Finite Group Databasebeta

← C2×#14763 #607697

Subgroups — C2×#14763 #607697

Order 512.

6814 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic, complemented
3C22×#389C212yesmaximal, complemented
4C22×#385C211yesmaximal, characteristic
5C22×C4×C22⋊C4C211yesmaximal, characteristic
6#14763C2116yesmaximal, direct product factor
7C2×#1446C211yesmaximal, characteristic, complemented
8C2×#1444C211yesmaximal, characteristic, complemented
9C2×#1441C211yesmaximal, characteristic, complemented
10C2×#1439C212yesmaximal
11C2×#1439C212yesmaximal
12C2×#1436C212yesmaximal, complemented
13C24×Q8C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C22×C22⋊Q8—28no—
16C22×C22⋊Q8—216no—
17C22×C22⋊Q8C2218yes—
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23C23×C4⋊C4C2212yes—
24C23×C22⋊C4C2212yes—
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2212yes—
27C23×C22⋊C4C2211yescharacteristic
28C23×C42C2212yes—
29C2×#395—28no—
30C2×#395C2218yescomplemented
31C2×#389—216no—
32C2×#389C22116yescomplemented
33C2×#385—28no—
34C2×#385C2218yes—
35C2×C4×C22⋊C4—28no—
36C2×C4×C22⋊C4C2218yes—
37C22×#95C2212yes—
38C22×#95C2212yes—
39C22×#95C2212yes—
40C22×#95C2212yes—
41C22×#95C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2212yes—
46C22×#95C2211yescharacteristic
47#1446C2218yescomplemented
48#1444C2218yescomplemented
49#1441C2218yescomplemented
50#1439C22116yes—
51#1439C22116yes—
52#1436C22116yescomplemented
53C26C2311yescharacteristic
54C23×Q8—28no—
55C23×Q8—24no—
56C23×Q8—24no—
57C23×Q8—24no—
58C23×Q8—44no—
59C23×Q8C2314yes—
60C24×C4—28no—
61C24×C4—24no—
62C24×C4—24no—
63C24×C4—44no—
64C24×C4D418yes—
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C24×C4C2312yes—
68C24×C4C2312yes—
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2314yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2312yes—
76C2×C22⋊Q8—2128no—
77C2×C22⋊Q8—232no—
78C2×C22⋊Q8—264no—
79C22×C4⋊C4—28no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—28no—
82C22×C4⋊C4—28no—
83C22×C4⋊C4—28no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4D4116yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2318yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2318yes—
100C22×C4⋊C4C2318yes—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4Q8116yescomplemented
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2318yes—
115C22×C22⋊C4C2314yes—
116C22×C42—28no—
117C22×C42—28no—
118C22×C42—44no—
119C22×C42C2318yes—
120#395—232no—
121#389—264no—
122#385—232no—
123C4×C22⋊C4—232no—
124C2×#95—28no—
125C2×#95—24no—
126C2×#95—24no—
127C2×#95—24no—
128C2×#95—28no—
129C2×#95—28no—
130C2×#95—28no—
131C2×#95—28no—
132C2×#95—44no—
133C2×#95D418yes—
134C2×#95C2318yes—
135C2×#95C2314yes—
136C2×#95C2318yes—
137C2×#95C2318yes—
138C2×#95C2318yes—
139C2×#95C2318yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2318yes—
143C2×#95C2314yes—
144C25—24no—
145C25—22no—
146C25—24no—
147C25—22no—
148C25—22no—
149C25—22no—
150C25—24no—
151C25—22no—
152C25—22no—
153C25C2×D414yes—
154C25C2×Q818yes—
155C25C2412yes—
156C25C2411yescharacteristic
157C22×Q8—232no—
158C22×Q8—216no—
159C22×Q8—28no—
160C22×Q8—28no—
161C22×Q8—28no—
162C22×Q8—48no—
163C22×Q8—432no—
164C22×Q8—432no—
165C22×Q8—416no—
166C22×Q8—416no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—232no—
205C23×C4—232no—
