Finite Group Databasebeta

← C2×#19708 #6357291

Subgroups — C2×#19708 #6357291

Order 512.

2130 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic, complemented
2C2×#3210C211yesmaximal, characteristic, complemented
3C2×#3207C211yesmaximal, characteristic, complemented
4C2×#3207C211yesmaximal, characteristic, complemented
5C2×#3169C211yesmaximal, characteristic, complemented
6C2×#3167C211yesmaximal, characteristic, complemented
7#19708C2116yesmaximal, direct product factor
8C2×#1722C211yesmaximal, characteristic, complemented
9C2×#1721C211yesmaximal, characteristic, complemented
10C2×#1720C211yesmaximal, characteristic, complemented
11C2×#1718C211yesmaximal, characteristic, complemented
12C2×#1717C211yesmaximal, characteristic, complemented
13C2×#1717C211yesmaximal, characteristic, complemented
14C2×#1715C211yesmaximal, characteristic, complemented
15C2×#1714C211yesmaximal, characteristic, complemented
16C2×#1452C211yesmaximal, characteristic, complemented
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×#532C2211yescharacteristic
21#3228—24no—
22#3228C2218yescomplemented
23#3210—24no—
24#3210C2218yescomplemented
25#3207—24no—
26#3207—24no—
27#3207C2218yescomplemented
28#3207C2218yescomplemented
29#3169—24no—
30#3169C2218yescomplemented
31#3167—24no—
32#3167C2218yescomplemented
33C2×#500C2211yescharacteristic
34C2×#500C2211yescharacteristic
35C2×#498C2211yescharacteristic
36C2×#498C2211yescharacteristic
37C2×#494C2211yescharacteristic
38C2×#493C2211yescharacteristic
39C2×#493C2211yescharacteristic
40C2×#493C2211yescharacteristic
41C2×#492C2211yescharacteristic
42C2×#492C2211yescharacteristic
43C2×#492C2211yescharacteristic
44C2×#492C2211yescharacteristic
45C2×#487C2211yescharacteristic
46C2×#487C2211yescharacteristic
47C2×#487C2211yescharacteristic
48C2×#487C2211yescharacteristic
49C2×#486C2211yescharacteristic
50C2×#486C2211yescharacteristic
51C2×#486C2211yescharacteristic
52C2×#486C2211yescharacteristic
53C2×#486C2211yescharacteristic
54C2×#486C2211yescharacteristic
55C2×#486C2211yescharacteristic
56C2×#486C2211yescharacteristic
57C2×#482—22no—
58C2×#478—22no—
59C2×#466—22no—
60C2×#465—22no—
61C2×#464—22no—
62C2×#463—22no—
63C2×#463—22no—
64C2×#452—22no—
65C2×#449—22no—
66C2×#449—22no—
67C2×C4×SD16—22no—
68C2×C4×D8—22no—
69C2×#431C2211yescharacteristic
70C2×#431C2211yescharacteristic
71C22×C4×C8C2211yescharacteristic
72#1722C2218yescomplemented
73#1721C2218yescomplemented
74#1720C2218yescomplemented
75#1718C2218yescomplemented
76#1717C2218yescomplemented
77#1717C2218yescomplemented
78#1715C2218yescomplemented
79#1714C2218yescomplemented
80#1452C2218yescomplemented
81C2×#322C2211yescharacteristic
82C2×#322C2211yescharacteristic
83C2×#322C2211yescharacteristic
84C2×#322C2211yescharacteristic
85C22×#41—21no—
86C22×#41—21no—
87C22×#41—21no—
88C22×Q16—22no—
89C22×SD16—22no—
90C22×SD16—22no—
91C22×SD16—22no—
92C22×SD16—22no—
93C22×D8—22no—
94C22×D8—22no—
95C22×D8—22no—
96C23×C8D412yes—
97#533—22no—
98#533—22no—
99#533—22no—
100#533C2314yes—
101#532—22no—
102#532—22no—
103#532—22no—
104#532—22no—
105#532—22no—
106#532—22no—
107#532—22no—
108#532—22no—
109#532—22no—
110#532C2314yes—
111#532C2314yes—
112#532C2314yes—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×#127C2311yescharacteristic
119C2×#127C2311yescharacteristic
120C2×#127C2311yescharacteristic
121C2×#125—21no—
122C2×#124—21no—
123C2×#124—21no—
124C2×#124—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×#121—21no—
128C2×#121—21no—
129C2×#121—21no—
130C2×#121—21no—
131C2×#121—21no—
132C2×#121—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4—21no—
140C2×C4×D4—21no—
141C2×#117—21no—
142C22×C42C2311yescharacteristic
143#500—22no—
144#500—22no—
145#500C2314yes—
146#500C2314yes—
147#498—22no—
148#498—22no—
149#498C2314yes—
150#498C2314yes—
151#494—22no—
152#494C2314yes—
153#493—22no—
154#493—22no—
155#493—22no—
156#493C2314yes—
157#493C2314yes—
158#493C2314yes—
159#492—22no—
160#492—22no—
