Finite Group Databasebeta

← C2×C4×#456 #6372698

Subgroups — C2×C4×#456 #6372698

Order 512.

4632 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×2−1+4C211yesmaximal, characteristic
2C22×#456C211yesmaximal, characteristic
3C2×C4×#137C212yesmaximal, complemented
4C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
5C4×#456C2116yesmaximal, direct product factor
6C2×#1886C211yesmaximal, characteristic, complemented
7C2×#1882C212yesmaximal, complemented
8C2×#1859C212yesmaximal, complemented
9C2×#1857C211yesmaximal, characteristic, complemented
10C2×#1793C211yesmaximal, characteristic
11C2×C4×#104C212yesmaximal, complemented
12C2×C4×#101C211yesmaximal, characteristic, complemented
13C22×2−1+4C2211yescharacteristic
14C22×#137C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C4×2−1+4—28no—
17C4×2−1+4—24no—
18C4×2−1+4C2218yes—
19C2×#520—22no—
20C2×#520—22no—
21C2×#520C2212yes—
22C2×#519—22no—
23C2×#519C2211yescharacteristic
24C2×C4×#41—22no—
25C2×C4×#41—22no—
26C2×C4×#41C2212yes—
27C22×C4×Q8—22no—
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30C2×#456C4116yesdirect product factor
31C2×#456C2218yes—
32C4×#137—28no—
33C4×#137C22116yescomplemented
34C2×#441—22no—
35C2×#440—22no—
36C2×C4×Q16—22no—
37C2×C4×SD16—22no—
38C2×#428C2212yes—
39C22×#104C2212yes—
40C22×#104C2212yes—
41C2×#417C2212yes—
42C2×#419C2212yes—
43C22×#101C2211yescharacteristic
44C22×#101C2211yescharacteristic
45C2×C4×M16C2212yes—
46C4×C4⋊Q8—24no—
47C4×C4⋊Q8C2218yescomplemented
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#389—22no—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4—22no—
53C2×C43C2211yescharacteristic
54#1886C2218yescomplemented
55#1882C22116yescomplemented
56#1859C22116yescomplemented
57#1857C2218yescomplemented
58#1793C2218yes—
59C4×#104C22116yescomplemented
60C4×#101C2218yescomplemented
61C2×#343C2212yes—
62C2×#339C2212yes—
63C2×#339C2212yes—
64C2×2−1+4—24no—
65C2×2−1+4—28no—
66C2×2−1+4—22no—
67C2×2−1+4—24no—
68C2×2−1+4C2×C418yes—
69C2×2−1+4C2314yes—
70C22×#41—22no—
71C22×#41—22no—
72C22×#41C2312yes—
73C23×Q8—22no—
74C23×Q8C2311yescharacteristic
75C23×Q8C2311yescharacteristic
76C2×#137—24no—
77C2×#137—28no—
78C2×#137C2×C4116yescomplemented
79C2×#137C2318yes—
80C22×Q16—22no—
81C22×SD16—22no—
82C22×M16C2312yes—
83C22×M16C2312yes—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8C2×C418yescomplemented
87C2×C4⋊Q8C2314yes—
88#520—216no—
89#520—216no—
90#520—24no—
91#520—216no—
92#520—24no—
93#520—216no—
94#520—24no—
95#520C2318yes—
96#519—216no—
97#519—22no—
98#519—216no—
99#519—24no—
100#519C2314yes—
101C4×#41—216no—
102C4×#41—216no—
103C4×#41—216no—
104C4×#41—216no—
105C4×#41—24no—
106C4×#41—24no—
107C4×#41D418yes—
108C4×#41C2318yes—
109C2×C4×Q8—22no—
110C2×C4×Q8—24no—
111C2×C4×Q8—216no—
112C2×C4×Q8—22no—
113C2×C4×Q8—22no—
114C2×C4×Q8—216no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—24no—
120C2×C4×Q8—22no—
121C2×C4×Q8—22no—
122C2×C4×Q8—22no—
123C2×C4×Q8—24no—
124C2×C4×Q8D412yes—
125C2×C4×Q8D412yes—
126C2×C4×Q8D414yes—
127C2×C4×Q8C2314yes—
128C2×C4×Q8C2314yes—
129C2×C4×D4—22no—
130C2×C4×D4—24no—
131C2×C4×D4—22no—
132C2×C4×D4—22no—
133C2×C4×D4—22no—
134C2×C4×D4—24no—
135C2×C4×D4—22no—
136C2×C4×D4D412yes—
137C2×C4×D4D412yes—
138C2×#117—24no—
139C2×#117—24no—
140C2×#117—22no—
141C2×#117—22no—
142C2×#117—22no—
143C2×#117C2312yes—
144C22×C4⋊C4—24no—
