Finite Group Databasebeta

← C4×#5831 #10955006

Subgroups — C4×#5831 #10955006

Order 768.

3232 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#41C211yesmaximal, characteristic, complemented
2C2×#5831C211yesmaximal, characteristic
3C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
4#84089C211yesmaximal, characteristic, complemented
5Q8×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6#82461C211yesmaximal, characteristic, complemented
7C4×Q8×Dic3C211yesmaximal, characteristic, complemented
8#82399C213yesmaximal, complemented
9C4×#938C213yesmaximal, complemented
10#82344C213yesmaximal, complemented
11C4×#936C213yesmaximal, complemented
12#82272C213yesmaximal, complemented
13#82243C213yesmaximal, complemented
14C4×#924C213yesmaximal, complemented
15C42×#67C213yesmaximal, complemented
16C4×#548—31nomaximal, Sylow
17C22×C3×#41C2211yescharacteristic
18C22×D3×Q8C2211yescharacteristic
19C3×C4×#41—28no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4C2213yes—
23C2×C3×C4×D4C2213yes—
24C2×C3×#117C2213yes—
25C22×C3×C42C2213yes—
26#5831C4116yesdirect product factor
27C2×Q8×Dic3C2211yescharacteristic
28C2×Q8×Dic3C2211yescharacteristic
29C2×Q8×Dic3C2211yescharacteristic
30C2×#938C2213yes—
31C2×#936C2213yes—
32C2×C4×#67C2213yes—
33C2×C4×#67C2213yes—
34C2×C4×#67C2213yes—
35C2×#924C2213yes—
36C4×D3×Q8—28no—
37C2×D3×C4⋊C4C2213yes—
38C2×#887C2213yes—
39C2×C42×D3C2213yes—
40C2×C4×Dic6C2213yes—
41#5221C2213yes—
42#5221C2213yes—
43#5212C2213yes—
44#5210C2216yes—
45C4×#217C2213yes—
46C4×#217C2213yes—
47C4×#217C2213yes—
48C4×#217C2216yes—
49#4987C2213yes—
50#4977C2213yes—
51Dic3×C4⋊C4C2213yes—
52Dic3×C4⋊C4C2213yes—
53#4973C2213yes—
54#4972C2213yes—
55#4956C2216yes—
56#4945C2216yes—
57#4945C2216yes—
58#4944C2216yes—
59C4×(C2×D3)⋊C4C2213yes—
60C4×(C2×D3)⋊C4C2213yes—
61C4×(C2×D3)⋊C4C2211yescharacteristic
62C4×C4⋊Dic3C2213yes—
63#4933C2216yes—
64C4×Dic3⋊C4C2213yes—
65C4×Dic3⋊C4C2213yes—
66C4×Dic3⋊C4C2213yes—
67C42×Dic3C2213yes—
68#4929C2213yes—
69#4928C2213yes—
70C2×#548—31no—
71C2×C4×#41D311yescharacteristic, complemented
72C22×C4×Q8—31no—
73#2367—33no—
74#2362—33no—
75#2361—33no—
76#2352—33no—
77#2351—31no—
78#2341—31no—
79Q8×C22⋊C4—31no—
80C4×#124—33no—
81C4×C22⋊Q8—33no—
82C4×#121—33no—
83C42×Q8—31no—
84C42×D4—33no—
85C2×C3×#41—24no—
86C2×C3×#41—24no—
87C2×C3×#41C2×C418yescomplemented
88C22×C3×Q8C2311yescharacteristic
89C22×C3×D4C2313yes—
90C23×C3×C4C2313yes—
91C23×C3×C4C2313yes—
92C23×C3×C4C2313yes—
