—34no—
179C22×C3×D3—62no—
180C22×C3×D3—64no—
181C22×C3×D3—68no—
182C22×C3×D3—68no—
183C22×C3×D3—616no—
184C22×C3×D3—68no—
185C22×C3×D3—68no—
186C22×C3×D3—128no—
187C22×C3×D3—124no—
188C22×C3×D3—124no—
189C22×C3×D3—128no—
190C32×D4—24no—
191C32×D4—28no—
192C32×D4—44no—
193C32×D4D1214yescomplemented
194C32×D4#6718yescomplemented
195C2×C32×C4—21no—
196C2×C32×C4—22no—
197C2×C32×C4—41no—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D312yes—
200#638—28no—
201C2×C3⋊Dic3—24no—
202C2×C3⋊Dic3C22×D312yes—
203C2×C3⋊Dic3C22×D311yescharacteristic
204C2×C3⋊Dic3C22×D312yes—
205C2×C3⋊Dic3C22×D311yescharacteristic
206C4⋊(C3⋊D3)—24no—
207C4×C3⋊D3—24no—
208C3×#67—616no—
209C3×#67—68no—
210C3×#67—68no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—34no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—61no—
217C2×C3×Dic3—62no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—62no—
220C2×C3×Dic3—68no—
221C2×C3×Dic3—124no—
222C3×D12—68no—
223C3×D12—64no—
224C3×D12—64no—
225C3×C4×D3—68no—
226C3×C4×D3—64no—
227C3×C4×D3—64no—
228#626—38no—
229#626—68no—
230#626—68no—
231#626—68no—
232#626—124no—
233#625—34no—
234#625—68no—
235#625—64no—
236#625—122no—
237D42—271noSylow
238C2×C33—24no—
239C2×C33C22×D411yescharacteristic
240C2×C33C22×D411yescharacteristic
241C2×C33H(5,𝔽2)11yescharacteristic
242C3×C3⋊D3—24no—
243C3×C3⋊D3—44no—
244C24×C3—94no—
245C23×D3—34no—
246C23×D3—34no—
247C23×D3—64no—
248C2×C3×D4—21no—
249C2×C3×D4—22no—
250C2×C3×D4—41no—
251C2×C3×D4—94no—
252C2×C3×D4—92no—
253C2×C3×D4—188no—
254C2×C3×D4D3211yescharacteristic
255C2×C3×D4D3212yes—
256C22×C3×C4—92no—
257C22×C3×C4—92no—
258C2×#67—34no—
259C2×#67—34no—
260C2×#67—64no—
261C2×#67—92no—
262C2×#67—94no—
263C2×#67—94no—
264C2×#67—94no—
265C2×#67—94no—
266C2×#67—1816no—
267C22×Dic3—34no—
268C22×Dic3—62no—
269D3×D4—616no—
270D3×D4—68no—
271D3×D4—68no—
272C2×D12—32no—
273C2×D12—32no—
274C2×D12—62no—
275C2×D12—91no—
276C2×D12—92no—
277C2×D12—94no—
278C2×D12—92no—
279C2×D12—188no—
280C2×C4×D3—32no—
281C2×C4×D3—32no—
282C2×C4×D3—62no—
283C3×C4⋊C4—91no—
284C3×C22⋊C4—92no—
285C3×C42—91no—
286#217—34no—
287#217—62no—
288#217—92no—
289#217—94no—
290(C2×D3)⋊C4—92no—
291(C2×D3)⋊C4—94no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—94no—
294C4⋊Dic3—32no—
295C4⋊Dic3—61no—
296C4⋊Dic3—91no—
297Dic3⋊C4—92no—
298C4×Dic3—32no—
299C4×Dic3—61no—
300C22×C32—24no—
301C22×C32—22no—
302C22×C32—24no—
303C22×C32—22no—
304C22×C32—21no—
305C22×C32—22no—
306C22×C32—24no—
307C22×C32—28no—
308C22×C32—41no—
309C22×C32—44no—
310C22×C32—42no—
311C22×C32—44no—
312C22×C32—44no—
313C22×C32—42no—
314C22×C32—82no—
