Finite Group Databasebeta

← C2×#9816 #206021

Subgroups — C2×#9816 #206021

Order 512.

1392 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#465C211yesmaximal, characteristic
2C2×#1810C211yesmaximal, characteristic, complemented
3C2×#1752C211yesmaximal, characteristic, complemented
4C2×#1508C211yesmaximal, characteristic, complemented
5C2×#1507C211yesmaximal, characteristic
6C2×#1472C211yesmaximal, characteristic, complemented
7C2×#1471C211yesmaximal, characteristic
8#9816C218yesmaximal, direct product factor
9C23×SD16C411yescharacteristic, complemented
10C24×C8C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12C22×#121C2211yescharacteristic
13C2×#465—21no—
14C2×#465—21no—
15C2×#465—22no—
16C2×#465—21no—
17C2×#465—21no—
18C2×#465—22no—
19C2×#465C414yescomplemented
20C2×#465C2214yes—
21C22×#108—21no—
22C22×#106C411yescharacteristic, complemented
23C22×#103C411yescharacteristic, complemented
24C22×#102C411yescharacteristic, complemented
25C2×#407C2211yescharacteristic
26C2×#407C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×#97—21no—
29C2×C4×C22⋊C4C2211yescharacteristic
30#1810—22no—
31#1810C2214yescomplemented
32#1752—22no—
33#1752C2214yescomplemented
34C2×#341—21no—
35C2×#336—21no—
36#1508—22no—
37#1508C2214yescomplemented
38#1507—22no—
39#1507C2214yes—
40C2×#328—22no—
41C2×#327—22no—
42#1472—22no—
43#1472C2214yescomplemented
44#1471—22no—
45#1471C2214yes—
46C2×#324—22no—
47C2×#323—22no—
48C23×Q8C2×C411yescharacteristic
49C23×D4—21no—
50C23×D4C2×C411yescharacteristic
51C24×C4C2311yescharacteristic
52C22×SD16—21no—
53C22×SD16—22no—
54C22×SD16—21no—
55C22×SD16—44no—
56C22×SD16C2×C414yescomplemented
57C22×M16—21no—
58C22×M16—21no—
59C23×C8—21no—
60C23×C8—44no—
61C23×C8C2×C411yescharacteristic
62C23×C8C2×C411yescharacteristic
63C23×C8C2×C412yes—
64C23×C8D411yescharacteristic
65C23×C8D411yescharacteristic
66C23×C8D412yes—
67C23×C8D411yescharacteristic
68C23×C8D411yescharacteristic
69C23×C8D412yes—
70C23×C8C2312yes—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8C2×C412yes—
82C2×C22⋊Q8C2312yes—
83C2×#121—21no—
84C2×#121—21no—
85C2×#121—21no—
86C2×#121—21no—
87C2×#121—21no—
88C2×#121—21no—
89C2×#121—21no—
90C2×#121—21no—
91C2×#121—21no—
92C2×#121—21no—
93C2×#121C2×C412yes—
94C2×#121C2312yes—
95C22×C4⋊C4—21no—
96C22×C4⋊C4C2×C411yescharacteristic
97C22×C4⋊C4C2×C411yescharacteristic
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C42—21no—
102#465—24no—
103#465—24no—
104#465—24no—
105#465—24no—
106#465—24no—
107#465—24no—
108#465—24no—
109#465—24no—
110C2×#108—24no—
111C2×#108—44no—
112C2×#106—21no—
113C2×#106—22no—
114C2×#106—21no—
115C2×#106C2×C414yescomplemented
116C2×#103—21no—
117C2×#103—21no—
118C2×#103—22no—
119C2×#103C2×C414yescomplemented
120C2×#102—21no—
121C2×#102—21no—
122C2×#102—22no—
123C2×#102C2×C414yescomplemented
124#407—21no—
125#407—21no—
126#407—21no—
127#407—21no—
128#407—21no—
129#407—21no—
130#407C2312yes—
131#407C2312yes—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8D412yes—
140C2×C22⋊C8D412yes—
141C2×C22⋊C8C2312yes—
142C2×#97—24no—
143C2×#97—44no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4D412yes—
147C4×C22⋊C4C2312yes—
148C2×#95D412yes—
149#341—24no—
150#336—24no—
151#328—28no—
152#327—28no—
153#324—28no—