145C22×C4⋊C4—22no—
146C22×C4⋊C4—22no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—22no—
149C22×C4⋊C4D412yes—
150C22×C4⋊C4C2311yescharacteristic
151C22×C4⋊C4C2311yescharacteristic
152C22×C42—24no—
153C22×C42—21no—
154C22×C42—22no—
155C22×C42—21no—
156C22×C42—21no—
157C22×C42D412yes—
158C22×C42C2311yescharacteristic
159C22×C42C2311yescharacteristic
160C22×C42C2311yescharacteristic
161#456C2×C4164yesdirect product factor
162#441—216no—
163#440—216no—
164C4×Q16—216no—
165C4×SD16—216no—
166#428—24no—
167#428C2318yes—
168C2×#107—22no—
169C2×#106—22no—
170C2×#104—28no—
171C2×#104C2×C4116yescomplemented
172C2×#104C2318yes—
173C2×#104C2318yes—
174#419—24no—
175#419C2318yes—
176#417—24no—
177#417C2318yes—
178C2×#103—22no—
179C2×#103—22no—
180C2×#103—22no—
181C2×#102—22no—
182C2×#101—24no—
183C2×#101C2×C418yescomplemented
184C2×#101C2314yes—
185C2×#101C2314yes—
186C4×M16—24no—
187C4×M16C2318yes—
188C2×#97—22no—
189C2×C4×C8—22no—
190#391—22no—
191#391—22no—
192#391C2314yes—
193#391C2314yes—
194#389—216no—
195#384—22no—
196#384C2314yes—
197C4×C4⋊C4—216no—
198C2×#95—22no—
199C2×#95—22no—
200C43—22no—
201C43C2314yes—
202#343C2318yes—
203#339C2318yes—
204#339C2318yes—
2052−1+4—216no—
2062−1+4—232no—
2072−1+4—232no—
2082−1+4—216no—
2092−1+4—216no—
2102−1+4—28no—
2112−1+4C22×C4116yes—
212C2×#41—28no—
213C2×#41—28no—
214C2×#41—216no—
215C2×#41—216no—
216C2×#41—216no—
217C2×#41—28no—
218C2×#41—216no—
219C2×#41—28no—
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—24no—
223C2×#41—24no—
224C2×#41C22×C418yes—
225C2×#41C2×D414yes—
226C2×#41#4118yes—
227C2×#41C2414yes—
228C22×Q8—22no—
229C22×Q8—28no—
230C22×Q8—216no—
231C22×Q8—216no—
232C22×Q8—28no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—24no—
241C22×Q8C22×C414yes—
242C22×Q8C22×C414yes—
243C22×Q8C2×D412yes—
244C22×Q8C2×D412yes—
245C22×Q8#4114yes—
246C22×Q8C2412yes—
247C22×Q8C2412yes—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4C2×D412yes—
251C23×C4—24no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—24no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—21no—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265#137—232no—
266#137—216no—
267#137C22×C4132yescomplemented
268C2×Q16—216no—
269C2×Q16—28no—
270C2×SD16—216no—
271C2×SD16—28no—
272C2×M16—22no—
273C2×M16—24no—
274C2×M16—22no—
275C2×M16—24no—
276C2×M16—24no—
277C2×M16C22×C418yes—
278C2×M16C2414yes—
279C2×M16C2414yes—
280C22×C8—22no—
281C22×C8—22no—
282C4⋊Q8—216no—
283C4⋊Q8—28no—
284C4⋊Q8C22×C4116yescomplemented
285C4×Q8—216no—
286C4×Q8—232no—
287C4×Q8—28no—
288C4×Q8—28no—
289C4×Q8—28no—
290C4×Q8—216no—
291C4×Q8—28no—
292C4×Q8—28no—
293C4×Q8—28no—
294C4×Q8—24no—
295C4×Q8—28no—
296C4×Q8—28no—
297C4×Q8—24no—
298C4×Q8—28no—
299C4×Q8—28no—
300C4×Q8—28no—
301C4×Q8—28no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×Q8C2×D418yes—
305C4×Q8C2×D418yes—
306C4×D4—216no—
307C4×D4—216no—
308C4×D4—232no—
309C4×D4—28no—
310C4×D4—216no—
311C4×D4—28no—
312C4×D4—28no—
313C4×D4—28no—
314C4×D4—28no—
315C4×D4—216no—
316C4×D4—28no—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4C2×D418yes—
320C4×D4C2×D418yes—
321#117—28no—
322#117—28no—
323#117—232no—
324#117—232no—
325#117—28no—
326#117—22no—
327#117—22no—
328#117—28no—
329#117—28no—
330#117—28no—
331#117C2×D414yes—