93C22×#67C2313yes—
94C2×D3×Q8—24no—
95C2×D3×Q8—24no—
96C2×D3×Q8C2×C418yescomplemented
97C22×C4×D3C2313yes—
98C22×C4×D3C2313yes—
99C22×C4×D3C2311yescharacteristic
100C22×Dic6C2313yes—
101C3×C4×Q8—24no—
102C3×C4×Q8D418yescomplemented
103C3×C4×D4—212no—
104C3×C4×D4—212no—
105C3×#117—212no—
106C2×C3×C4⋊C4C2313yes—
107C2×C3×C4⋊C4C2313yes—
108C2×C3×C22⋊C4C2316yes—
109C2×C3×C42—212no—
110C2×C3×C42C2313yes—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2313yes—
113C2×C3×C42C2313yes—
114C2×#217C2313yes—
115C2×#217C2313yes—
116C2×#217C2316yes—
117C2×#217C2316yes—
118C2×#217C2313yes—
119Q8×Dic3—24no—
120Q8×Dic3—24no—
121Q8×Dic3C2×C418yescomplemented
122#938C2×C4124yescomplemented
123#936C2×C4124yescomplemented
124C4×#67—212no—
125C4×#67—212no—
126C4×#67C2×C4124yescomplemented
127C2×(C2×D3)⋊C4C2313yes—
128C2×(C2×D3)⋊C4C2311yescharacteristic
129C2×(C2×D3)⋊C4C2311yescharacteristic
130C2×C4⋊Dic3C2313yes—
131C2×C4⋊Dic3C2313yes—
132C2×C4⋊Dic3C2313yes—
133#924C2×C4124yescomplemented
134C2×Dic3⋊C4C2313yes—
135C2×Dic3⋊C4C2313yes—
136C2×Dic3⋊C4C2316yes—
137C2×Dic3⋊C4C2313yes—
138C2×C4×Dic3C2311yescharacteristic
139C2×C4×Dic3C2313yes—
140C2×C4×Dic3C2313yes—
141C2×C4×Dic3C2313yes—
142C2×C4×Dic3C2313yes—
143C2×C4×Dic3C2313yes—
144C2×C4×Dic3C2313yes—
145C2×C4×Dic3C2313yes—
146C2×C4×Dic3C2313yes—
147D3×C4⋊C4—212no—
148#887—212no—
149C42×D3—212no—
150C4×Dic6—212no—
151C3⋊#95C2316yes—
152C3⋊#95C2313yes—
153C3⋊#95C2316yes—
154C3⋊#95C2311yescharacteristic
155C3⋊#95C2311yescharacteristic
156C3⋊#95C2313yes—
157C3⋊#95C2313yes—
158C3⋊#95C2313yes—
159C3⋊#95C2316yes—
160C22×#41C2×D311yescharacteristic
161C23×Q8—31no—
162#548—316no—
163C2×#124—33no—
164C2×C22⋊Q8—33no—
165C2×#121—33no—
166C4×#41—28no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×Q8—33no—
171C2×C4×Q8—33no—
172C2×C4×Q8—68no—
173C2×C4×Q8C2×D311yescharacteristic
174C2×C4×Q8C2×D311yescharacteristic
175C2×C4×D4—33no—
176C2×C4×D4—33no—
177C2×C4×D4—33no—
178C2×C4×D4C2×D313yes—
179C2×C4×D4C2×D313yes—
180C2×#117C2×D313yes—
181C22×C4⋊C4—33no—
182C22×C42—33no—
183C22×C42C2×D313yes—
184#391—33no—
185#391—33no—
186#390—33no—
187#390—33no—
188#389—33no—
189#389—33no—
190#388—36no—
191#388—36no—
192#388—36no—
193#387—36no—
194#386—36no—
195#385—36no—
196#382—33no—
197#380—33no—
198#380—33no—
199C4×C4⋊C4—33no—
200C4×C4⋊C4—33no—
201C4×C4⋊C4—33no—
202C4×C4⋊C4—33no—
203C4×C4⋊C4—33no—
204C4×C4⋊C4—33no—
205C4×C22⋊C4—33no—
206C4×C22⋊C4—33no—
207C4×C22⋊C4—33no—
208C4×C22⋊C4—36no—
209C4×C22⋊C4—33no—
