Finite Group Databasebeta

← C2×#19709 #6357292

Subgroups — C2×#19709 #6357292

Order 512.

2028 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3231C211yesmaximal, characteristic, complemented
2C2×#3228C211yesmaximal, characteristic, complemented
3C2×#3210C211yesmaximal, characteristic, complemented
4C2×#3207C211yesmaximal, characteristic
5C2×#3167C211yesmaximal, characteristic, complemented
6C2×#3167C211yesmaximal, characteristic, complemented
7#19709C2116yesmaximal, direct product factor
8C2×#1720C211yesmaximal, characteristic, complemented
9C2×#1719C211yesmaximal, characteristic, complemented
10C2×#1718C211yesmaximal, characteristic, complemented
11C2×#1717C211yesmaximal, characteristic, complemented
12C2×#1716C211yesmaximal, characteristic, complemented
13C2×#1715C211yesmaximal, characteristic, complemented
14C2×#1714C211yesmaximal, characteristic, complemented
15C2×#1714C211yesmaximal, characteristic, complemented
16C2×#1452C211yesmaximal, characteristic, complemented
17C2×#533C2211yescharacteristic
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×#532C2211yescharacteristic
21#3231—24no—
22#3231C2218yescomplemented
23#3228—24no—
24#3228C2218yescomplemented
25#3210—24no—
26#3210C2218yescomplemented
27#3207—24no—
28#3207C2218yes—
29#3167—24no—
30#3167—24no—
31#3167C2218yescomplemented
32#3167C2218yescomplemented
33C2×#500C2211yescharacteristic
34C2×#500C2211yescharacteristic
35C2×#498C2211yescharacteristic
36C2×#498C2211yescharacteristic
37C2×#498C2211yescharacteristic
38C2×#498C2211yescharacteristic
39C2×#494C2211yescharacteristic
40C2×#493C2211yescharacteristic
41C2×#492C2211yescharacteristic
42C2×#492C2211yescharacteristic
43C2×#492C2211yescharacteristic
44C2×#492C2211yescharacteristic
45C2×#487C2211yescharacteristic
46C2×#487C2211yescharacteristic
47C2×#487C2211yescharacteristic
48C2×#487C2211yescharacteristic
49C2×#487C2211yescharacteristic
50C2×#487C2211yescharacteristic
51C2×#486C2211yescharacteristic
52C2×#486C2211yescharacteristic
53C2×#486C2211yescharacteristic
54C2×#486C2211yescharacteristic
55C2×#486C2211yescharacteristic
56C2×#486C2211yescharacteristic
57C2×#485—22no—
58C2×#482—22no—
59C2×#479—22no—
60C2×#478—22no—
61C2×#465—22no—
62C2×#465—22no—
63C2×#464—22no—
64C2×#463—22no—
65C2×#452—22no—
66C2×#449—22no—
67C2×C4×SD16—22no—
68C2×C4×SD16—22no—
69C2×#431C2211yescharacteristic
70C2×#431C2211yescharacteristic
71C22×C4×C8C2211yescharacteristic
72#1720C2218yescomplemented
73#1719C2218yescomplemented
74#1718C2218yescomplemented
75#1717C2218yescomplemented
76#1716C2218yescomplemented
77#1715C2218yescomplemented
78#1714C2218yescomplemented
79#1714C2218yescomplemented
80#1452C2218yescomplemented
81C2×#322C2211yescharacteristic
82C2×#322C2211yescharacteristic
83C2×#322C2211yescharacteristic
84C2×#322C2211yescharacteristic
85C22×#41—21no—
86C22×#41—21no—
87C22×Q16—22no—
88C22×SD16—22no—
89C22×SD16—22no—
90C22×SD16—22no—
91C22×SD16—22no—
92C22×D8—22no—
93C23×C8D412yes—