315C22×C32—82no—
316C22×C32C2×D1214yescomplemented
317C22×C32C2×#6718yescomplemented
318C22×C32C23×D311yescharacteristic
319C22×C32C23×D312yes—
320C2×C3⋊D3—22no—
321C2×C3⋊D3—22no—
322C2×C3⋊D3—24no—
323C2×C3⋊D3—24no—
324C2×C3⋊D3—24no—
325C2×C3⋊D3—416no—
326C2×C3⋊D3D3×D414yescomplemented
327C2×C3×D3—34no—
328C2×C3×D3—34no—
329C2×C3×D3—64no—
330C2×C3×D3—64no—
331C2×C3×D3—64no—
332C2×C3×D3—62no—
333C2×C3×D3—62no—
334C2×C3×D3—62no—
335C2×C3×D3—62no—
336C2×C3×D3—68no—
337C2×C3×D3—64no—
338C2×C3×D3—64no—
339C2×C3×D3—64no—
340C2×C3×D3—64no—
341C2×C3×D3—64no—
342C2×C3×D3—64no—
343C2×C3×D3—124no—
344C2×C3×D3—124no—
345C2×C3×D3—1216no—
346C2×C3×D3—1216no—
347C2×C3×D3—1232no—
348C32×C4—22no—
349C32×C4—24no—
350C32×C4—42no—
351C32×C4C2×D1212yescomplemented
352C32×C4C2×#6714yescomplemented
353C3⋊Dic3—22no—
354C3⋊Dic3—22no—
355C3⋊Dic3—21no—
356C3⋊Dic3D3×D412yescomplemented
357C3×Dic3—34no—
358C3×Dic3—62no—
359C3×Dic3—64no—
360C3×Dic3—62no—
361C3×Dic3—64no—
362C3×Dic3—62no—
363C3×Dic3—62no—
364C3×Dic3—121no—
365C3×Dic3—122no—
366C22×D4—92no—
367C22×D4—272no—
368#127—271no—
369#121—272no—
370#121—272no—
371#120—274no—
372C4×D4—91no—
373C4×D4—271no—
374C33D4211yescharacteristic, complemented, Sylow
375C23×C3—22no—
376C23×C3—24no—
377C23×C3—42no—
378C23×C3—92no—
379C23×C3—94no—
380C23×C3—188no—
381C23×C3—188no—
382C23×C3—364no—
383C23×C3C2×D3212yes—
384C23×C3C2×D3214yes—
385C22×D3—32no—
386C22×D3—32no—
387C22×D3—34no—
388C22×D3—34no—
389C22×D3—62no—
390C22×D3—64no—
391C22×D3—68no—
392C22×D3—68no—
393C22×D3—616no—
394C22×D3—68no—
395C22×D3—68no—
396C22×D3—128no—
397C22×D3—124no—
398C22×D3—124no—
399C22×D3—128no—
400C3×D4—24no—
401C3×D4—28no—
402C3×D4—44no—
403C3×D4—188no—
404C3×D4—184no—
405C3×D4#62514yescomplemented
406C3×D4#62618yescomplemented
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—41no—
410C2×C3×C4—91no—
411C2×C3×C4—92no—
412C2×C3×C4—91no—
413C2×C3×C4—92no—
414C2×C3×C4—184no—
415C2×C3×C4—184no—
416C2×C3×C4C2×D3211yescharacteristic
417C2×C3×C4C2×D3212yes—
418#67—616no—
419#67—68no—
420#67—68no—
421#67—98no—
422#67—188no—
423#67—188no—
424#67—188no—
425#67—364no—
426C2×Dic3—32no—
427C2×Dic3—32no—
428C2×Dic3—32no—
429C2×Dic3—34no—
430C2×Dic3—32no—
431C2×Dic3—61no—
432C2×Dic3—62no—
433C2×Dic3—61no—
434C2×Dic3—62no—
435C2×Dic3—68no—
436C2×Dic3—124no—
437D12—68no—
438D12—64no—
439D12—64no—
440D12—94no—
441D12—184no—
442D12—188no—
443D12—362no—
444C4×D3—68no—
445C4×D3—64no—
446C4×D3—64no—
447C2×C32—28no—
448C2×C32—24no—
449C2×C32—21no—
450C2×C32—22no—
451C2×C32—21no—
452C2×C32—22no—
453C2×C32—21no—
454C2×C32—22no—
455C2×C32—41no—
456C2×C32—41no—
