Finite Group Databasebeta

← C4×#5838 #11659430

Subgroups — C4×#5838 #11659430

Order 768.

2658 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×2−1+4C211yesmaximal, characteristic, complemented
2C2×#5838C211yesmaximal, characteristic
3C4×#1010C211yesmaximal, characteristic, complemented
4#84111C211yesmaximal, characteristic, complemented
5#82571C213yesmaximal, complemented
6#82570C211yesmaximal, characteristic, complemented
7#82562C211yesmaximal, characteristic, complemented
8#82557C213yesmaximal, complemented
9#82549C213yesmaximal, complemented
10#82547C213yesmaximal, complemented
11C4×#951C213yesmaximal, complemented
12C4×#948C211yesmaximal, characteristic, complemented
13C4×#950C213yesmaximal, complemented
14C4×#942C213yesmaximal, complemented
15C2×C4×#216C213yesmaximal, complemented
16C4×#578—31nomaximal, Sylow
17C2×C3×2−1+4C2211yescharacteristic
18C2×#1010C2211yescharacteristic
19C3×#520—23no—
20C3×#520C2213yes—
21C3×#520C2211yescharacteristic
22C3×#519—21no—
23C3×#519C2213yes—
24C3×C4×#41—23no—
25C3×C4×#41C2213yes—
26C3×C4×#41C2211yescharacteristic
27C2×C3×C4×Q8—21no—
28C2×C3×C4×Q8C2213yes—
29#5838C4116yesdirect product factor
30#5827C2211yescharacteristic
31C2×#951C2213yes—
32C2×#950C2213yes—
33C2×#948C2211yescharacteristic
34C2×#948C2211yescharacteristic
35C22×#216C2213yes—
36C2×#942C2213yes—
37#5572—21no—
38C4×D3×Q8—21no—
39#5546—23no—
40C4×#237—23no—
41#5523C2213yes—
42#5516C2213yes—
43C4×#235C2213yes—
44C2×C4×Dic6C2213yes—
45#5233C2213yes—
46#5232C2213yes—
47#5225C2213yes—
48C2×#835C2213yes—
49#5030C2216yes—
50#5030C2216yes—
51C4×#216C2213yes—
52C4×#216C2213yes—
53C4×#216C2216yes—
54C4×#216C2216yes—
55C4×#216C2216yes—
56#5026C2216yes—
57C4×#215C2213yes—
58C4×#215C2213yes—
59#5018C2216yes—
60C4×#214C2213yes—
61C4×#214C2213yes—
62C4×#213C2211yescharacteristic
63C4×#213C2213yes—
64#5004C2213yes—
65#5002C2213yes—
66#4998C2213yes—
67#4997C2213yes—
68#4997C2213yes—
69#4995C2213yes—
70C2×#810C2213yes—
71#4966C2213yes—
72#4964C2213yes—
73C2×#807C2213yes—
74C4×#208C2213yes—
75C2×C4×C3⋊C8C2213yes—
76C2×#578—31no—
77C4×2−1+4—31no—
78C4×2−1+4D311yescharacteristic, complemented
79#2951—33no—
80#2950—33no—
81#2947—33no—
82#2947—33no—
83C4×#137—33no—
84C4×#137—33no—
85C4×#135—33no—
86C2×C4×Q16—33no—
87#2913—31no—
88#2898—31no—
89#2898—31no—
90C4×#131—31no—
91C3×2−1+4—24no—
92C3×2−1+4C2×C418yescomplemented
93C2×C3×#41—23no—
