Finite Group Databasebeta

← C2×#16816 #611586

Subgroups — C2×#16816 #611586

Order 512.

5690 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#394C212yesmaximal
2C22×#386C211yesmaximal, characteristic
3C22×#380C211yesmaximal, characteristic, complemented
4#16816C2116yesmaximal, direct product factor
5C2×#1444C211yesmaximal, characteristic, complemented
6C2×#1442C211yesmaximal, characteristic, complemented
7C2×#1442C212yesmaximal, complemented
8C2×#1440C212yesmaximal, complemented
9C2×#1440C212yesmaximal, complemented
10C2×#1439C211yesmaximal, characteristic, complemented
11C2×#1439C211yesmaximal, characteristic, complemented
12C2×C22⋊#95C211yesmaximal, characteristic, complemented
13C24×D4C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2212yes—
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C2×#394—216no—
26C2×#394C22116yes—
27C2×#386—28no—
28C2×#386C2218yes—
29C2×#380—28no—
30C2×#380C2218yescomplemented
31C22×#95C2212yes—
32C22×#95C2211yescharacteristic
33C22×#95C2211yescharacteristic
34C22×#95C2212yes—
35C22×#95C2211yescharacteristic
36C22×#95C2212yes—
37C22×#95C2211yescharacteristic
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42#1444C2218yescomplemented
43#1442C22116yescomplemented
44#1442C2218yescomplemented
45#1440C22116yescomplemented
46#1440C22116yescomplemented
47#1439C2218yescomplemented
48#1439C2218yescomplemented
49C22⋊#95C2218yescomplemented
50C26C2311yescharacteristic
51C26C2311yescharacteristic
52C23×D4—24no—
53C23×D4—24no—
54C23×D4—44no—
55C23×D4—44no—
56C23×D4D418yes—
57C23×D4C2314yes—
58C24×C4—24no—
59C24×C4—24no—
60C24×C4—44no—
61C24×C4—44no—
62C24×C4D418yes—
63C24×C4C2312yes—
64C24×C4C2312yes—
65C24×C4C2311yescharacteristic
66C24×C4C2312yes—
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2314yes—
73C22×C4⋊C4—28no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4—44no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4—44no—
80C22×C4⋊C4C2318yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C22⋊C4—28no—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—28no—
87C22×C22⋊C4—28no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4C2318yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2318yes—
106C22×C22⋊C4C2318yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109#394—264no—
110#386—232no—
111#380—232no—
112C2×#95—28no—
113C2×#95—28no—
114C2×#95—24no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—28no—
118C2×#95—24no—
119C2×#95—44no—
120C2×#95—44no—
121C2×#95—44no—
122C2×#95—44no—
123C2×#95C2318yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2318yes—
127C2×#95C2318yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2318yes—
131C2×#95C2318yes—
132C2×#95C2318yes—
133C2×#95C2318yes—
134C25—24no—
135C25—22no—
136C25—22no—
137C25—24no—
138C25—24no—
139C25—24no—
140C25—22no—
141C25—22no—
142C25—22no—
143C25—24no—
144C25—22no—
145C25—22no—
146C25—22no—
147C25—24no—
148C25—24no—
149C25—24no—
150C25—22no—
151C25—22no—
152C25C2×D414yes—
153C25C2×D414yes—
154C25C2×D414yes—
155C25#4114yes—
156C25C2412yes—
157C25C2412yes—
158C25C2411yescharacteristic
159C22×D4—232no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—432no—
163C22×D4—48no—
164C22×D4—48no—
165C22×D4—416no—
166C22×D4—416no—
167C22×D4—432no—
168C22×D4C2×D4116yes—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—232no—
186C23×C4—28no—
187C23×C4—28no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—416no—
220C23×C4—416no—
221C23×C4—48no—
222C23×C4—48no—
223C23×C4—432no—
224C23×C4—432no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D4116yes—
229C23×C4#4118yes—
230C23×C4#4114yes—
231C23×C4#4114yes—
232C23×C4C2414yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—432no—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—216no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—216no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—432no—
271C2×C22⋊C4—432no—
272C2×C22⋊C4—416no—
273C2×C22⋊C4—432no—
274C2×C22⋊C4—416no—
275C2×C22⋊C4—432no—
276C2×C22⋊C4—416no—
277#95—216no—
278#95—28no—
279#95—28no—
280#95—216no—
281#95—28no—
282#95—216no—
283#95—28no—
284#95—48no—
285#95—48no—
286#95—48no—
287#95—48no—
288C24—28no—
289C24—28no—
290C24—28no—
291C24—28no—
292C24—216no—
293C24—28no—
294C24—28no—
295C24—216no—
296C24—216no—
297C24—28no—
298C24—28no—
299C24—28no—
300C24—28no—
301C24—216no—
302C24—24no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—22no—
309C24—22no—
310C24—28no—
311C24—216no—
312C24—216no—
313C24—28no—
314C24—28no—
315C24—22no—
316C24—24no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—48no—
326C24—48no—
327C24—48no—
328C24—48no—
329C24—48no—
330C24—48no—
331C24—48no—
332C24—48no—
333C24—44no—
334C24—44no—
335C24—44no—
336C24—44no—
337C24—44no—
338C24—44no—
339C24—48no—
340C24—48no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24—44no—
345C24—44no—
346C24—48no—
347C24—48no—
348C24—48no—
349C24—48no—
350C24—48no—
351C24#121116yes—
352C24C22×D414yes—