210C4×C22⋊C4—33no—
211C4×C22⋊C4—31no—
212C43—33no—
213C24×C3C2413yes—
214C23×D3C2411yescharacteristic
215C3×#41—232no—
216C2×C3×Q8C22×C414yes—
217C2×C3×Q8C2×D414yes—
218C2×C3×Q8#4114yes—
219C2×C3×D4—26no—
220C2×C3×D4—26no—
221C2×C3×D4C22×C4112yes—
222C22×C3×C4—26no—
223C22×C3×C4—26no—
224C22×C3×C4—26no—
225C22×C3×C4—26no—
226C22×C3×C4—26no—
227C22×C3×C4—26no—
228C22×C3×C4—26no—
229C22×C3×C4—43no—
230C22×C3×C4C22×C4112yes—
231C22×C3×C4C2413yes—
232C22×C3×C4C2411yescharacteristic
233C22×C3×C4C2413yes—
234C22×C3×C4C2411yescharacteristic
235C22×C3×C4C2413yes—
236C22×C3×C4C2411yescharacteristic
237C2×#67—26no—
238C2×#67—26no—
239C2×#67C22×C4112yes—
240C22×Dic3C2413yes—
241C22×Dic3C2413yes—
242C22×Dic3C2411yescharacteristic
243C22×Dic3C2413yes—
244C22×Dic3C2413yes—
245C22×Dic3C2411yescharacteristic
246C22×Dic3C2411yescharacteristic
247D3×Q8—232no—
248C2×C4×D3—22no—
249C2×C4×D3—26no—
250C2×C4×D3—26no—
251C2×C4×D3—22no—
252C2×C4×D3—26no—
253C2×C4×D3—26no—
254C2×C4×D3—43no—
255C2×C4×D3C22×C4112yes—
256C2×C4×D3#4114yescomplemented
257C2×Dic6—26no—
258C2×Dic6—26no—
259C2×Dic6C22×C4112yes—
260C3×C4⋊C4—26no—
261C3×C4⋊C4C2×D4112yes—
262C3×C22⋊C4—212no—
263C3×C42—22no—
264C3×C42—26no—
265C3×C42—26no—
266C3×C42C2×D4112yes—
267#217—26no—
268#217—26no—
269#217—46no—
270#217C22×C4124yes—
271#217C22×C4112yes—
272(C2×D3)⋊C4—26no—
273(C2×D3)⋊C4C22×C414yes—
274(C2×D3)⋊C4#4114yescomplemented
275C4⋊Dic3—26no—
276C4⋊Dic3—26no—
277C4⋊Dic3C22×C4112yes—
278Dic3⋊C4—212no—
279Dic3⋊C4—26no—
280Dic3⋊C4—26no—
281Dic3⋊C4C22×C4112yes—
282C4×Dic3—26no—
283C4×Dic3—26no—
284C4×Dic3—26no—
285C4×Dic3—26no—
286C4×Dic3—26no—
287C4×Dic3—22no—
288C4×Dic3—26no—
289C4×Dic3—43no—
290C4×Dic3C22×C4112yes—
291C2×#41—24no—
292C2×#41—24no—
293C2×#41C4×D318yescomplemented
294C22×Q8—38no—
295C22×Q8—33no—
296C22×Q8—64no—
297C22×Q8—64no—
298C22×Q8C22×D311yescharacteristic
299C22×D4—33no—
300C22×D4C22×D313yes—
301C23×C4—33no—
302C23×C4—31no—
303C23×C4—33no—
304C23×C4C22×D313yes—
305C23×C4C22×D313yes—
306C23×C4C22×D313yes—
307#124—324no—
308C22⋊Q8—324no—
309#121—324no—
310C4×Q8—24no—
311C4×Q8—38no—
312C4×Q8—64no—
313C4×Q8—64no—
314C4×Q8—612no—
315C4×Q8—612no—
316C4×Q8#6718yescomplemented
317C4×D4—212no—
318C4×D4—212no—
319C4×D4—324no—
320C4×D4—612no—
321C4×D4—612no—
322#117—212no—
323C2×C4⋊C4—33no—
324C2×C4⋊C4—33no—
325C2×C4⋊C4—33no—
326C2×C4⋊C4—36no—
327C2×C4⋊C4—33no—
328C2×C4⋊C4—33no—
329C2×C4⋊C4—33no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4C22×D313yes—