161#492—22no—
162#492—22no—
163#492C2314yes—
164#492C2314yes—
165#492C2314yes—
166#492C2314yes—
167#487—22no—
168#487—22no—
169#487—22no—
170#487—22no—
171#487C2314yes—
172#487C2314yes—
173#487C2314yes—
174#487C2314yes—
175#486—22no—
176#486—22no—
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#486C2314yes—
190#486C2314yes—
191#482—216no—
192#478—216no—
193#466—216no—
194#465—216no—
195#464—216no—
196#463—216no—
197#463—216no—
198#452—216no—
199#449—216no—
200#449—216no—
201C4×SD16—216no—
202C4×D8—216no—
203#431D414yes—
204#431D414yes—
205#431C2314yes—
206#431C2314yes—
207C2×#107—22no—
208C2×#107—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×#102—22no—
218C2×#102—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—24no—
243C2×#41—42no—
244C2×#41—42no—
245C2×#41—42no—
246C2×#41—84no—
247C2×#41—84no—
248C2×#41—84no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×D4—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—41no—
264C22×D4—41no—
265C22×D4—41no—
266C23×C4—41no—
267C23×C4—41no—
268C23×C4—41no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C2×Q16—28no—
273C2×Q16—84no—
274C2×SD16—28no—
275C2×SD16—28no—
276C2×SD16—28no—
277C2×SD16—28no—
278C2×SD16—48no—
279C2×SD16—84no—
280C2×SD16—84no—
281C2×SD16—84no—
282C2×D8—28no—
283C2×D8—28no—
284C2×D8—28no—
285C2×D8—48no—
286C2×D8—84no—
287C2×D8—84no—
288C22×C8—44no—
289C22×C8C2×D412yes—
290C22×C8C2×D412yes—
291C22×C8C2×D412yes—
292C22×C8C2×D412yes—
293C22×C8C2×D412yes—
294C22×C8C2×D412yes—
295C22×C8C2×D418yes—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—21no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
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⋊Q8C2412yes—
312C4⋊Q8C2412yes—
313C4⋊Q8C2412yes—
314C4⋊Q8C2412yes—
315C4⋊Q8C2412yes—
316#127—21no—
317#127—21no—
318#127—21no—
319#127—21no—
320#127—21no—
321#127—21no—
322#127—21no—
323#127—21no—
324#127—21no—
325#127C2412yes—
326#127C2412yes—
327#127C2412yes—
328#125—24no—
329#125—42no—
330#124—24no—
331#124—24no—
332#124—24no—
333#124—42no—
334#124—42no—
335#124—42no—
336C22⋊Q8—24no—
337C22⋊Q8—24no—
338C22⋊Q8—42no—
339C22⋊Q8—42no—
340#121—24no—
341#121—24no—
342#121—24no—
343#121—24no—
344#121—24no—
345#121—24no—
346#121—42no—
347#121—42no—
348#121—42no—
349#121—42no—
350#121—42no—
351#121—42no—
352C4×Q8—24no—
353C4×Q8—24no—
354C4×Q8—42no—
355C4×Q8—42no—
356C4×D4—24no—
357C4×D4—24no—
358C4×D4—24no—
359C4×D4—24no—
360C4×D4—24no—
361C4×D4—24no—
362C4×D4—42no—
363C4×D4—42no—
364C4×D4—42no—
365C4×D4—42no—
366C4×D4—42no—
367C4×D4—42no—
368#117—24no—
369#117—42no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—41no—
381C2×C4⋊C4—41no—
382C2×C22⋊C4—41no—
383C2×C22⋊C4—41no—
384C2×C22⋊C4—41no—
385C2×C22⋊C4—41no—
386C2×C22⋊C4—41no—
387C2×C22⋊C4—41no—
388C2×C22⋊C4—41no—
389C2×C42—41no—
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D412yes—
394C2×C42C2×D412yes—
395C2×C42C2×D412yes—
396C2×C42C2411yescharacteristic
397C2×C42C2412yes—
398#107—28no—
399#107—28no—
400#106—28no—
401#106—28no—
402C4⋊C8C2×D418yes—
403C4⋊C8C2×D418yes—
404#103—28no—
405#103—28no—
406#103—28no—
407#103—28no—
408#102—28no—
409#102—28no—
410#102—28no—
411#102—28no—
412#102—28no—
413#102—28no—
414#102—28no—
415#102—28no—
416C22⋊C8C2×D418yes—
417C22⋊C8C2×D418yes—
418C4×C8C2×D418yes—
419C4×C8C2×D412yes—
420C4×C8C2×D412yes—
421C4×C8C2×D412yes—
422C4×C8C2×D412yes—
423C4×C8C2×D412yes—
424C4×C8C2×D412yes—
425C4×C8C2412yes—
426C4×C8C2412yes—
427C4×C8C2412yes—
428C4×C8C2412yes—
429C4×C8C2412yes—
430C4×C8C2412yes—
431C24—41no—
432C24—41no—
433C24—41no—
434C24#12711yescharacteristic
435#41—88no—
436#41—88no—
437#41—88no—
438#41—88no—
439#41—88no—
440#41—88no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—41no—
447C2×Q8—41no—
448C2×Q8—41no—
449C2×Q8—41no—
450C2×Q8—41no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8—44no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—44no—
478C2×D4—44no—
479C2×D4—44no—
480C2×D4—44no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—44no—
