Finite Group Databasebeta

← C2×#19619 #6357097

Subgroups — C2×#19619 #6357097

Order 512.

2400 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×#3169C211yesmaximal, characteristic, complemented
2C2×#3169C211yesmaximal, characteristic, complemented
3C2×#3167C211yesmaximal, characteristic, complemented
4C2×#3167C211yesmaximal, characteristic, complemented
5C2×#3147C211yesmaximal, characteristic, complemented
6C2×#3131C211yesmaximal, characteristic, complemented
7#19619C2116yesmaximal, direct product factor
8C2×#1723C211yesmaximal, characteristic, complemented
9C2×#1721C211yesmaximal, characteristic, complemented
10C2×#1720C211yesmaximal, characteristic, complemented
11C2×#1719C211yesmaximal, characteristic, complemented
12C2×#1717C211yesmaximal, characteristic, complemented
13C2×#1717C211yesmaximal, characteristic, complemented
14C2×#1715C211yesmaximal, characteristic, complemented
15C2×#1714C211yesmaximal, characteristic, complemented
16C22×#322C211yesmaximal, characteristic, complemented
17C22×#135—22no—
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×#532C2211yescharacteristic
21C2×#532C2211yescharacteristic
22#3169—24no—
23#3169—24no—
24#3169C2218yescomplemented
25#3169C2218yescomplemented
26#3167—24no—
27#3167—24no—
28#3167C2218yescomplemented
29#3167C2218yescomplemented
30C2×#500C2211yescharacteristic
31C2×#500C2211yescharacteristic
32C2×#498C2211yescharacteristic
33C2×#498C2211yescharacteristic
34#3147—24no—
35#3147C2218yescomplemented
36C2×#495—22no—
37C2×#494C2211yescharacteristic
38C2×#493C2211yescharacteristic
39C2×#493C2211yescharacteristic
40C2×#493C2211yescharacteristic
41C2×#492C2211yescharacteristic
42C2×#492C2211yescharacteristic
43C2×#492C2211yescharacteristic
44C2×#492C2211yescharacteristic
45#3131—24no—
46#3131C2218yescomplemented
47C2×#488—22no—
48C2×#487C2211yescharacteristic
49C2×#487C2211yescharacteristic
50C2×#487C2211yescharacteristic
51C2×#487C2211yescharacteristic
52C2×#486C2211yescharacteristic
53C2×#486C2211yescharacteristic
54C2×#486C2211yescharacteristic
55C2×#486C2211yescharacteristic
56C2×#486C2211yescharacteristic
57C2×#486C2211yescharacteristic
58C2×#486C2211yescharacteristic
59C2×#486C2211yescharacteristic
60C2×#466—22no—
61C2×#466—22no—
62C2×#465—22no—
63C2×#465—22no—
64C2×C4×SD16—22no—
65C2×C4×SD16—22no—
66C2×C4×D8—22no—
67C2×C4×D8—22no—
68C2×#416—22no—
69C22×C4×C8C2211yescharacteristic
70C22×C4×C8C2211yescharacteristic
71C22×C4×C8C2211yescharacteristic
72#1723C2218yescomplemented
73#1721C2218yescomplemented
74#1720C2218yescomplemented
75#1719C2218yescomplemented
76#1717C2218yescomplemented
77#1717C2218yescomplemented
78#1715C2218yescomplemented
79#1714C2218yescomplemented
80C2×#322C2211yescharacteristic
81C2×#322C2211yescharacteristic
82C2×#322C2211yescharacteristic
83C2×#322C2211yescharacteristic
84C2×#322C2218yescomplemented
85C22×#41—21no—
86C22×#41—21no—
87C22×#41—21no—
88C2×#135—216no—
89C2×#135—416no—
90C22×Q16—22no—
91C22×SD16—22no—
92C22×SD16—22no—
93C22×SD16—22no—
94C22×SD16—22no—
95C22×D8—22no—
96C22×D8—22no—
97C22×D8—22no—
98C23×C8D412yes—
99C23×C8D412yes—
100C23×C8D412yes—
101#533—22no—
102#533—22no—
103#533—22no—
104#533C2314yes—
105#532—22no—
106#532—22no—
107#532—22no—
108#532—22no—
109#532—22no—
110#532—22no—
111#532—22no—
112#532—22no—
113#532—22no—