206C23×C4—216no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—416no—
220C23×C4—48no—
221C23×C4—432no—
222C23×C4—432no—
223C23×C4C2×D418yes—
224C23×C4C2×D418yes—
225C23×C4C2×D4116yes—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D418yes—
231C23×C4C2×Q818yes—
232C23×C4C2×Q818yes—
233C23×C4#4114yes—
234C23×C4#4114yes—
235C23×C4#4114yes—
236C23×C4#4118yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2414yes—
247C22⋊Q8—2256no—
248C2×C4⋊C4—264no—
249C2×C4⋊C4—232no—
250C2×C4⋊C4—264no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—216no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—216no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—432no—
267C2×C4⋊C4—416no—
268C2×C4⋊C4—416no—
269C2×C4⋊C4C2×D4132yes—
270C2×C22⋊C4—232no—
271C2×C22⋊C4—264no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—264no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—416no—
284C2×C22⋊C4C2×Q8132yescomplemented
285C2×C42—216no—
286C2×C42—216no—
287C2×C42—48no—
288C2×C42—432no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—216no—
293#95—216no—
294#95—216no—
295#95—216no—
296#95—216no—
297#95—48no—
298#95C2×D4116yes—
299C24—28no—
300C24—28no—
301C24—216no—
302C24—28no—
303C24—28no—
304C24—216no—
305C24—28no—
306C24—28no—
307C24—216no—
308C24—216no—
309C24—216no—
310C24—216no—
311C24—24no—
312C24—22no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—48no—
321C24—48no—
322C24—48no—
323C24—48no—
324C24—48no—
325C24—48no—
326C24—44no—
327C24—44no—
328C24—44no—
329C24—44no—
330C24C4⋊Q8116yes—
331C24C22×D414yes—
332C24C22×D412yes—
333C24C22×D414yes—
334C24C22×D412yes—
335C24C22×Q818yes—
336C24C22×Q814yes—
337C24C2×#4112yes—
338C24C2×#4114yes—
339C24C2×#4112yes—
340C24C2×#4112yes—
341C24H(5,𝔽2)14yes—
342C242−1+412yes—
343C242−1+412yes—
344C24C2511yescharacteristic
345C2×Q8—232no—
346C2×Q8—432no—
347C2×Q8—432no—
348C2×Q8—4128no—
349C2×Q8—416no—
350C2×Q8—416no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—216no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—216no—
357C22×C4—216no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
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—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—232no—
417C22×C4—232no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—416no—
429C22×C4—4128no—
430C22×C4—432no—
431C22×C4—432no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4#120116yes—
462C22×C4#121116yes—
463C22×C4C22⋊Q8132yescomplemented
464C22×C4C22×D418yes—
465C22×C4C22×D418yes—
466C22×C4C22×D414yes—
467C22×C4C22×D418yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D418yes—
471C22×C4C22×Q818yes—
472C22×C4C22×Q818yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4118yes—
477C4⋊C4—264no—
478C4⋊C4—232no—
479C4⋊C4—264no—
480C4⋊C4—432no—
481C4⋊C4—416no—
482C4⋊C4—416no—
483C22⋊C4—232no—
484C22⋊C4—264no—
485C22⋊C4—264no—
486C22⋊C4—416no—
487C42—432no—
488C23—232no—
489C23—232no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—232no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—416no—
504C23—416no—
505C23—416no—
506C23—44no—
507C23—416no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—432no—
517C23—42no—
518C23—432no—
519C23—432no—
520C23—432no—
521C23C2×#12014yes—
522C23C2×#12114yes—
523C23C2×C22⋊Q818yes—
524C23#52514yes—
525C23#52518yes—
526C23C2×C4⋊Q818yes—