94C2×C3×#41C2313yes—
95C2×C3×#41C2311yescharacteristic
96C22×C3×Q8—21no—
97C22×C3×Q8C2313yes—
98#1010—24no—
99#1010C2×C418yescomplemented
100C2×D3×Q8—21no—
101C2×#237—23no—
102C2×#235C2313yes—
103C22×Dic6C2313yes—
104C3×C4×Q8—21no—
105C3×C4×Q8—23no—
106C3×C4×Q8—23no—
107C3×C4×Q8—23no—
108C3×C4×Q8—23no—
109C3×C4×Q8D411yescharacteristic
110C3×C4×Q8D411yescharacteristic
111C3×C4×Q8C2313yes—
112C3×C4×Q8C2313yes—
113C3×C4×Q8C2316yes—
114C3×C4×D4—23no—
115C3×C4×D4—23no—
116C3×C4×D4—23no—
117C3×C4×D4—26no—
118C3×C4×D4D413yes—
119C3×C4×D4D413yes—
120C3×C4×D4C2311yescharacteristic
121C3×C4×D4C2313yes—
122C3×#117—23no—
123C3×#117—26no—
124C3×#117—23no—
125C3×#117C2313yes—
126C3×#117C2313yes—
127C2×C3×C4⋊C4—23no—
128C2×C3×C4⋊C4—23no—
129C2×C3×C4⋊C4C2313yes—
130C2×C3×C42—23no—
131C2×C3×C42—23no—
132C2×C3×C42C2313yes—
133#951C2×C4124yescomplemented
134#950C2×C4124yescomplemented
135#948—24no—
136#948C2×C418yescomplemented
137Q8×Dic3—21no—
138C2×#216C2×C4124yescomplemented
139C2×#216C2313yes—
140C2×#216C2316yes—
141#942C2×C4124yescomplemented
142C2×#215C2313yes—
143D4×Dic3—23no—
144C2×#214C2313yes—
145C2×#213C2311yescharacteristic
146C4×#67—23no—
147C4×#67—23no—
148#926C2313yes—
149C2×C4⋊Dic3C2313yes—
150C2×Dic3⋊C4—23no—
151C2×C4×Dic3—23no—
152C2×#208C2313yes—
153C2×#208C2313yes—
154C22×C3⋊C8C2313yes—
155C22×C3⋊C8C2313yes—
156#892—23no—
157D3×C4⋊C4—23no—
158#887—23no—
159#887—23no—
160#67⋊C4—26no—
161#877—26no—
162C4×D12C2311yescharacteristic
163C4×D12C2313yes—
164#872—23no—
165C42×D3—23no—
166C4×Dic6—23no—
167C4×Dic6—23no—
168C4×Dic6C2313yes—
169C4×Dic6C2316yes—
170C4×Dic6C2313yes—
171#835C2313yes—
172#835C2316yes—
173#835C2313yes—
174#832C2313yes—
175#832C2311yescharacteristic
176#810C2316yes—
177#810C2313yes—
178#810C2313yes—
179#809C2313yes—
180#809C2311yescharacteristic
181#808C2313yes—
182#808C2313yes—
183#807C2313yes—
184#807C2316yes—
185#807C2311yescharacteristic
186#803C2316yes—
187C4×C3⋊C8C2311yescharacteristic
188C4×C3⋊C8C2313yes—
189C4×C3⋊C8C2313yes—
190C4×C3⋊C8C2313yes—
191C2×2−1+4—31no—
192C2×2−1+4C2×D311yescharacteristic
193#578—316no—
194C2×#137—33no—
195C2×#137—33no—
196C2×#135—33no—
197C22×Q16—33no—
198C2×#131—31no—
199C2×#131—31no—
200#520—23no—
201#520—31no—
202#520—33no—
203#520—63no—
204#520C2×D313yes—
205#520C2×D311yescharacteristic
206#519—21no—
207#519—33no—
208#519—61no—
209#519C2×D313yes—
210C4×#41—23no—