457C2×C32—41no—
458C2×C32—48no—
459C2×C32—44no—
460C2×C32—84no—
461C2×C32C22×D1211yescharacteristic
462C2×C32C2×D3×D412yes—
463C2×C32C2×D3×D411yescharacteristic
464C2×C32#100412yes—
465C2×C32#100911yescharacteristic
466C2×C32C22×#6712yes—
467C3⋊D3—24no—
468C3⋊D3—44no—
469C3×D3—68no—
470C3×D3—64no—
471C3×D3—64no—
472C3×D3—128no—
473C3×D3—124no—
474C3×D3—124no—
475C24—94no—
476C2×D4—94no—
477C2×D4—92no—
478C2×D4—188no—
479C2×D4—271no—
480C2×D4—274no—
481C2×D4—272no—
482C2×D4—272no—
483C2×D4—548no—
484C2×D4#111311yescharacteristic
485C22×C4—92no—
486C22×C4—92no—
487C4⋊C4—91no—
488C4⋊C4—271no—
489C22⋊C4—92no—
490C22⋊C4—272no—
491C22⋊C4—274no—
492C42—91no—
493C22×C3—24no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—24no—
497C22×C3—21no—
498C22×C3—22no—
499C22×C3—28no—
500C22×C3—24no—
501C22×C3—42no—
502C22×C3—44no—
503C22×C3—44no—
504C22×C3—42no—
505C22×C3—41no—
506C22×C3—44no—
507C22×C3—82no—
508C22×C3—82no—
509C22×C3—94no—
510C22×C3—182no—
511C22×C3—182no—
512C22×C3—184no—
513C22×C3—184no—
514C22×C3—184no—
515C22×C3—3616no—
516C22×C3C2×#62514yescomplemented
517C22×C3C2×#62618yescomplemented
518C22×C3C22×D3211yescharacteristic
519C22×C3C22×D3212yes—
520C2×D3—34no—
521C2×D3—34no—
522C2×D3—64no—
523C2×D3—64no—
524C2×D3—64no—
525C2×D3—62no—
526C2×D3—62no—
527C2×D3—62no—
528C2×D3—62no—
529C2×D3—68no—
530C2×D3—64no—
531C2×D3—64no—
532C2×D3—64no—
533C2×D3—64no—
534C2×D3—64no—
535C2×D3—64no—
536C2×D3—124no—
537C2×D3—124no—
538C2×D3—1216no—
539C2×D3—1216no—
540C2×D3—1232no—
541C3×C4—24no—
542C3×C4—22no—
543C3×C4—42no—
544C3×C4—92no—
545C3×C4—182no—
546C3×C4—181no—
547C3×C4—182no—
548C3×C4C2×#62512yescomplemented
549C3×C4C2×#62614yescomplemented
550Dic3—34no—
551Dic3—62no—
552Dic3—64no—
553Dic3—64no—
554Dic3—62no—
555Dic3—62no—
556Dic3—62no—
557Dic3—122no—
558Dic3—121no—
559C32—21no—
560C32—22no—
561C32—41no—
562C32D4×D1211yescharacteristic, complemented
563C32D4×#6712yescomplemented
564C23—92no—
565C23—94no—
566C23—188no—
567C23—188no—
568C23—364no—
569C23C2×#111312yes—
570D4—188no—
571D4—184no—
572D4—274no—
573D4—544no—
574D4—548no—
575D4—1082no—
576D4#629514yesdirect product factor
577C2×C4—91no—
578C2×C4—91no—
579C2×C4—92no—
580C2×C4—92no—
581C2×C4—184no—
582C2×C4—184no—
583C2×C4C2×#111311yescharacteristic
584C2×C3—24no—
585C2×C3—28no—
586C2×C3—21no—
587C2×C3—22no—
588C2×C3—21no—
589C2×C3—22no—
590C2×C3—22no—
591C2×C3—21no—
592C2×C3—48no—
593C2×C3—44no—
594C2×C3—41no—
595C2×C3—41no—
596C2×C3—41no—
597C2×C3—84no—
598C2×C3—184no—
599C2×C3—364no—
600C2×C3D32×D411yescharacteristic
601C2×C3D32×D412yes—
602C2×C3#6431311yescharacteristic