94#533—22no—
95#533—22no—
96#533—22no—
97#533—22no—
98#533—22no—
99#533—22no—
100#533C2314yes—
101#533C2314yes—
102#532—22no—
103#532—22no—
104#532—22no—
105#532—22no—
106#532—22no—
107#532—22no—
108#532C2314yes—
109#532C2314yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×#127C2311yescharacteristic
117C2×#127C2311yescharacteristic
118C2×#125—21no—
119C2×#125—21no—
120C2×#124—21no—
121C2×#124—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×#121—21no—
127C2×#121—21no—
128C2×#121—21no—
129C2×#121—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×#117—21no—
139C2×#117—21no—
140C22×C42C2311yescharacteristic
141#500—22no—
142#500—22no—
143#500C2314yes—
144#500C2314yes—
145#498—22no—
146#498—22no—
147#498—22no—
148#498—22no—
149#498C2314yes—
150#498C2314yes—
151#498C2314yes—
152#498C2314yes—
153#494—22no—
154#494C2314yes—
155#493—22no—
156#493C2314yes—
157#492—22no—
158#492—22no—
159#492—22no—
160#492—22no—
161#492C2314yes—
162#492C2314yes—
163#492C2314yes—
164#492C2314yes—
165#487—22no—
166#487—22no—
167#487—22no—
168#487—22no—
169#487—22no—
170#487—22no—
171#487C2314yes—
172#487C2314yes—
173#487C2314yes—
174#487C2314yes—
175#487C2314yes—
176#487C2314yes—
177#486—22no—
178#486—22no—
179#486—22no—
180#486—22no—
181#486—22no—
182#486—22no—
183#486C2314yes—
184#486C2314yes—
185#486C2314yes—
186#486C2314yes—
187#486C2314yes—
188#486C2314yes—
189#485—216no—
190#482—216no—
191#479—216no—
192#478—216no—
193#465—216no—
194#465—216no—
195#464—216no—
196#463—216no—
197#452—216no—
198#449—216no—
199C4×SD16—216no—
200C4×SD16—216no—
201#431D414yes—
202#431D414yes—
203#431C2314yes—
204#431C2314yes—
205C2×#107—22no—
206C2×#107—22no—
207C2×#106—22no—
208C2×#106—22no—
209C2×#106—22no—
210C2×#106—22no—
211C2×C4⋊C8D412yes—
212C2×C4⋊C8D412yes—
213C2×#103—22no—
214C2×#103—22no—
215C2×#103—22no—
216C2×#103—22no—
217C2×#103—22no—
218C2×#103—22no—
219C2×#102—22no—
220C2×#102—22no—
221C2×#102—22no—
222C2×#102—22no—
223C2×#102—22no—
224C2×#102—22no—
225C2×C22⋊C8D412yes—
226C2×C22⋊C8D412yes—
227C2×C4×C8D414yes—
228C2×C4×C8D412yes—
229C2×C4×C8C2311yescharacteristic
230C2×C4×C8C2311yescharacteristic
231C2×C4×C8C2314yes—
232C2×C4×C8C2311yescharacteristic
233C2×C4×C8C2311yescharacteristic
234C2×C4×C8C2311yescharacteristic
235C2×C4×C8C2311yescharacteristic
236#322C2314yes—
237#322C2314yes—
238#322C2314yes—
239#322C2314yes—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—42no—
243C2×#41—42no—
244C2×#41—84no—
245C2×#41—84no—
246C22×Q8—21no—
247C22×Q8—21no—
248C22×Q8—21no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×D4—21no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—41no—
259C22×D4—41no—
260C23×C4—41no—
261C23×C4—41no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C2×Q16—28no—
266C2×Q16—84no—
267C2×SD16—28no—
268C2×SD16—28no—
269C2×SD16—28no—
270C2×SD16—28no—
271C2×SD16—48no—
272C2×SD16—48no—
273C2×SD16—84no—
274C2×SD16—84no—
275C2×D8—28no—
276C2×D8—84no—
277C22×C8—44no—