211C4×#41—33no—
212C4×#41—63no—
213C4×#41C2×D313yes—
214C4×#41C2×D311yescharacteristic
215C2×C4×Q8—21no—
216C2×C4×Q8—33no—
217C2×C4×Q8—61no—
218C2×C4×Q8C2×D313yes—
219#441—36no—
220#441—36no—
221#440—36no—
222#440—36no—
223C4×Q16—33no—
224C4×Q16—36no—
225C4×Q16—36no—
226C4×Q16—36no—
227C4×Q16—33no—
228C4×SD16—33no—
229C4×SD16—33no—
230C4×SD16—33no—
231C4×SD16—33no—
232C4×D8—31no—
233C4×D8—33no—
234#428—33no—
235#428—33no—
236#427—33no—
237C2×#107—33no—
238#419—33no—
239#419—33no—
240#417—33no—
241#417—33no—
242#416—33no—
243#416—33no—
244C2×#103—33no—
245C2×#103—33no—
246#405—33no—
247#405—33no—
248C4×M16—33no—
249C2×C4×C8—33no—
250C3×#41—224no—
251C3×#41—26no—
252C3×#41C22×C4112yes—
253C3×#41C22×C414yes—
254C3×#41#4114yes—
255C2×C3×Q8—28no—
256C2×C3×Q8—23no—
257C2×C3×Q8—26no—
258C2×C3×Q8C22×C4112yes—
259C2×C3×Q8C2×D411yescharacteristic
260C2×C3×Q8C2413yes—
261C2×C3×D4—23no—
262C2×C3×D4C2×D413yes—
263C2×C3×D4C2411yescharacteristic
264C22×C3×C4—23no—
265C22×C3×C4—23no—
266C22×C3×C4—23no—
267C22×C3×C4—21no—
268C22×C3×C4—23no—
269C22×C3×C4C2×D413yes—
270C22×C3×C4C2×D413yes—
271C22×C3×C4C2413yes—
272C2×#67—23no—
273C22×Dic3—23no—
274C22×Dic3—23no—
275D3×Q8—28no—
276#237—224no—
277#235—26no—
278#235C22×C4112yes—
279C2×D12C2411yescharacteristic
280C2×C4×D3—23no—
281C2×C4×D3—21no—
282C2×C4×D3—23no—
283C2×Dic6—23no—
284C2×Dic6—26no—
285C2×Dic6C22×C4112yes—
286C2×Dic6C2413yes—
287C3×C4⋊C4—26no—
288C3×C4⋊C4—23no—
289C3×C4⋊C4—23no—
290C3×C4⋊C4—23no—
291C3×C4⋊C4C2×D413yes—
292C3×C4⋊C4C2×D413yes—
293C3×C4⋊C4C2413yes—
294C3×C4⋊C4C2411yescharacteristic
295C3×C22⋊C4—23no—
296C3×C22⋊C4—21no—
297C3×C22⋊C4—23no—
298C3×C22⋊C4C2×D416yes—
299C3×C42—23no—
300C3×C42—23no—
301C3×C42—26no—
302C3×C42—23no—
303C3×C42C2×D413yes—
304C3×C42C2×D413yes—
305C3×C42C2411yescharacteristic
306C3×C42C2413yes—
307#217—23no—
308#216C22×C4124yes—
309#216C22×C4112yes—
310#215C22×C4112yes—
311#214C22×C4112yes—
312#213C22×C414yes—
313(C2×D3)⋊C4—21no—
314(C2×D3)⋊C4—23no—
315C4⋊Dic3—23no—
316C4⋊Dic3C2413yes—
317C4⋊Dic3C2411yescharacteristic
318Dic3⋊C4—23no—
319Dic3⋊C4—26no—
320Dic3⋊C4—23no—
321C4×Dic3—23no—
322C4×Dic3—26no—
323C4×Dic3—23no—
324C4×Dic3—23no—
325#208—26no—
326#208C22×C4112yes—
327C2×C3⋊C8—26no—
328C2×C3⋊C8C22×C4112yes—
329C2×C3⋊C8C2411yescharacteristic
330C2×C3⋊C8C2413yes—
331C2×C3⋊C8C2411yescharacteristic
332C2×C3⋊C8C2413yes—
3332−1+4—24no—
3342−1+4—38no—