114#532C2314yes—
115#532C2314yes—
116#532C2314yes—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×#127C2311yescharacteristic
123C2×#127C2311yescharacteristic
124C2×#127C2311yescharacteristic
125C2×#125—21no—
126C2×#124—21no—
127C2×#124—21no—
128C2×#124—21no—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—21no—
131C2×#121—21no—
132C2×#121—21no—
133C2×#121—21no—
134C2×#121—21no—
135C2×#121—21no—
136C2×#121—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8—21no—
139C2×C4×D4—21no—
140C2×C4×D4—21no—
141C2×C4×D4—21no—
142C2×C4×D4—21no—
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×#117—21no—
146C22×C42C2311yescharacteristic
147#500—22no—
148#500—22no—
149#500C2314yes—
150#500C2314yes—
151#498—22no—
152#498—22no—
153#498C2314yes—
154#498C2314yes—
155#495—216no—
156#494—22no—
157#494C2314yes—
158#493—22no—
159#493—22no—
160#493—22no—
161#493C2314yes—
162#493C2314yes—
163#493C2314yes—
164#492—22no—
165#492—22no—
166#492—22no—
167#492—22no—
168#492C2314yes—
169#492C2314yes—
170#492C2314yes—
171#492C2314yes—
172#488—216no—
173#487—22no—
174#487—22no—
175#487—22no—
176#487—22no—
177#487C2314yes—
178#487C2314yes—
179#487C2314yes—
180#487C2314yes—
181#486—22no—
182#486—22no—
183#486—22no—
184#486—22no—
185#486—22no—
186#486—22no—
187#486—22no—
188#486—22no—
189#486C2314yes—
190#486C2314yes—
191#486C2314yes—
192#486C2314yes—
193#486C2314yes—
194#486C2314yes—
195#486C2314yes—
196#486C2314yes—
197#466—216no—
198#466—216no—
199#465—216no—
200#465—216no—
201C4×SD16—216no—
202C4×SD16—216no—
203C4×D8—216no—
204C4×D8—216no—
205C2×#107—22no—
206C2×#107—22no—
207C2×#106—22no—
208C2×#106—22no—
209#416—216no—
210C2×#103—22no—
211C2×#103—22no—
212C2×#103—22no—
213C2×#103—22no—
214C2×#102—22no—
215C2×#102—22no—
216C2×#102—22no—
217C2×#102—22no—
218C2×#102—22no—
219C2×#102—22no—
220C2×#102—22no—
221C2×#102—22no—
222C2×C4×C8D412yes—
223C2×C4×C8D414yes—
224C2×C4×C8D414yes—
225C2×C4×C8D414yes—
226C2×C4×C8D412yes—
227C2×C4×C8D412yes—
228C2×C4×C8C2311yescharacteristic
229C2×C4×C8C2311yescharacteristic
230C2×C4×C8C2311yescharacteristic
231C2×C4×C8C2311yescharacteristic
232C2×C4×C8C2311yescharacteristic
233C2×C4×C8C2314yes—
234C2×C4×C8C2311yescharacteristic
235C2×C4×C8C2314yes—
236C2×C4×C8C2314yes—
237#322C2314yes—
238#322C2314yes—
239#322C2314yes—
240#322C2314yes—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—42no—
245C2×#41—42no—
246C2×#41—42no—
247C2×#41—48no—
248C2×#41—48no—
249C2×#41—48no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—21no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4—21no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4—41no—
265C22×D4—41no—
266C22×D4—41no—
267C23×C4—41no—
268C23×C4—41no—
269C23×C4—41no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273#135—464no—
274C2×Q16—28no—
275C2×Q16—48no—
276C2×SD16—28no—
277C2×SD16—28no—
278C2×SD16—28no—
279C2×SD16—28no—
280C2×SD16—48no—
281C2×SD16—48no—
282C2×SD16—48no—
283C2×SD16—48no—
284C2×D8—28no—
285C2×D8—28no—
286C2×D8—28no—
287C2×D8—48no—
288C2×D8—48no—
289C2×D8—48no—
290C22×C8—28no—
291C22×C8—28no—
292C22×C8—44no—
293C22×C8C2×D412yes—
294C22×C8C2×D412yes—
295C22×C8C2×D412yes—
296C22×C8C2×D412yes—
297C22×C8C2×D418yes—
298C22×C8C2×D412yes—
299C22×C8C2×D418yes—
300C22×C8C2×D418yes—