278C22×C8C2×D412yes—
279C22×C8C2×D412yes—
280C22×C8C2×D412yes—
281C22×C8C2×D412yes—
282C22×C8C2×D412yes—
283C22×C8C2×D412yes—
284C22×C8C2×D418yes—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8C2412yes—
304C4⋊Q8C2412yes—
305C4⋊Q8C2412yes—
306C4⋊Q8C2412yes—
307C4⋊Q8C2412yes—
308C4⋊Q8C2412yes—
309#127—21no—
310#127—21no—
311#127—21no—
312#127—21no—
313#127—21no—
314#127—21no—
315#127C2412yes—
316#127C2412yes—
317#125—24no—
318#125—24no—
319#125—42no—
320#125—42no—
321#124—24no—
322#124—24no—
323#124—42no—
324#124—42no—
325C22⋊Q8—24no—
326C22⋊Q8—24no—
327C22⋊Q8—24no—
328C22⋊Q8—24no—
329C22⋊Q8—42no—
330C22⋊Q8—42no—
331C22⋊Q8—42no—
332C22⋊Q8—42no—
333#121—24no—
334#121—24no—
335#121—24no—
336#121—24no—
337#121—42no—
338#121—42no—
339#121—42no—
340#121—42no—
341C4×Q8—24no—
342C4×Q8—24no—
343C4×Q8—24no—
344C4×Q8—24no—
345C4×Q8—42no—
346C4×Q8—42no—
347C4×Q8—42no—
348C4×Q8—42no—
349C4×D4—24no—
350C4×D4—24no—
351C4×D4—24no—
352C4×D4—24no—
353C4×D4—42no—
354C4×D4—42no—
355C4×D4—42no—
356C4×D4—42no—
357#117—24no—
358#117—24no—
359#117—42no—
360#117—42no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4—41no—
375C2×C4⋊C4—41no—
376C2×C4⋊C4—41no—
377C2×C22⋊C4—41no—
378C2×C22⋊C4—41no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4—41no—
381C2×C22⋊C4—41no—
382C2×C22⋊C4—41no—
383C2×C42—41no—
384C2×C42—41no—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D412yes—
389C2×C42C2×D412yes—
390C2×C42C2×D412yes—
391C2×C42C2411yescharacteristic
392C2×C42C2412yes—
393#107—28no—
394#107—28no—
395#106—28no—
396#106—28no—
397#106—28no—
398#106—28no—
399C4⋊C8C2×D418yes—
400C4⋊C8C2×D418yes—
401#103—28no—
402#103—28no—
403#103—28no—
404#103—28no—
405#103—28no—
406#103—28no—
407#102—28no—
408#102—28no—
409#102—28no—
410#102—28no—
411#102—28no—
412#102—28no—
413C22⋊C8C2×D418yes—
414C22⋊C8C2×D418yes—
415C4×C8C2×D418yes—
416C4×C8C2×D412yes—
417C4×C8C2×D412yes—
418C4×C8C2×D412yes—
419C4×C8C2×D412yes—
420C4×C8C2×D412yes—
421C4×C8C2×D412yes—
422C4×C8C2412yes—
423C4×C8C2412yes—
424C4×C8C2412yes—
425C4×C8C2412yes—
426C4×C8C2412yes—
427C4×C8C2412yes—
428C24—41no—
429C24—41no—
430C24#12711yescharacteristic
431#41—88no—
432#41—88no—
433#41—88no—
434#41—88no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—41no—
442C2×Q8—41no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—41no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—84no—
474C2×D4—84no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—84no—
487C22×C4—84no—
488C22×C4#12712yes—
489C22×C4#12712yes—
490C22×C4#12712yes—
491C22×C4#12711yescharacteristic
492C22×C4#12711yescharacteristic
493C22×C4#12711yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500Q16—88no—
501SD16—416no—
502SD16—416no—
503SD16—88no—
504SD16—88no—
505D8—88no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—48no—
513C2×C8C22×D414yes—
514C2×C8C22×D414yes—
515C2×C8C22×D414yes—
516C2×C8C22×D414yes—
517C2×C8C22×D414yes—
518C2×C8C22×D414yes—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—41no—