3352−1+4—64no—
3362−1+4C4×D318yescomplemented
337C2×#41—23no—
338C2×#41—33no—
339C2×#41—63no—
340C2×#41C22×D313yes—
341C2×#41C22×D311yescharacteristic
342C22×Q8—21no—
343C22×Q8—33no—
344C22×Q8—61no—
345C22×Q8C22×D313yes—
346#137—324no—
347#137—324no—
348#135—324no—
349C2×Q16—324no—
350C2×Q16—33no—
351C2×Q16—36no—
352C2×SD16—33no—
353C2×SD16—33no—
354C2×D8—31no—
355#131—38no—
356#131—64no—
357C2×M16—33no—
358C2×M16—33no—
359C22×C8—33no—
360C22×C8—33no—
361C4×Q8—21no—
362C4×Q8—23no—
363C4×Q8—23no—
364C4×Q8—23no—
365C4×Q8—23no—
366C4×Q8—33no—
367C4×Q8—36no—
368C4×Q8—33no—
369C4×Q8—61no—
370C4×Q8—63no—
371C4×Q8—63no—
372C4×Q8—63no—
373C4×Q8—63no—
374C4×Q8#6711yescharacteristic
375C4×Q8#6711yescharacteristic
376C4×Q8C22×D313yes—
377C4×Q8C22×D316yes—
378C4×Q8C22×D313yes—
379C4×D4—23no—
380C4×D4—23no—
381C4×D4—23no—
382C4×D4—26no—
383C4×D4—31no—
384C4×D4—33no—
385C4×D4—63no—
386C4×D4—63no—
387C4×D4—63no—
388C4×D4—66no—
389C4×D4#6713yes—
390C4×D4#6713yes—
391C4×D4C22×D311yescharacteristic
392C4×D4C22×D313yes—
393#117—23no—
394#117—26no—
395#117—23no—
396#117—33no—
397#117—63no—
398#117—66no—
399#117—63no—
400#117C22×D313yes—
401#117C22×D313yes—
402C2×C4⋊C4—23no—
403C2×C4⋊C4—23no—
404C2×C4⋊C4—33no—
405C2×C4⋊C4—63no—
406C2×C4⋊C4—63no—
407C2×C4⋊C4C22×D313yes—
408C2×C42—23no—
409C2×C42—23no—
410C2×C42—63no—
411C2×C42—63no—
412C2×C42C22×D313yes—
413#107—33no—
414#107—36no—
415#107—31no—
416#106—33no—
417#106—33no—
418#103—33no—
419#103—36no—
420#103—33no—
421#103—33no—
422#103—36no—
423#103—33no—
424#102—31no—
425#102—33no—
426#102—31no—
427#102—33no—
428#97—36no—
429C4×C8—31no—
430C4×C8—33no—
431C4×C8—33no—
432C4×C8—33no—
433C23×C3—21no—
434C23×C3C22×D413yes—
435C22×D3—21no—
436C3×Q8—212no—
437C3×Q8—23no—
438C3×Q8C4×D414yes—
439C3×Q8C23×C416yes—
440C3×D4—212no—
441C3×D4—21no—
442C3×D4C4×D4112yes—
443C3×D4C23×C412yes—
444C2×C3×C4—212no—
445C2×C3×C4—23no—
446C2×C3×C4—212no—
447C2×C3×C4—26no—
448C2×C3×C4—21no—
449C2×C3×C4—23no—
450C2×C3×C4—43no—
451C2×C3×C4—42no—
452C2×C3×C4—83no—
453C2×C3×C4—83no—
454C2×C3×C4C23×C416yes—
455C2×C3×C4C22×D413yes—
456C2×C3×C4C22×D413yes—
457C2×C3×C4C22×D411yescharacteristic
458C2×C3×C4C22×D413yes—
459C2×C3×C4C22×D411yescharacteristic
460C2×C3×C4C2×#4116yes—
461C2×C3×C4C2511yescharacteristic
462#67—212no—
463C2×Dic3—212no—
464C2×Dic3—23no—
465C2×Dic3—21no—
466C2×Dic3—23no—
467C2×Dic3—83no—
468D12—21no—