603C2×C3#6431412yes—
604C2×C3C22×#62511yescharacteristic
605C2×C3C22×#62612yes—
606D3—68no—
607D3—64no—
608D3—64no—
609D3—128no—
610D3—124no—
611D3—124no—
612C22—22no—
613C22—22no—
614C22—94no—
615C22—182no—
616C22—182no—
617C22—184no—
618C22—184no—
619C22—184no—
620C22—3616no—
621C22C2×#629514yescomplemented
622C22C22×#111311yescharacteristic
623C4—92no—
624C4—182no—
625C4—181no—
626C4—182no—
627C4C2×#629512yescomplemented
628C3—22no—
629C3—21no—
630C3—41no—
631C3D4×#62511yescharacteristic, complemented
632C3D4×#62612yescomplemented
633C2—24no—
634C2—184no—
635C2—364no—
636C2D4×#111311yescharacteristic
637C2#1172462311yescharacteristic
638C2C22×#629511yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33D42
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×C3⋊D3C2semidirect
2. C22×#6295C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. #11723727C2semidirect
5. #11723717C2semidirect
6. #11723697C2semidirect
7. #11723696C2semidirect
8. C4×#6295C2semidirect
11. C23×C3×C3⋊D3C22semidirect
13. C2×C3×#638C22semidirect
14. C22×C3×C3⋊Dic3C22semidirect
16. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C2×C3×C4×C3⋊D3C22semidirect
20. C2×#6295C22semidirect
21. C2×#6295C22semidirect
22. C2×#6295C22semidirect
23. C3×#3757C22semidirect
24. C3×C4⋊(C3⋊Dic3)C22semidirect
25. C3×C4×C3⋊Dic3C22semidirect
26. #88815C22semidirect
27. #88814C22semidirect
28. #88814C22semidirect
29. C2×D4×C3⋊D3D3semidirect
35. D4×C3⋊Dic3D3semidirect
56. C33×D4D4semidirect
69. #6295D4direct (N × H)
72. C23×C3⋊D3C2×D3semidirect
81. C2×#638C2×D3semidirect
82. C22×C3⋊Dic3C2×D3semidirect
84. C2×C4⋊(C3⋊D3)C2×D3semidirect
85. C2×C4×C3⋊D3C2×D3semidirect
111. #3757C2×D3semidirect
112. C4⋊(C3⋊Dic3)C2×D3semidirect
113. C4×C3⋊Dic3C2×D3semidirect
130. C22×C33C2×D4semidirect
138. C2×C3×C3⋊D3C2×D4semidirect
139. C33×C4C2×D4semidirect
143. C3×C3⋊Dic3C2×D4semidirect
193. C32×D4D12semidirect
194. C32×D4#67semidirect
316. C22×C32C2×D12semidirect
317. C22×C32C2×#67semidirect
326. C2×C3⋊D3D3×D4semidirect
351. C32×C4C2×D12semidirect
352. C32×C4C2×#67semidirect
356. C3⋊Dic3D3×D4semidirect
374. C33D42semidirect
405. C3×D4#625semidirect
406. C3×D4#626semidirect
516. C22×C3C2×#625semidirect
517. C22×C3C2×#626semidirect
548. C3×C4C2×#625semidirect
549. C3×C4C2×#626semidirect
562. C32D4×D12semidirect
563. C32D4×#67semidirect
576. D4#6295direct (N × H)
621. C22C2×#6295semidirect
627. C4C2×#6295semidirect
631. C3D4×#625semidirect
632. C3D4×#626semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C22×#62952
C3×D4×C3⋊Dic32
#117237272
#117237172
#117236972
#117236962
C4×#62952
D4×#6263
D4×#6253