332C2×C4⋊C4C22×D313yes—
333C2×C22⋊C4—33no—
334C2×C22⋊C4—36no—
335C2×C22⋊C4—31no—
336C2×C22⋊C4—31no—
337C2×C22⋊C4—33no—
338C2×C22⋊C4—33no—
339C2×C22⋊C4—33no—
340C2×C22⋊C4—36no—
341C2×C22⋊C4C22×D316yes—
342C2×C42—212no—
343C2×C42—33no—
344C2×C42—33no—
345C2×C42—33no—
346C2×C42—33no—
347C2×C42—33no—
348C2×C42—33no—
349C2×C42—33no—
350C2×C42—31no—
351C2×C42—33no—
352C2×C42—612no—
353C2×C42C22×D313yes—
354C2×C42C22×D313yes—
355C2×C42C22×D311yescharacteristic
356C2×C42C22×D313yes—
357#95—31no—
358#95—31no—
359#95—33no—
360#95—33no—
361#95—33no—
362#95—33no—
363#95—36no—
364#95—36no—
365#95—36no—
366C23×C3—23no—
367C23×C3—23no—
368C23×C3—23no—
369C23×C3—23no—
370C23×C3—23no—
371C23×C3—23no—
372C23×C3C23×C416yes—
373C23×C3C2511yescharacteristic
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—21no—
378C22×D3C23×C412yes—
379C22×D3C2×#4112yes—
380C22×D3C2×#4112yes—
381C3×Q8C4×D4116yescomplemented
382C3×D4—224no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—23no—
386C2×C3×C4—21no—
387C2×C3×C4—23no—
388C2×C3×C4—21no—
389C2×C3×C4—224no—
390C2×C3×C4—23no—
391C2×C3×C4—23no—
392C2×C3×C4—23no—
393C2×C3×C4—412no—
394C2×C3×C4—412no—
395C2×C3×C4C23×C412yes—
396C2×C3×C4C23×C416yes—
397C2×C3×C4C22×D416yes—
398C2×C3×C4C22×D416yes—
399C2×C3×C4C22×D412yes—
400C2×C3×C4C2×#4116yes—
401C2×C3×C4C2×#4112yes—
402C2×C3×C4C2×#4112yes—
403C2×C3×C42−1+412yes—
404#67—224no—
405C2×Dic3—21no—
406C2×Dic3—23no—
407C2×Dic3—21no—
408C2×Dic3—23no—
409C2×Dic3—23no—
410C2×Dic3—23no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—23no—
416C2×Dic3—23no—
417C2×Dic3—23no—
418C2×Dic3—23no—
419C2×Dic3—23no—
420C2×Dic3—23no—
421C2×Dic3C23×C412yes—
422C2×Dic3C23×C416yes—
423C2×Dic3C23×C416yes—
424C2×Dic3C2×#4112yes—
425C2×Dic3C2×#4112yes—
426C4×D3—224no—
427C4×D3—28no—
428C4×D3—412no—
429Dic6—224no—
430C24—31no—
431C24C23×D313yes—
432#41—232no—
433C2×Q8—312no—
434C2×Q8—632no—
435C2×Q8—66no—
436C2×Q8—66no—
437C2×Q8C2×C4×D314yes—
438C2×Q8#23514yes—
439C2×Q8C2×#6714yes—
440C2×D4—26no—
441C2×D4—26no—
442C2×D4—312no—
443C2×D4—66no—
444C2×D4—66no—
445C2×D4C2×C4×D3112yes—
446C22×C4—26no—
447C22×C4—26no—
448C22×C4—26no—
449C22×C4—26no—
450C22×C4—26no—
451C22×C4—26no—
452C22×C4—26no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—33no—
456C22×C4—33no—
457C22×C4—31no—
458C22×C4—312no—
459C22×C4—34no—
460C22×C4—33no—
461C22×C4—33no—
462C22×C4—43no—
463C22×C4—66no—
464C22×C4—66no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—66no—