469D12C23×C412yes—
470C4×D3—212no—
471C4×D3—42no—
472C4×D3—83no—
473Dic6—212no—
474Dic6—23no—
475Dic6C23×C416yes—
476C3⋊C8—21no—
477C3⋊C8—23no—
478C3⋊C8C23×C412yes—
479C3⋊C8C23×C416yes—
480#41—224no—
481#41—26no—
482#41—312no—
483#41—624no—
484#41—66no—
485#41C2×C4×D3112yes—
486#41C2×C4×D314yes—
487#41#23514yes—
488C2×Q8—28no—
489C2×Q8—26no—
490C2×Q8—23no—
491C2×Q8—312no—
492C2×Q8—33no—
493C2×Q8—68no—
494C2×Q8—66no—
495C2×Q8—63no—
496C2×Q8C2×C4×D3112yes—
497C2×Q8C2×#6711yescharacteristic
498C2×Q8C23×D313yes—
499C2×D4—23no—
500C2×D4—31no—
501C2×D4—63no—
502C2×D4C2×#6713yes—
503C2×D4C23×D311yescharacteristic
504C22×C4—23no—
505C22×C4—23no—
506C22×C4—23no—
507C22×C4—21no—
508C22×C4—23no—
509C22×C4—63no—
510C22×C4—63no—
511C22×C4—63no—
512C22×C4—61no—
513C22×C4—63no—
514C22×C4C2×#6713yes—
515C22×C4C2×#6713yes—
516C22×C4C23×D313yes—
517Q16—312no—
518Q16—324no—
519SD16—312no—
520SD16—312no—
521D8—34no—
522M16—312no—
523M16—66no—
524C2×C8—312no—
525C2×C8—31no—
526C2×C8—33no—
527C2×C8—31no—
528C2×C8—33no—
529C2×C8—66no—
530C4⋊C4—26no—
531C4⋊C4—23no—
532C4⋊C4—23no—
533C4⋊C4—23no—
534C4⋊C4—31no—
535C4⋊C4—33no—
536C4⋊C4—63no—
537C4⋊C4—66no—
538C4⋊C4—63no—
539C4⋊C4—63no—
540C4⋊C4C2×#6713yes—
541C4⋊C4C2×#6713yes—
542C4⋊C4C23×D311yescharacteristic
543C4⋊C4C23×D313yes—
544C22⋊C4—23no—
545C22⋊C4—21no—
546C22⋊C4—23no—
547C22⋊C4—63no—
548C22⋊C4—61no—
549C22⋊C4—63no—
550C22⋊C4C2×#6716yes—
551C42—23no—
552C42—26no—
553C42—23no—
554C42—23no—
555C42—63no—
556C42—63no—
557C42—66no—
558C42—63no—
559C42C2×#6713yes—
560C42C2×#6713yes—
561C42C23×D313yes—
562C42C23×D311yescharacteristic
563C22×C3—23no—
564C22×C3—23no—
565C22×C3—22no—
566C22×C3—41no—
567C22×C3—41no—
568C22×C3C2×C4×D416yes—
569C22×C3C23×D411yescharacteristic
570C2×D3—22no—
571C2×D3—41no—
572C2×D3—41no—
573C3×C4—23no—
574C3×C4—21no—
575C3×C4—26no—
576C3×C4—23no—
577C3×C4—43no—
578C3×C4—41no—
579C3×C4C2×C4×D416yes—
580C3×C4#57812yescomplemented
581C3×C4C24×C411yescharacteristic
582C3×C4C22×#4111yescharacteristic
583Dic3—26no—
584Dic3—43no—
585Dic3—41no—
586C23—21no—
587C23—61no—
588C23C22×#6713yes—
589Q8—23no—
590Q8—212no—
591Q8—36no—
592Q8—63no—
593Q8—612no—
594Q8C4×#6714yes—
595Q8C22×C4×D316yes—
596D4—212no—
597D4—21no—
598D4—32no—
599D4—612no—
600D4—61no—
601D4C4×#67112yes—
602D4C22×C4×D312yes—
603C2×C4—23no—
604C2×C4—212no—
605C2×C4—212no—
606C2×C4—26no—
607C2×C4—21no—
608C2×C4—23no—
609C2×C4—43no—