154#323—28no—
155C25—21no—
156C25#4111yescharacteristic
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8C22×C412yes—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—84no—
179C22×D4C22×C412yes—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4#4111yescharacteristic
197C23×C4#4111yescharacteristic
198C23×C4#4111yescharacteristic
199C23×C4C2411yescharacteristic
200C2×SD16—24no—
201C2×SD16—24no—
202C2×SD16—44no—
203C2×SD16—48no—
204C2×SD16—48no—
205C2×SD16—44no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—41no—
209C2×M16—41no—
210C2×M16—42no—
211C2×M16—42no—
212C2×M16—42no—
213C2×M16—42no—
214C22×C8—24no—
215C22×C8—24no—
216C22×C8—24no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—44no—
221C22×C8—44no—
222C22×C8—44no—
223C22×C8—44no—
224C22×C8—42no—
225C22×C8—42no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8—42no—
229C22×C8—41no—
230C22×C8—41no—
231C22×C8C22⋊C411yescharacteristic
232C22×C8C22⋊C412yes—
233C22×C8C22⋊C411yescharacteristic
234C22×C8C22⋊C411yescharacteristic
235C22×C8C22⋊C412yes—
236C22×C8C22⋊C412yes—
237C22×C8C22⋊C412yes—
238C22×C8C22⋊C412yes—
239C22×C8C22⋊C412yes—
240C22×C8C22⋊C411yescharacteristic
241C22×C8C22×C412yes—
242C22×C8C22×C412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D412yes—
245C22×C8C2×D412yes—
246C22×C8C2×D412yes—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8—22no—
255C22⋊Q8—41no—
256C22⋊Q8—41no—
257C22⋊Q8—41no—
258C22⋊Q8—41no—
259C22⋊Q8—41no—
260C22⋊Q8—41no—
261C22⋊Q8—41no—
262C22⋊Q8—41no—
263#121—22no—
264#121—22no—
265#121—22no—
266#121—22no—
267#121—22no—
268#121—22no—
269#121—22no—
270#121—22no—
271#121—41no—
272#121—41no—
273#121—41no—
274#121—41no—
275#121—41no—
276#121—41no—
277#121—41no—
278#121—41no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4C22×C412yes—
296C2×C4⋊C4C22×C412yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4—41no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4—41no—
308C2×C22⋊C4—41no—
309C2×C22⋊C4—41no—
310C2×C22⋊C4—41no—
311C2×C22⋊C4—41no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—42no—
315C2×C42—42no—
316#108—48no—
317#106—24no—
318#106—24no—
319#103—24no—
320#103—24no—
321#102—24no—
322#102—24no—
323C22⋊C8—22no—
324C22⋊C8—22no—
325C22⋊C8—22no—
326C22⋊C8—22no—
327C22⋊C8—22no—
328C22⋊C8—22no—
329C22⋊C8—22no—
330C22⋊C8—41no—
331C22⋊C8—41no—
332C22⋊C8—41no—
333C22⋊C8—41no—
334C22⋊C8C2×D414yes—
335#97—48no—
336#95—22no—
337#95C2×D414yes—
338C24—24no—
339C24—24no—
340C24—22no—
341C24—42no—
342C24—41no—
343C24—41no—
344C24—44no—
345C24—82no—
346C24—82no—
347C24C4×D411yescharacteristic
348C24C4×D411yescharacteristic
349C24#12111yescharacteristic
350C24#12111yescharacteristic
351C24#12411yescharacteristic
352C24#12411yescharacteristic
353C24C2×#4111yescharacteristic
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×Q8—82no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—82no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—84no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C4×D411yescharacteristic
434C22×C4C4×D411yescharacteristic
435C22×C4C4×D411yescharacteristic
436C22×C4C4×D411yescharacteristic
437C22×C4#12111yescharacteristic
438C22×C4#12111yescharacteristic
439C22×C4#12111yescharacteristic
440C22×C4#12111yescharacteristic