468C22×C4—66no—
469C22×C4—123no—
470C22×C4C2×C4×D3112yes—
471C22×C4C23×D313yes—
472C22×C4C23×D311yescharacteristic
473C22×C4C23×D311yescharacteristic
474C22×C4C23×D313yes—
475C22×C4C23×D313yes—
476C22×C4C23×D311yescharacteristic
477C4⋊C4—26no—
478C4⋊C4—312no—
479C4⋊C4—312no—
480C4⋊C4—66no—
481C4⋊C4—66no—
482C4⋊C4—66no—
483C4⋊C4—612no—
484C4⋊C4—66no—
485C4⋊C4C2×#67112yes—
486C22⋊C4—212no—
487C22⋊C4—312no—
488C22⋊C4—324no—
489C22⋊C4—34no—
490C22⋊C4—34no—
491C22⋊C4—66no—
492C22⋊C4—66no—
493C22⋊C4—66no—
494C22⋊C4—126no—
495C42—22no—
496C42—26no—
497C42—26no—
498C42—312no—
499C42—66no—
500C42—66no—
501C42—66no—
502C42—62no—
503C42—66no—
504C42—66no—
505C42—66no—
506C42—123no—
507C42C2×#67112yes—
508C22×C3—26no—
509C22×C3—26no—
510C22×C3—26no—
511C22×C3—43no—
512C22×C3—43no—
513C22×C3—43no—
514C22×C3—43no—
515C22×C3C24×C411yescharacteristic
516C22×C3C23×D411yescharacteristic
517C22×C3C22×#4111yescharacteristic
518C22×C3C22×#4111yescharacteristic
519C22×C3C22×#4111yescharacteristic
520C22×C3C2×2−1+411yescharacteristic
521C22×C3#58511yescharacteristic
522C2×D3—22no—
523C2×D3—22no—
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—22no—
527C2×D3—22no—
528C2×D3C4×#4114yescomplemented
529C3×C4—22no—
530C3×C4—26no—
531C3×C4—22no—
532C3×C4—43no—
533C3×C4C2×C4×D4112yes—
534C3×C4#54814yescomplemented
535Dic3—26no—
536Dic3—26no—
537Dic3—22no—
538Dic3—22no—
539Dic3—22no—
540Dic3—43no—
541Dic3—43no—
542C23—23no—
543C23—23no—
544C23—23no—
545C23—23no—
546C23—23no—
547C23—23no—
548C23—32no—
549C23—32no—
550C23—32no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23C22×C4×D316yes—
556C23C24×D311yescharacteristic
557Q8—624no—
558Q8C4×#67116yescomplemented
559D4—224no—
560D4—624no—
561C2×C4—21no—
562C2×C4—23no—
563C2×C4—23no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—224no—
568C2×C4—23no—
569C2×C4—23no—
570C2×C4—23no—
571C2×C4—32no—
572C2×C4—36no—
573C2×C4—36no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—412no—
577C2×C4—412no—
578C2×C4—61no—
579C2×C4—63no—
580C2×C4—63no—
581C2×C4—63no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—63no—
588C2×C4—624no—
589C2×C4—68no—
590C2×C4—63no—
591C2×C4—63no—
592C2×C4—63no—
593C2×C4—63no—
594C2×C4—63no—
595C2×C4—63no—
596C2×C4—1212no—
597C2×C4C22×C4×D312yes—
598C2×C4C22×C4×D316yes—
599C2×C4C2×#23516yes—
600C2×C4D3×#4112yes—
601C2×C4D3×#4112yes—
602C2×C4#101012yes—
603C2×C4C22×#6716yes—
604C2×C4C22×#6716yes—
605C2×C4C22×#6712yes—
606C2×C3—46no—