484C2×D4—44no—
485C2×D4—44no—
486C2×D4—84no—
487C2×D4—84no—
488C2×D4—84no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—84no—
501C22×C4—84no—
502C22×C4—84no—
503C22×C4#12712yes—
504C22×C4#12712yes—
505C22×C4#12712yes—
506C22×C4#12711yescharacteristic
507C22×C4#12711yescharacteristic
508C22×C4#12711yescharacteristic
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515Q16—88no—
516SD16—416no—
517SD16—88no—
518SD16—88no—
519SD16—88no—
520D8—416no—
521D8—88no—
522D8—88no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—48no—
530C2×C8C22×D414yes—
531C2×C8C22×D414yes—
532C2×C8C22×D414yes—
533C2×C8C22×D414yes—
534C2×C8C22×D414yes—
535C2×C8C22×D414yes—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C4⋊C4—41no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C42—44no—
566C42#12712yes—
567C42#12712yes—
568C42#12712yes—
569C42C22×D412yes—
570C42C22×D411yescharacteristic
571C42C22×D411yescharacteristic
572C42C22×D411yescharacteristic
573C42C22×D412yes—
574C42C22×D412yes—
575C42C2511yescharacteristic
576C23—22no—
577C23—22no—
578C23—22no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—82no—
583C23—82no—
584C23—82no—
585C23—82no—
586C23—82no—
587C23—82no—
588C23—82no—
589C23—82no—
590C23—82no—
591C23C2×#12712yes—
592C23C2×#12711yescharacteristic
593Q8—44no—
594Q8—44no—
595Q8—44no—
596Q8—44no—
597Q8—44no—
598Q8—82no—
599Q8—82no—
600Q8—82no—
601Q8—82no—
602Q8—82no—
603D4—44no—
604D4—44no—
605D4—44no—
606D4—44no—
607D4—44no—
608D4—44no—
609D4—44no—
610D4—44no—
611D4—44no—
612D4—88no—
613D4—88no—
614D4—88no—
615D4—82no—
616D4—82no—
617D4—82no—
618D4—82no—
619D4—82no—
620D4—82no—
621D4—82no—
622D4—82no—
623D4—82no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
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—42no—
637C2×C4—42no—
638C2×C4—88no—
639C2×C4—88no—
640C2×C4—88no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4C2×#12712yes—
647C2×C4C2×#12712yes—
648C2×C4C2×#12712yes—
649C2×C4C2×#12711yescharacteristic
650C2×C4C2×#12711yescharacteristic
651C2×C4C2×#12711yescharacteristic
652C2×C4#57514yes—
653C2×C4#57614yes—
654C2×C4#57714yes—
655C2×C4C23×D411yescharacteristic
656C2×C4C23×D411yescharacteristic
657C2×C4C23×D411yescharacteristic
658C8—24no—
659C8—24no—
660C8—24no—
661C8—24no—
662C8—24no—
663C8—24no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—44no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—164no—
678C22—164no—
679C22—164no—
680C22#315012yes—
681C22#315412yes—
682C22#315512yes—
683C22C22×#12711yescharacteristic
684C4—22no—
685C4—22no—
686C4—22no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4#315012yes—
693C4#315412yes—
694C4#315512yes—
695C4C2×#57512yes—
696C4C2×#57612yes—
697C4C2×#57712yes—
698C2—42no—
699C2—162no—
700C2—162no—
701C2—162no—
702C2#1970814yesdirect product factor
703C2C2×#315011yescharacteristic
704C2C2×#315411yescharacteristic
705C2C2×#315511yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC42C25
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×#3228C2semidirect
2. C2×#3210C2semidirect
3. C2×#3207C2semidirect
4. C2×#3207C2semidirect
5. C2×#3169C2semidirect
6. C2×#3167C2semidirect
7. #19708C2direct (N × H)
8. C2×#1722C2semidirect
9. C2×#1721C2semidirect
10. C2×#1720C2semidirect
11. C2×#1718C2semidirect
12. C2×#1717C2semidirect
13. C2×#1717C2semidirect
14. C2×#1715C2semidirect
15. C2×#1714C2semidirect
16. C2×#1452C2semidirect
22. #3228C22semidirect
24. #3210C22semidirect
27. #3207C22semidirect
28. #3207C22semidirect
30. #3169C22semidirect
32. #3167C22semidirect
72. #1722C22semidirect
73. #1721C22semidirect
74. #1720C22semidirect
75. #1718C22semidirect
76. #1717C22semidirect
77. #1717C22semidirect
78. #1715C22semidirect
79. #1714C22semidirect
80. #1452C22semidirect
702. C2#19708direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32282
C2×#32102
C2×#32072
C2×#31692
C2×#31672
#197082
C2×#17222
C2×#17212
C2×#17202
C2×#17182
C2×#17172
C2×#17152
C2×#17142
C2×#14522