Finite Group Databasebeta

← C2×#28208 #6463807

Subgroups — C2×#28208 #6463807

Order 512.

2258 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic
3C2×C4×#125C211yesmaximal, characteristic, complemented
4C2×C4×#124C211yesmaximal, characteristic
5#28208C2116yesmaximal, direct product factor
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2085C211yesmaximal, characteristic, complemented
8C2×#2042C211yesmaximal, characteristic
9C2×#2042C211yesmaximal, characteristic
10C2×#1928C211yesmaximal, characteristic, complemented
11C2×#1926C211yesmaximal, characteristic
12C2×#1927C211yesmaximal, characteristic, complemented
13C2×#1925C211yesmaximal, characteristic
14C2×#1831C211yesmaximal, characteristic, complemented
15C2×#1805C211yesmaximal, characteristic, complemented
16C2×#1804C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#479—28no—
22C2×#479C2218yescomplemented
23C2×#478—28no—
24C2×#478C2218yes—
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C4×#125—24no—
40C4×#125C2218yescomplemented
41C4×#124—24no—
42C4×#124C2218yes—
43C2×#391C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#388—21no—
48C2×#388—21no—
49C2×#387—21no—
50C2×#387—21no—
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×C4×C4⋊C4—21no—
54C2×C4×C4⋊C4—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4—21no—
58C2×C4×C22⋊C4—21no—
59C2×C43C2211yescharacteristic
60#2085C2218yescomplemented
61#2085C2218yescomplemented
62#2042C2218yes—
63#2042C2218yes—
64#1928C2218yescomplemented
65#1927C2218yescomplemented
66#1926C2218yes—
67#1925C2218yes—
68#1831C2218yescomplemented
69#1805C2218yescomplemented
70#1804C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#125—22no—
87C2×#125—24no—
88C2×#125D418yes—
89C2×#125C2314yes—
90C2×#124—24no—
91C2×#124—24no—
92C2×#124—22no—
93C2×#124C2314yes—
94C2×C4×Q8—22no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8C2314yes—
98C2×C4×D4—22no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#479—232no—
129#478—232no—
130C2×#107—24no—
131C2×#107—24no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106—24no—
135C2×#106—24no—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8—24no—
139C2×C4⋊C8—24no—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#103—24no—
143C2×#103—24no—
144C2×#103—24no—
145C2×#103—24no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102—24no—
151C2×#102—24no—
152C2×#102—24no—
153C2×#102—24no—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158#391—22no—
159#391C2314yes—
160#390—22no—
161#390C2314yes—
162#389—22no—
163#389—22no—
164#389C2314yes—
165#389C2314yes—
166#388—28no—
167#388—28no—
168#387—28no—
169#387—28no—
170#382—22no—
171#382—22no—
172#382C2314yes—
173#382C2314yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—28no—
177C4×C4⋊C4—28no—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—28no—
181C4×C22⋊C4—28no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C43D414yes—
193C43C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202C25—21no—
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8#4114yes—
210C22×Q8C2412yes—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4#4114yes—
218C22×D4C2412yes—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—24no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2411yescharacteristic
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253#125—216no—
254#125—28no—
255#125C2×D4116yes—
256#124—216no—
257#124—28no—
258#124—28no—
259C4×Q8—28no—
260C4×Q8—28no—
261C4×Q8—48no—
262C4×D4—28no—
263C4×D4—28no—
264C4×D4—48no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42C2×D412yes—
330C2×C42C2×D414yes—
331C2×C42#4114yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335#107—28no—
336#107—28no—
337#106—28no—
338#106—28no—
339C4⋊C8—28no—
340C4⋊C8—28no—
341#103—28no—
342#103—28no—
343#103—28no—
344#103—28no—
345#102—28no—
346#102—28no—
347#102—28no—
348#102—28no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359C24—24no—
360C24—22no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24C22×D411yescharacteristic
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8C2×#4114yes—
381C2×D4—24no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4C2×#4114yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—48no—
440C22×C4—816no—
441C22×C4#12114yes—
442C22×C4#12114yes—
443C22×C4#13514yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C42−1+412yes—
456C22×C42−1+412yes—
457C22×C4C2511yescharacteristic
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—44no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8—44no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—48no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4C22×D414yes—
496C4⋊C4C22×D414yes—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C42—24no—
503C42—24no—
504C42—28no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—44no—
510C42—44no—
511C42—44no—
512C42—48no—
513C42—48no—
514C42—48no—
515C42#12118yes—
516C42C22×D414yes—
517C42C2×#4114yes—
518C23—24no—
519C23—48no—
520C23—48no—
521C23—42no—
522C23—42no—
523C23—48no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23C2×#12112yes—
533C23#52512yes—
534C23#53612yes—
535C23C2×#13512yes—
536C23#57512yes—
537C23#57512yes—
538C23#57512yes—
539C23C23×D411yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—42no—
543Q8—42no—
544Q8—48no—
545D4—24no—
546D4—24no—
547D4—42no—
548D4—42no—
549D4—48no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—816no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4C2×#12114yes—
601C2×C4C2×#12114yes—
602C2×C4C2×#12112yes—
603C2×C4#52512yes—
604C2×C4#53612yes—
605C2×C4#54814yes—
606C2×C4#54814yes—
607C2×C4#55214yes—
608C2×C4#55214yes—
609C2×C4C2×#13514yes—
610C2×C4C23×D411yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×2−1+411yescharacteristic
615C2×C4C2×2−1+411yescharacteristic
616C8—42no—
617C8—42no—
618C8—42no—
619C8—42no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—816no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22#253614yes—
629C22#303514yes—
630C22#304014yes—
631C22#310214yes—
632C22#311114yes—
633C22#323214yes—
634C22#324014yes—
635C22C22×#12111yescharacteristic
636C22C2×#52511yescharacteristic
637C22C2×#53611yescharacteristic
638C22C22×#13511yescharacteristic
639C22C2×#57511yescharacteristic
640C22C2×#57511yescharacteristic
641C22C2×#57511yescharacteristic
642C4—24no—
643C4—24no—
644C4—24no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—44no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4#253614yes—
656C4C2×#54811yescharacteristic
657C4C2×#54811yescharacteristic
658C4C2×#55211yescharacteristic
659C4C2×#55211yescharacteristic
660C2—84no—
661C2#2820818yesdirect product factor
662C2C2×#253611yescharacteristic
663C2C2×#303511yescharacteristic
664C2C2×#304011yescharacteristic
665C2C2×#310211yescharacteristic
666C2C2×#311111yescharacteristic
667C2C2×#323211yescharacteristic
668C2C2×#324011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#479C2semidirect
3. C2×C4×#125C2semidirect
5. #28208C2direct (N × H)
6. C2×#2085C2semidirect
7. C2×#2085C2semidirect
10. C2×#1928C2semidirect
12. C2×#1927C2semidirect
14. C2×#1831C2semidirect
15. C2×#1805C2semidirect
22. C2×#479C22semidirect
40. C4×#125C22semidirect
60. #2085C22semidirect
61. #2085C22semidirect
64. #1928C22semidirect
65. #1927C22semidirect
68. #1831C22semidirect
69. #1805C22semidirect
661. C2#28208direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4792
C22×#4782
C2×C4×#1252
C2×C4×#1242
#282082
C2×#20852
C2×#20422
C2×#19282
C2×#19262
C2×#19272
C2×#19252
C2×#18312
C2×#18052
C2×#18042