332#117C2414yes—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—232no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—216no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—28no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—28no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—48no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4C2×D418yes—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—232no—
368C2×C42—28no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—24no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—24no—
377C2×C42—22no—
378C2×C42—28no—
379C2×C42—28no—
380C2×C42—24no—
381C2×C42—24no—
382C2×C42—21no—
383C2×C42—24no—
384C2×C42—44no—
385C2×C42—41no—
386C2×C42—42no—
387C2×C42—41no—
388C2×C42C22×C414yes—
389C2×C42C2×D418yes—
390C2×C42C2×D412yes—
391C2×C42C2×D412yes—
392C2×C42C2×D414yes—
393C2×C42C2×D412yes—
394C2×C42C2412yes—
395C2×C42C2412yes—
396C2×C42C2412yes—
397#107—28no—
398#106—28no—
399#104—216no—
400#104C22×C4132yescomplemented
401#103—28no—
402#103—28no—
403#103—28no—
404#102—28no—
405#101—28no—
406#101C22×C4116yescomplemented
407#97—28no—
408C4×C8—28no—
409#95—28no—
410#95—28no—
411C24—22no—
412C24C22×D411yescharacteristic
413#41—232no—
414#41—264no—
415#41—232no—
416#41—264no—
417#41—232no—
418#41—232no—
419#41—28no—
420#41—28no—
421#41—28no—
422#41—28no—
423#41—28no—
424#41—24no—
425#41—24no—
426#41C4×D4116yes—
427#41C23×C418yes—
428#41C2×#4118yes—
429C2×Q8—28no—
430C2×Q8—216no—
431C2×Q8—232no—
432C2×Q8—264no—
433C2×Q8—232no—
434C2×Q8—28no—
435C2×Q8—28no—
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—28no—
441C2×Q8—24no—
442C2×Q8—28no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—28no—
447C2×Q8—28no—
448C2×Q8—24no—
449C2×Q8—24no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8C4×D418yes—
455C2×Q8C4×D418yes—
456C2×Q8C23×C414yes—
457C2×Q8C23×C414yes—
458C2×Q8C22×D414yes—
459C2×Q8C2×#4114yes—
460C2×D4—28no—
461C2×D4—216no—
462C2×D4—28no—
463C2×D4—24no—
464C2×D4—28no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—24no—
468C2×D4C4×D418yes—
469C2×D4C22×D414yes—
470C22×C4—216no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—232no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—28no—
481C22×C4—24no—
482C22×C4—28no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—216no—
493C22×C4—22no—
494C22×C4—28no—
495C22×C4—28no—
496C22×C4—22no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—48no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—41no—
507C22×C4—41no—
508C22×C4—84no—
509C22×C4—88no—
510C22×C4C4×D418yes—
511C22×C4#12011yescharacteristic
512C22×C4#12012yes—
513C22×C4#12012yes—
514C22×C4#12012yes—
515C22×C4#12011yescharacteristic
516C22×C4C23×C412yes—
517C22×C4C22×D414yes—
518C22×C4C22×D412yes—
519C22×C4C22×D414yes—
520C22×C4C22×D412yes—
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C2×#4114yes—
525C22×C4C2×#4112yes—
526C22×C4C2511yescharacteristic
527Q16—232no—
528SD16—232no—
529M16—28no—
530M16—24no—
531M16—24no—
532M16—24no—
533M16—44no—
534M16C23×C418yes—
535C2×C8—28no—
536C2×C8—24no—
537C2×C8—24no—
538C2×C8—44no—
539C4⋊C4—216no—
540C4⋊C4—28no—
541C4⋊C4—28no—
542C4⋊C4—232no—
543C4⋊C4—28no—
544C4⋊C4—216no—
545C4⋊C4—24no—
546C4⋊C4—22no—
547C4⋊C4—28no—
548C4⋊C4—22no—
549C4⋊C4—24no—
550C4⋊C4—24no—
551C4⋊C4—416no—
552C4⋊C4—42no—
553C4⋊C4—48no—
554C4⋊C4C22×D414yes—
555C4⋊C4C22×D414yes—
556C22⋊C4—216no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—28no—