527C23C2×C4⋊Q814yes—
528C23#53214yes—
529C23#53218yes—
530C23#53318yes—
531C23#53414yes—
532C23#53614yes—
533C23#53918yes—
534C23#54218yes—
535C23D4218yes—
536C23#54618yes—
537C23#54714yes—
538C23#54814yes—
539C23D4×Q818yes—
540C23D4×Q818yes—
541C23#55214yes—
542C23#55418yes—
543C23#55418yes—
544C23#55614yes—
545C23C23×D411yescharacteristic
546C23C23×D412yes—
547C23C23×D411yescharacteristic
548C23C23×Q812yes—
549C23C22×#4111yescharacteristic
550C23C22×#4112yes—
551C23C22×#4111yescharacteristic
552C23C22×#4111yescharacteristic
553C23C2×H(5,𝔽2)12yes—
554C23C2×2−1+411yescharacteristic
555C23C2×2−1+411yescharacteristic
556Q8—464no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—232no—
560C2×C4—232no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—264no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—464no—
589C2×C4—432no—
590C2×C4—416no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—432no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—416no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—432no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—88no—
624C2×C4—88no—
625C2×C4C2×#12018yes—
626C2×C4C2×#12118yes—
627C2×C4C2×C22⋊Q8116yescomplemented
628C22—28no—
629C22—28no—
630C22—28no—
631C22—48no—
632C22—44no—
633C22—44no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—44no—
638C22—464no—
639C22—44no—
640C22#238818yes—
641C22#241018yes—
642C22#241518yes—
643C22#243418yes—
644C22#2449116yes—
645C22#2452116yes—
646C22#2465116yes—
647C22#2556116yes—
648C22#258318yes—
649C22#266318yes—
650C22#285718yes—
651C22C22×#12011yescharacteristic
652C22C22×#12111yescharacteristic
653C22C22×C22⋊Q812yes—
654C22C2×#52511yescharacteristic
655C22C2×#52512yes—
656C22C22×C4⋊Q811yescharacteristic
657C22C2×#53212yes—
658C22C2×#53211yescharacteristic
659C22C2×#53312yes—
660C22C2×#53411yescharacteristic
661C22C2×#53611yescharacteristic
662C22C2×#53912yes—
663C22C2×#54212yes—
664C22C2×D4212yes—
665C22C2×#54612yes—
666C22C2×#54711yescharacteristic
667C22C2×#54811yescharacteristic
668C22C2×D4×Q812yes—
669C22C2×D4×Q812yes—
670C22C2×#55211yescharacteristic
671C22C2×#55412yes—
672C22C2×#55412yes—
673C22C2×#55611yescharacteristic
674C4—28no—
675C4—28no—
676C4—216no—
677C4—48no—
678C4—48no—
679C4—44no—
680C4—48no—
681C4—44no—
682C4—82no—
683C4—82no—
684C2—48no—
685C2#14763116yesdirect product factor
686C2C2×#238811yescharacteristic
687C2C2×#266311yescharacteristic
688C2C2×#241011yescharacteristic
689C2C2×#244912yes—
690C2C2×#245212yes—
691C2C2×#243411yescharacteristic
692C2C2×#246512yes—
693C2C2×#241511yescharacteristic
694C2C2×#255612yes—
695C2C2×#258311yescharacteristic
696C2C2×#285711yescharacteristic
subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#395C2semidirect
3. C22×#389C2semidirect
6. #14763C2direct (N × H)
7. C2×#1446C2semidirect
8. C2×#1444C2semidirect
9. C2×#1441C2semidirect
12. C2×#1436C2semidirect
30. C2×#395C22semidirect
32. C2×#389C22semidirect
47. #1446C22semidirect
48. #1444C22semidirect
49. #1441C22semidirect
52. #1436C22semidirect
111. C22×C22⋊C4Q8semidirect
284. C2×C22⋊C4C2×Q8semidirect
463. C22×C4C22⋊Q8semidirect
627. C2×C4C2×C22⋊Q8semidirect
685. C2#14763direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C22×#3952
C22×#3892
C22×#3852
C22×C4×C22⋊C42
#147632
C2×#14462
C2×#14442
C2×#14412
C2×#14392
C2×#14362