353C24C22×D414yes—
354C24C22×D414yes—
355C24C22×D412yes—
356C24C22×D412yes—
357C24C2×#4114yes—
358C24C2×#4112yes—
359C24C2×#4112yes—
360C24C2×#4112yes—
361C24C2×#4114yes—
362C24H(5,𝔽2)14yes—
363C24H(5,𝔽2)14yes—
364C24H(5,𝔽2)14yes—
365C242−1+412yes—
366C242−1+412yes—
367C24C2511yescharacteristic
368C2×D4—232no—
369C2×D4—432no—
370C2×D4—4128no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—432no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—232no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—4128no—
437C22×C4—432no—
438C22×C4—432no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4#123116yes—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C2×#4118yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4114yes—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—432no—
489C22⋊C4—432no—
490C22⋊C4—432no—
491C22⋊C4—416no—
492C22⋊C4—432no—
493C22⋊C4—416no—
494C22⋊C4—432no—
495C22⋊C4—416no—
496C23—232no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—24no—
504C23—28no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—232no—
512C23—232no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—416no—
519C23—416no—
520C23—416no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—432no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—432no—
529C23—416no—
530C23—44no—
531C23—44no—
532C23—432no—
533C23—432no—
534C23—416no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—432no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—432no—
553C23—416no—
554C23—44no—
555C23—44no—
556C23—416no—
557C23—88no—
558C23—88no—
559C23—88no—
560C23—88no—
561C23—816no—
562C23—816no—
563C23C2×#12118yes—
564C23C2×#12114yes—
565C23C2×#12314yes—
566C23#53214yes—
567C23#53414yes—
568C23#53414yes—
569C23#53418yes—
570C23#53514yes—
571C23#53614yes—
572C23#53818yes—
573C23#53818yes—
574C23#53818yes—
575C23#53918yes—
576C23#54218yes—
577C23#54618yes—
578C23#54714yes—
579C23#54714yes—
580C23#55214yes—
581C23#55214yes—
582C23#55318yes—
583C23#55318yes—
584C23#55518yes—
585C23#56218yes—
586C23#56218yes—
587C23C23×D411yescharacteristic
588C23C23×D411yescharacteristic
589C23C22×#4111yescharacteristic
590C23C22×#4111yescharacteristic
591C23C22×#4111yescharacteristic
592C23C22×#4112yes—
593C23C2×H(5,𝔽2)12yes—
594C23C2×H(5,𝔽2)12yes—
595C23C2×H(5,𝔽2)12yes—
596C23C2×2−1+411yescharacteristic
597C23C2×2−1+411yescharacteristic
598D4—464no—
599C2×C4—232no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—464no—
613C2×C4—416no—
614C2×C4—416no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—432no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—88no—
657C2×C4—816no—
658C2×C4—816no—
659C2×C4—88no—
660C2×C4—816no—
661C2×C4C2×#12318yes—
662C22—28no—
663C22—28no—
664C22—28no—
665C22—44no—
666C22—44no—
667C22—48no—
668C22—48no—
669C22—464no—
670C22—44no—
671C22—44no—
672C22—48no—
673C22—48no—
674C22—44no—
675C22—48no—
676C22—48no—
677C22—44no—
678C22—44no—
679C22—82no—
680C22—82no—
681C22—832no—
682C22—82no—
683C22—82no—
684C22—84no—
685C22—84no—
686C22#248618yes—
687C22#250018yes—
688C22#254318yes—
689C22#261718yes—
690C22#262218yes—
691C22#266318yes—
692C22#2686116yes—
693C22#2708116yes—
694C22#2779116yes—
695C22#2781116yes—
696C22#281618yes—
697C22C22×#12111yescharacteristic
698C22C22×#12311yescharacteristic
699C22C2×#53211yescharacteristic
700C22C2×#53411yescharacteristic
701C22C2×#53411yescharacteristic
702C22C2×#53412yes—
703C22C2×#53511yescharacteristic
704C22C2×#53611yescharacteristic
705C22C2×#53812yes—
706C22C2×#53812yes—
707C22C2×#53812yes—
708C22C2×#53912yes—
709C22C2×#54212yes—
710C22C2×#54612yes—
711C22C2×#54711yescharacteristic
712C22C2×#54711yescharacteristic
713C22C2×#55211yescharacteristic
714C22C2×#55211yescharacteristic
715C22C2×#55312yes—
716C22C2×#55312yes—
717C22C2×#55512yes—
718C22C2×#56212yes—
719C22C2×#56212yes—
720C4—28no—
721C4—48no—
722C4—44no—
723C4—44no—
724C4—84no—
725C4—82no—
726C4—84no—
727C4—82no—
728C4—84no—
729C2—48no—
730C2—84no—
731C2#16816116yesdirect product factor
732C2C2×#261711yescharacteristic
733C2C2×#262211yescharacteristic
734C2C2×#266311yescharacteristic
735C2C2×#248611yescharacteristic
736C2C2×#250011yescharacteristic
737C2C2×#254311yescharacteristic
738C2C2×#281611yescharacteristic
739C2C2×#268612yes—
740C2C2×#270812yes—
741C2C2×#277912yes—
742C2C2×#278112yes—
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#380C2semidirect
4. #16816C2direct (N × H)
5. C2×#1444C2semidirect
6. C2×#1442C2semidirect
7. C2×#1442C2semidirect
8. C2×#1440C2semidirect
9. C2×#1440C2semidirect
10. C2×#1439C2semidirect
11. C2×#1439C2semidirect
12. C2×C22⋊#95C2semidirect
30. C2×#380C22semidirect
42. #1444C22semidirect
43. #1442C22semidirect
44. #1442C22semidirect
45. #1440C22semidirect
46. #1440C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. C22⋊#95C22semidirect
731. C2#16816direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3942
C22×#3862
C22×#3802
#168162
C2×#14442
C2×#14422
C2×#14402
C2×#14392
C2×C22⋊#952