441C22×C4#12111yescharacteristic
442C22×C4#12111yescharacteristic
443C22×C4#12411yescharacteristic
444C22×C4#12411yescharacteristic
445C22×C4#12711yescharacteristic
446C22×C4#12711yescharacteristic
447C22×C4#12711yescharacteristic
448C22×C4#12711yescharacteristic
449C22×C4C23×C411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4111yescharacteristic
455SD16—48no—
456SD16—48no—
457M16—42no—
458M16—42no—
459M16—42no—
460M16—42no—
461M16—82no—
462M16—82no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—44no—
481C2×C8—42no—
482C2×C8—44no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—82no—
486C2×C8—82no—
487C2×C8#9914yes—
488C2×C8C2×C22⋊C412yes—
489C2×C8C2×C22⋊C412yes—
490C2×C8C2×C22⋊C412yes—
491C2×C8C2×C22⋊C412yes—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C42—42no—
515C42—42no—
516C42—44no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—416no—
537C23—42no—
538C23—44no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23—82no—
545C23—82no—
546C23—82no—
547C23—168no—
548C23#39011yescharacteristic
549C23#39011yescharacteristic
550C23#39011yescharacteristic
551C23#39011yescharacteristic
552C23#44414yes—
553C23#47114yes—
554C23C2×C4×D411yescharacteristic
555C23C2×#12111yescharacteristic
556C23C2×#12411yescharacteristic
557Q8—42no—
558Q8—42no—
559Q8—82no—
560Q8—82no—
561Q8—82no—
562D4—42no—
563D4—42no—
564D4—82no—
565D4—82no—
566D4—84no—
567D4—84no—
568D4—84no—
569D4—84no—
570D4—82no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—88no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4#39011yescharacteristic
621C2×C4#39011yescharacteristic
622C2×C4#39011yescharacteristic
623C2×C4#39011yescharacteristic
624C2×C4C2×#9912yes—
625C2×C4#41012yes—
626C2×C4C22×C22⋊C411yescharacteristic
627C2×C4C2×C4×D411yescharacteristic
628C2×C4C2×#12111yescharacteristic
629C2×C4C2×#12711yescharacteristic
630C8—22no—
631C8—22no—
632C8—22no—
633C8—44no—
634C8—82no—
635C8—82no—
636C8C2×#9914yes—
637C22—28no—
638C22—28no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—164no—
662C22—164no—
663C22—164no—
664C22#196412yes—
665C22#197512yes—
666C22#197512yes—
667C22#197712yes—
668C22#197712yes—
669C22C2×#39011yescharacteristic
670C22C2×#44412yes—
671C22C2×#47112yes—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—84no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#196412yes—
683C4C22×#9911yescharacteristic
684C4C2×#41011yescharacteristic
685C2—22no—
686C2—22no—
687C2—22no—
688C2—22no—
689C2—41no—
690C2—41no—
691C2—162no—
692C2#981614yesdirect product factor
693C2C2×#196411yescharacteristic
694C2C2×#197511yescharacteristic
695C2C2×#197711yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1810C2semidirect
3. C2×#1752C2semidirect
4. C2×#1508C2semidirect
6. C2×#1472C2semidirect
8. #9816C2direct (N × H)
9. C23×SD16C4semidirect
19. C2×#465C4semidirect
22. C22×#106C4semidirect
23. C22×#103C4semidirect
24. C22×#102C4semidirect
31. #1810C22semidirect
33. #1752C22semidirect
37. #1508C22semidirect
43. #1472C22semidirect
56. C22×SD16C2×C4semidirect
115. C2×#106C2×C4semidirect
119. C2×#103C2×C4semidirect
123. C2×#102C2×C4semidirect
692. C2#9816direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4652
C2×#18102
C2×#17522
C2×#15082
C2×#15072
C2×#14722
C2×#14712
#98162