542C4⋊C4—41no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C42—44no—
554C42—44no—
555C42#12712yes—
556C42#12712yes—
557C42#12712yes—
558C42C22×D412yes—
559C42C22×D411yescharacteristic
560C42C22×D411yescharacteristic
561C42C22×D411yescharacteristic
562C42C22×D412yes—
563C42C22×D412yes—
564C42C2511yescharacteristic
565C23—22no—
566C23—22no—
567C23—22no—
568C23—42no—
569C23—42no—
570C23—82no—
571C23—82no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23—82no—
576C23C2×#12712yes—
577C23C2×#12711yescharacteristic
578Q8—44no—
579Q8—44no—
580Q8—44no—
581Q8—44no—
582Q8—44no—
583Q8—44no—
584Q8—82no—
585Q8—82no—
586Q8—82no—
587Q8—82no—
588Q8—82no—
589Q8—82no—
590D4—44no—
591D4—44no—
592D4—44no—
593D4—44no—
594D4—44no—
595D4—44no—
596D4—88no—
597D4—88no—
598D4—82no—
599D4—82no—
600D4—82no—
601D4—82no—
602D4—82no—
603D4—82no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—88no—
620C2×C4—88no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4C2×#12712yes—
628C2×C4C2×#12712yes—
629C2×C4C2×#12712yes—
630C2×C4C2×#12711yescharacteristic
631C2×C4C2×#12711yescharacteristic
632C2×C4C2×#12711yescharacteristic
633C2×C4#57514yes—
634C2×C4#57714yes—
635C2×C4#57714yes—
636C2×C4C23×D411yescharacteristic
637C2×C4C23×D411yescharacteristic
638C2×C4C23×D411yescharacteristic
639C8—24no—
640C8—24no—
641C8—24no—
642C8—24no—
643C8—24no—
644C8—24no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—44no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—164no—
656C22—164no—
657C22#315012yes—
658C22#315412yes—
659C22#315412yes—
660C22C22×#12711yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#315012yes—
671C4#315412yes—
672C4#315412yes—
673C4C2×#57512yes—
674C4C2×#57712yes—
675C4C2×#57712yes—
676C2—42no—
677C2—162no—
678C2—162no—
679C2#1970914yesdirect product factor
680C2C2×#315011yescharacteristic
681C2C2×#315411yescharacteristic
682C2C2×#315411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC42C25
prime radicalC2×#3207C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3231C2semidirect
2. C2×#3228C2semidirect
3. C2×#3210C2semidirect
5. C2×#3167C2semidirect
6. C2×#3167C2semidirect
7. #19709C2direct (N × H)
8. C2×#1720C2semidirect
9. C2×#1719C2semidirect
10. C2×#1718C2semidirect
11. C2×#1717C2semidirect
12. C2×#1716C2semidirect
13. C2×#1715C2semidirect
14. C2×#1714C2semidirect
15. C2×#1714C2semidirect
16. C2×#1452C2semidirect
22. #3231C22semidirect
24. #3228C22semidirect
26. #3210C22semidirect
31. #3167C22semidirect
32. #3167C22semidirect
72. #1720C22semidirect
73. #1719C22semidirect
74. #1718C22semidirect
75. #1717C22semidirect
76. #1716C22semidirect
77. #1715C22semidirect
78. #1714C22semidirect
79. #1714C22semidirect
80. #1452C22semidirect
679. C2#19709direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32312
C2×#32282
C2×#32102
C2×#32072
C2×#31672
#197092
C2×#17202
C2×#17192
C2×#17182
C2×#17172
C2×#17162
C2×#17152
C2×#17142
C2×#14522