301C22×C8C2×D412yes—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8C2412yes—
318C4⋊Q8C2412yes—
319C4⋊Q8C2412yes—
320C4⋊Q8C2412yes—
321C4⋊Q8C2412yes—
322#127—21no—
323#127—21no—
324#127—21no—
325#127—21no—
326#127—21no—
327#127—21no—
328#127—21no—
329#127—21no—
330#127—21no—
331#127C2412yes—
332#127C2412yes—
333#127C2412yes—
334#125—24no—
335#125—42no—
336#124—24no—
337#124—24no—
338#124—24no—
339#124—42no—
340#124—42no—
341#124—42no—
342C22⋊Q8—24no—
343C22⋊Q8—24no—
344C22⋊Q8—42no—
345C22⋊Q8—42no—
346#121—24no—
347#121—24no—
348#121—24no—
349#121—24no—
350#121—24no—
351#121—24no—
352#121—42no—
353#121—42no—
354#121—42no—
355#121—42no—
356#121—42no—
357#121—42no—
358C4×Q8—24no—
359C4×Q8—24no—
360C4×Q8—42no—
361C4×Q8—42no—
362C4×D4—24no—
363C4×D4—24no—
364C4×D4—24no—
365C4×D4—24no—
366C4×D4—24no—
367C4×D4—24no—
368C4×D4—42no—
369C4×D4—42no—
370C4×D4—42no—
371C4×D4—42no—
372C4×D4—42no—
373C4×D4—42no—
374#117—24no—
375#117—42no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—41no—
387C2×C4⋊C4—41no—
388C2×C22⋊C4—41no—
389C2×C22⋊C4—41no—
390C2×C22⋊C4—41no—
391C2×C22⋊C4—41no—
392C2×C22⋊C4—41no—
393C2×C22⋊C4—41no—
394C2×C22⋊C4—41no—
395C2×C42—41no—
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D412yes—
398C2×C42C2×D412yes—
399C2×C42C2×D412yes—
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2412yes—
404#107—28no—
405#107—28no—
406#106—28no—
407#106—28no—
408#103—28no—
409#103—28no—
410#103—28no—
411#103—28no—
412#102—28no—
413#102—28no—
414#102—28no—
415#102—28no—
416#102—28no—
417#102—28no—
418#102—28no—
419#102—28no—
420C4×C8C2×D418yes—
421C4×C8C2×D418yes—
422C4×C8C2×D412yes—
423C4×C8C2×D412yes—
424C4×C8C2×D412yes—
425C4×C8C2×D412yes—
426C4×C8C2×D412yes—
427C4×C8C2×D418yes—
428C4×C8C2×D412yes—
429C4×C8C2412yes—
430C4×C8C2412yes—
431C4×C8C2412yes—
432C4×C8C2412yes—
433C4×C8C2412yes—
434C4×C8C2412yes—
435C24—41no—
436C24—41no—
437C24—41no—
438C24#12711yescharacteristic
439#41—416no—
440#41—416no—
441#41—416no—
442#41—88no—
443#41—88no—
444#41—88no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—41no—
451C2×Q8—41no—
452C2×Q8—41no—
453C2×Q8—41no—
454C2×Q8—41no—
455C2×Q8—44no—
456C2×Q8—44no—
457C2×Q8—44no—
458C2×Q8—44no—
459C2×Q8—44no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—41no—
480C2×D4—41no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—44no—
484C2×D4—44no—
485C2×D4—44no—
486C2×D4—44no—
487C2×D4—44no—
488C2×D4—44no—
489C2×D4—44no—
490C2×D4—84no—
491C2×D4—84no—
492C2×D4—84no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—84no—
504C22×C4—84no—
505C22×C4—84no—
506C22×C4#12711yescharacteristic
507C22×C4#12711yescharacteristic
508C22×C4#12712yes—
509C22×C4#12712yes—
510C22×C4#12712yes—
511C22×C4#12711yescharacteristic
512C22×C4AGL(1,ℤ/8)18yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519Q16—416no—
520SD16—416no—
521SD16—416no—
522SD16—416no—
523SD16—416no—
524D8—416no—
525D8—416no—
526D8—416no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8—216no—
531C2×C8—24no—
532C2×C8—24no—
533C2×C8—216no—
534C2×C8—48no—
535C2×C8#13518yescomplemented
536C2×C8C22×D414yes—
537C2×C8C22×D414yes—
538C2×C8C22×D414yes—
539C2×C8C22×D414yes—
540C2×C8C22×D414yes—
541C2×C8C22×D414yes—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C4⋊C4—41no—
556C4⋊C4—41no—
557C4⋊C4—41no—
558C4⋊C4—41no—
559C4⋊C4—41no—
560C4⋊C4—41no—