607C2×C3C22×C4×D411yescharacteristic
608C2×C3C2×C4×#4111yescharacteristic
609C2×C3C4×2−1+411yescharacteristic
610C2×C3C2×#54811yescharacteristic
611C2×C3D4×#4111yescharacteristic
612C2×C3#348011yescharacteristic
613C2×C3#348211yescharacteristic
614D3—24no—
615C22—26no—
616C22—26no—
617C22—26no—
618C22—34no—
619C22—43no—
620C22—43no—
621C22—43no—
622C22—43no—
623C22—62no—
624C22—62no—
625C22—62no—
626C22—62no—
627C22—62no—
628C22—62no—
629C22C23×C4×D311yescharacteristic
630C22C22×#23511yescharacteristic
631C22C2×D3×#4111yescharacteristic
632C22C2×D3×#4111yescharacteristic
633C22C2×#101011yescharacteristic
634C22#596411yescharacteristic
635C22C23×#6711yescharacteristic
636C4—22no—
637C4—22no—
638C4—26no—
639C4—43no—
640C4—62no—
641C4—66no—
642C4—62no—
643C4—66no—
644C4—62no—
645C4—123no—
646C4—123no—
647C4C2×C4×#67112yes—
648C4#583114yesdirect product factor
649C3C4×#54811yescharacteristic, complemented, Sylow
650C2—46no—
651C2—64no—
652C2C4×#101011yescharacteristic
653C2C4×D3×#4111yescharacteristic
654C2#8584211yescharacteristic
655C2#8584411yescharacteristic
656C2C22×C4×#6711yescharacteristic
657C2C2×#583111yescharacteristic
658C2#41×#6711yescharacteristic
subgroupG/H
centerC22×C4C23×D3
commutatorC22×C3C24×C4
FrattiniC23C24×D3
FittingC2×C3×C4×#41C2
socle, cosocleC23×C3C25
prime radicalC2×#5831C2
hypercenterC2×C4×#41D3
hypocenterC3C4×#548
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×#41C2semidirect
3. C2×C4×D3×Q8C2semidirect
4. #84089C2semidirect
5. Q8×(C2×D3)⋊C4C2semidirect
6. #82461C2semidirect
7. C4×Q8×Dic3C2semidirect
8. #82399C2semidirect
9. C4×#938C2semidirect
10. #82344C2semidirect
11. C4×#936C2semidirect
12. #82272C2semidirect
13. #82243C2semidirect
14. C4×#924C2semidirect
15. C42×#67C2semidirect
26. #5831C4direct (N × H)
71. C2×C4×#41D3semidirect
87. C2×C3×#41C2×C4semidirect
96. C2×D3×Q8C2×C4semidirect
102. C3×C4×Q8D4semidirect
121. Q8×Dic3C2×C4semidirect
122. #938C2×C4semidirect
123. #936C2×C4semidirect
126. C4×#67C2×C4semidirect
133. #924C2×C4semidirect
256. C2×C4×D3#41semidirect
274. (C2×D3)⋊C4#41semidirect
293. C2×#41C4×D3semidirect
316. C4×Q8#67semidirect
381. C3×Q8C4×D4semidirect
528. C2×D3C4×#41semidirect
534. C3×C4#548semidirect
558. Q8C4×#67semidirect
648. C4#5831direct (N × H)
649. C3C4×#548semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×#412
C2×#58312
C2×C4×D3×Q82
#840892
Q8×(C2×D3)⋊C42
#824612
C4×Q8×Dic32
#823992
C4×#9382
#823442
C4×#9362
#822722
#822432
C4×#9242
C42×#672
C4×#5483