560C22⋊C4—28no—
561C42—216no—
562C42—216no—
563C42—216no—
564C42—24no—
565C42—28no—
566C42—28no—
567C42—28no—
568C42—24no—
569C42—28no—
570C42—24no—
571C42—22no—
572C42—24no—
573C42—48no—
574C42—44no—
575C42—44no—
576C42—42no—
577C42#12014yes—
578C42#12014yes—
579C42C23×C414yes—
580C42C22×D414yes—
581C42C22×D414yes—
582C23—22no—
583C23—22no—
584C23—24no—
585C23—28no—
586C23—44no—
587C23—44no—
588C23C2×C4×D412yes—
589C23C2×#12011yescharacteristic
590C23C2×#12011yescharacteristic
591C23#52512yes—
592C23C23×D411yescharacteristic
593Q8—232no—
594Q8—28no—
595Q8—24no—
596Q8—28no—
597Q8—28no—
598Q8—216no—
599Q8—28no—
600Q8—44no—
601Q8—48no—
602Q8C2×C4×D418yes—
603D4—232no—
604D4—28no—
605D4—28no—
606D4—216no—
607D4—24no—
608D4—28no—
609D4C2×C4×D418yes—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—216no—
613C2×C4—264no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—216no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—48no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—416no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—88no—
650C2×C4—82no—
651C2×C4—816no—
652C2×C4—82no—
653C2×C4—82no—
654C2×C4#45614yesdirect product factor
655C2×C4C2×C4×D418yes—
656C2×C4C2×C4×D414yes—
657C2×C4C2×C4×D414yes—
658C2×C4C2×C4×D412yes—
659C2×C4C2×#12012yes—
660C2×C4C2×#12014yes—
661C2×C4C2×#12012yes—
662C2×C4C2×#12012yes—
663C2×C4#52514yes—
664C2×C4#52514yes—
665C2×C4#52512yes—
666C2×C4C24×C411yescharacteristic
667C2×C4C23×D412yes—
668C2×C4C22×#4112yes—
669C2×C4C22×#4111yescharacteristic
670C8—28no—
671C8—44no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—24no—
677C22—24no—
678C22—28no—
679C22—44no—
680C22—44no—
681C22—416no—
682C22C4×#12012yes—
683C22C4×#12012yes—
684C22C2×#45612yes—
685C22#301812yes—
686C22C22×C4×D411yescharacteristic
687C22C22×#12011yescharacteristic
688C22C2×#52511yescharacteristic
689C4—28no—
690C4—22no—
691C4—24no—
692C4—28no—
693C4—28no—
694C4—44no—
695C4—44no—
696C4—44no—
697C4—44no—
698C4—44no—
699C4—82no—
700C4C4×#12014yes—
701C4C2×#45614yesdirect product factor
702C4C22×C4×D412yes—
703C4C2×#52512yes—
704C2—21no—
705C2—21no—
706C2—22no—
707C2—48no—
708C2C4×#45614yesdirect product factor
709C2C2×C4×#12011yescharacteristic
710C2C22×#45611yescharacteristic
711C2C2×#301811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C2×#120
prime radicalC22×2−1+4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×#137C2semidirect
4. C2×C4×C4⋊Q8C2semidirect
5. C4×#456C2direct (N × H)
6. C2×#1886C2semidirect
7. C2×#1882C2semidirect
8. C2×#1859C2semidirect
9. C2×#1857C2semidirect
11. C2×C4×#104C2semidirect
12. C2×C4×#101C2semidirect
30. C2×#456C4direct (N × H)
33. C4×#137C22semidirect
47. C4×C4⋊Q8C22semidirect
54. #1886C22semidirect
55. #1882C22semidirect
56. #1859C22semidirect
57. #1857C22semidirect
59. C4×#104C22semidirect
60. C4×#101C22semidirect
78. C2×#137C2×C4semidirect
86. C2×C4⋊Q8C2×C4semidirect
161. #456C2×C4direct (N × H)
171. C2×#104C2×C4semidirect
183. C2×#101C2×C4semidirect
267. #137C22×C4semidirect
284. C4⋊Q8C22×C4semidirect
400. #104C22×C4semidirect
406. #101C22×C4semidirect
654. C2×C4#456direct (N × H)
701. C4C2×#456direct (N × H)
708. C2C4×#456direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×2−1+42
C22×#4562
C2×C4×#1372
C2×C4×C4⋊Q82
C4×#4562
C2×#18862
C2×#18822
C2×#18592
C2×#18572
C2×#17932
C2×C4×#1042
C2×C4×#1012