561C4⋊C4—41no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C42—44no—
572C42#12712yes—
573C42#12712yes—
574C42#12712yes—
575C42C22×D412yes—
576C42C22×D412yes—
577C42C22×D412yes—
578C42C22×D411yescharacteristic
579C42C22×D411yescharacteristic
580C42C22×D411yescharacteristic
581C42C2511yescharacteristic
582C23—22no—
583C23—22no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—82no—
588C23—82no—
589C23—82no—
590C23—82no—
591C23—82no—
592C23—82no—
593C23—82no—
594C23—82no—
595C23—82no—
596C23#49614yes—
597C23C2×#12712yes—
598C23C2×#12711yescharacteristic
599Q8—44no—
600Q8—44no—
601Q8—44no—
602Q8—44no—
603Q8—44no—
604Q8—82no—
605Q8—82no—
606Q8—82no—
607Q8—82no—
608Q8—82no—
609D4—44no—
610D4—44no—
611D4—44no—
612D4—44no—
613D4—44no—
614D4—44no—
615D4—44no—
616D4—44no—
617D4—44no—
618D4—88no—
619D4—88no—
620D4—88no—
621D4—82no—
622D4—82no—
623D4—82no—
624D4—82no—
625D4—82no—
626D4—82no—
627D4—82no—
628D4—82no—
629D4—82no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—88no—
647C2×C4—88no—
648C2×C4—88no—
649C2×C4#49614yes—
650C2×C4#49618yes—
651C2×C4C2×#12712yes—
652C2×C4C2×#12712yes—
653C2×C4C2×#12711yescharacteristic
654C2×C4C2×#12712yes—
655C2×C4C2×#12711yescharacteristic
656C2×C4C2×#12711yescharacteristic
657C2×C4C2×#13514yes—
658C2×C4C2×AGL(1,ℤ/8)14yes—
659C2×C4C2×AGL(1,ℤ/8)18yes—
660C2×C4#57514yes—
661C2×C4C23×D411yescharacteristic
662C2×C4C23×D411yescharacteristic
663C2×C4C23×D411yescharacteristic
664C8—24no—
665C8—24no—
666C8—24no—
667C8—24no—
668C8—24no—
669C8C2×#13518yescomplemented
670C22—28no—
671C22—22no—
672C22—22no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—164no—
683C22—164no—
684C22—164no—
685C22#314712yes—
686C22C2×#49614yes—
687C22C2×#49612yes—
688C22#315012yes—
689C22C22×#12711yescharacteristic
690C4—22no—
691C4—22no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4#314712yes—
698C4C2×#49614yes—
699C4C2×#49612yes—
700C4#315012yes—
701C4C22×#13512yes—
702C4C22×AGL(1,ℤ/8)12yes—
703C4C2×#57512yes—
704C2—24no—
705C2—162no—
706C2—162no—
707C2—162no—
708C2#1961914yesdirect product factor
709C2C2×#314711yescharacteristic
710C2C22×#49611yescharacteristic
711C2C2×#315011yescharacteristic
subgroupG/H
centerC22×C4#127
commutator, Frattini, cosocleC42C25
socleC23C2×#127
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3169C2semidirect
2. C2×#3169C2semidirect
3. C2×#3167C2semidirect
4. C2×#3167C2semidirect
5. C2×#3147C2semidirect
6. C2×#3131C2semidirect
7. #19619C2direct (N × H)
8. C2×#1723C2semidirect
9. C2×#1721C2semidirect
10. C2×#1720C2semidirect
11. C2×#1719C2semidirect
12. C2×#1717C2semidirect
13. C2×#1717C2semidirect
14. C2×#1715C2semidirect
15. C2×#1714C2semidirect
16. C22×#322C2semidirect
24. #3169C22semidirect
25. #3169C22semidirect
28. #3167C22semidirect
29. #3167C22semidirect
35. #3147C22semidirect
46. #3131C22semidirect
72. #1723C22semidirect
73. #1721C22semidirect
74. #1720C22semidirect
75. #1719C22semidirect
76. #1717C22semidirect
77. #1717C22semidirect
78. #1715C22semidirect
79. #1714C22semidirect
84. C2×#322C22semidirect
535. C2×C8#135semidirect
669. C8C2×#135semidirect
708. C2#19619direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31692
C2×#31672
C2×#31472
C2×#31312
#196192
C2×#17232
C2×#17212
C2×#17202
C2×#17192
C2×#17172
C2×#17152
C2×#17142
C22×#3222