610C2×C4—42no—
611C2×C4—63no—
612C2×C4—612no—
613C2×C4—612no—
614C2×C4—61no—
615C2×C4—63no—
616C2×C4—83no—
617C2×C4—83no—
618C2×C4—122no—
619C2×C4—243no—
620C2×C4—243no—
621C2×C4C22×C4×D316yes—
622C2×C4C2×#23516yes—
623C2×C4C22×#6713yes—
624C2×C4C22×#6713yes—
625C2×C4C22×#6711yescharacteristic
626C2×C4C22×#6713yes—
627C2×C4C22×#6711yescharacteristic
628C2×C4C24×D311yescharacteristic
629C8—32no—
630C8—36no—
631C8—61no—
632C8—63no—
633C2×C3—26no—
634C2×C3—42no—
635C2×C3C22×C4×D411yescharacteristic
636C2×C3C2×#57811yescharacteristic
637C2×C3#358411yescharacteristic
638D3—42no—
639C22—23no—
640C22—23no—
641C22—22no—
642C22—41no—
643C22—41no—
644C22—62no—
645C22—121no—
646C22—121no—
647C22C2×C4×#6716yes—
648C22C23×#6711yescharacteristic
649C4—21no—
650C4—23no—
651C4—26no—
652C4—23no—
653C4—43no—
654C4—41no—
655C4—66no—
656C4—123no—
657C4—121no—
658C4C2×C4×#6716yes—
659C4#583812yesdirect product factor
660C4C23×C4×D311yescharacteristic
661C4C22×#23511yescharacteristic
662C3C4×#57811yescharacteristic, complemented, Sylow
663C2—26no—
664C2—42no—
665C2—122no—
666C2C22×C4×#6711yescharacteristic
667C2C2×#583811yescharacteristic
668C2#8729711yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC3×C4×2−1+4C2
socleC22×C3C23×D4
prime radicalC2×#5838C2
cosocleC2×C3×C4C25
hypercenterC4×2−1+4D3
hypocenterC3C4×#578
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C4×2−1+4

Complements

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

NH ≅ G/Ntype
1. C3×C4×2−1+4C2semidirect
3. C4×#1010C2semidirect
4. #84111C2semidirect
5. #82571C2semidirect
6. #82570C2semidirect
7. #82562C2semidirect
8. #82557C2semidirect
9. #82549C2semidirect
10. #82547C2semidirect
11. C4×#951C2semidirect
12. C4×#948C2semidirect
13. C4×#950C2semidirect
14. C4×#942C2semidirect
15. C2×C4×#216C2semidirect
29. #5838C4direct (N × H)
78. C4×2−1+4D3semidirect
92. C3×2−1+4C2×C4semidirect
99. #1010C2×C4semidirect
133. #951C2×C4semidirect
134. #950C2×C4semidirect
136. #948C2×C4semidirect
138. C2×#216C2×C4semidirect
141. #942C2×C4semidirect
336. 2−1+4C4×D3semidirect
580. C3×C4#578semidirect
659. C4#5838direct (N × H)
662. C3C4×#578semidirect

Maximal subgroups

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

subgroupindex
C3×C4×2−1+42
C2×#58382
C4×#10102
#841112
#825712
#825702
#825622
#825572
#825492
#825472
C4×#9512
C4×#9482
C4×#9502
C4×#9422
C2×C4×#2162
C4×#5783