Finite Group Databasebeta

← #401608

Subgroups — #401608

Order 512.

2042 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
1C2×Q8×C22⋊C4C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic, complemented
3C2×#1803C211yesmaximal, characteristic, complemented
4C2×#1787C212yesmaximal
5#15150C211yesmaximal, characteristic
6#11199C212yesmaximal
7#11190C212yesmaximal, complemented
8#11077C211yesmaximal, characteristic, complemented
9#11042C211yesmaximal, characteristic
10#10789C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C24×Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2212yes—
21C2×#397—21no—
22C2×#397—21no—
23Q8×C22⋊C4—24no—
24Q8×C22⋊C4—24no—
25C2×#391—21no—
26C2×#391—21no—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#380—21no—
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34C2×C4×C22⋊C4C2211yescharacteristic
35#2072—24no—
36#1803—24no—
37#1787—28no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C23×Q8—22no—
49C23×Q8—22no—
50C23×Q8—22no—
51C23×Q8—22no—
52C23×Q8—21no—
53C23×Q8—22no—
54C23×Q8—24no—
55C23×Q8—24no—
56C23×Q8—42no—
57C23×Q8—42no—
58C23×Q8D414yes—
59C23×Q8C2311yescharacteristic
60C23×Q8C2311yescharacteristic
61C24×C4—21no—
62C24×C4C2311yescharacteristic
63C23×C8C2312yes—
64C23×C8C2312yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—24no—
69C2×C4×Q8—24no—
70C2×C4×Q8—42no—
71C2×C4×Q8—42no—
72C2×C4×Q8D414yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—42no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#103—42no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—42no—
105C2×C22⋊C8—42no—
106#397—44no—
107#397—44no—
108#391—28no—
109#391—28no—
110#391—22no—
111#391—22no—
112#391—22no—
113#391D414yes—
114#384—22no—
115#384—22no—
116#380—28no—
117#380—22no—
118#380—22no—
119#380—24no—
120#380—24no—
121C4×C22⋊C4—28no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95D411yescharacteristic
135C2×#95D411yescharacteristic
136#340—24no—
137#340—24no—
138#340—42no—
139#340—42no—
140#336—22no—
141#336—42no—
142C25C2×D411yescharacteristic
143C22×Q8—24no—
144C22×Q8—28no—
145C22×Q8—28no—
146C22×Q8—24no—
147C22×Q8—28no—
148C22×Q8—28no—
149C22×Q8—24no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—216no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—44no—
164C22×Q8—44no—
165C22×Q8—48no—
166C22×Q8—48no—
167C22×Q8—48no—
168C22×Q8—48no—
169C22×Q8—48no—
170C22×Q8—48no—
171C22×Q8—48no—
172C22×Q8—44no—
173C22×Q8—44no—
174C22×Q8—48no—
175C22×Q8—44no—
176C22×Q8—44no—
177C22×Q8—48no—
178C22×Q8C2×D412yes—
179C22×Q8C2×D412yes—
180C22×Q8C2×D412yes—
181C22×Q8C2×D412yes—
182C22×Q8#4112yes—
183C22×Q8#4112yes—
184C22×Q8#4114yes—
185C22×Q8#4114yes—
186C23×C4—21no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D412yes—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4#4112yes—
212C23×C4C2411yescharacteristic
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C4×Q8—28no—
228C4×Q8—44no—
229C4×Q8—48no—
230C4×Q8—48no—
231C4×Q8—44no—
232C4×Q8—44no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4SD1614yes—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C4⋊C8—44no—
288#103—48no—
289#103—48no—
290#103—48no—
291#103—48no—
292C22⋊C8—48no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—22no—
300#95—42no—
301#95—44no—
302#95—44no—
303#95—42no—
304#95—44no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24#12112yes—
314C24C2×SD1612yes—
315C24AGL(1,ℤ/8)12yes—
316C24C22×D411yescharacteristic
317C2×Q8—216no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—416no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—416no—
344C2×Q8—432no—
345C2×Q8—416no—
346C2×Q8—416no—
347C2×Q8—432no—
348C2×Q8—416no—
349C2×Q8—416no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—42no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8#12114yes—
359C2×Q8#12114yes—
360C2×Q8#12318yes—
361C2×Q8#12314yes—
362C2×Q8#12414yes—
363C2×Q8#12618yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—216no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—416no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—416no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
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—42no—
440C22×C4#12012yes—
441C22×C4#12012yes—
442C22×C4#12112yes—
443C22×C4C2×SD1612yes—
444C22×C4C2×SD1612yes—
445C22×C4#13712yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4111yescharacteristic
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)12yes—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C22⋊C4—28no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—88no—
492C42—22no—
493C42—22no—
494C42—24no—
495C42—48no—
496C42—48no—
497C42—41no—
498C42—41no—
499C42—42no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23#46014yes—
523C23C2×#12011yescharacteristic
524C23C2×#12111yescharacteristic
525C23#52511yescharacteristic
526C23#54612yes—
527C23#54612yes—
528C23C22×SD1611yescharacteristic
529C23C2×AGL(1,ℤ/8)11yescharacteristic
530C23C2×#13711yescharacteristic
531C23#57511yescharacteristic
532C23#57812yes—
533C23#57812yes—
534Q8—24no—
535Q8—24no—
536Q8—28no—
537Q8—28no—
538Q8—24no—
539Q8—28no—
540Q8—432no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—44no—
545Q8—42no—
546Q8#39718yescomplemented
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—432no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4#44814yes—
613C2×C4C2×#12111yescharacteristic
614C2×C4C2×#12111yescharacteristic
615C2×C4C2×#12312yes—
616C2×C4C2×#12311yescharacteristic
617C2×C4#52511yescharacteristic
618C2×C4#52511yescharacteristic
619C2×C4C2×#12411yescharacteristic
620C2×C4C2×#12612yes—
621C2×C4#53411yescharacteristic
622C2×C4#53412yes—
623C2×C4#53511yescharacteristic
624C2×C4#53812yes—
625C2×C4#54612yes—
626C2×C4#54612yes—
627C2×C4#55112yes—
628C2×C4#55112yes—
629C2×C4#55112yes—
630C2×C4#55112yes—
631C8—82no—
632C8—82no—
633C22—24no—
634C22—28no—
635C22—42no—
636C22—42no—
637C22—44no—
638C22—44no—
639C22—42no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22#240511yescharacteristic
645C22#241212yes—
646C22#243511yescharacteristic
647C22#248012yes—
648C22#284711yescharacteristic
649C22C2×#44811yescharacteristic
650C22#300611yescharacteristic
651C22#301111yescharacteristic
652C22#301111yescharacteristic
653C22C2×#46011yescharacteristic
654C22#303711yescharacteristic
655C22#304412yes—
656C22#310311yescharacteristic
657C22#311211yescharacteristic
658C22#312712yes—
659C22#329812yes—
660C22#329812yes—
661C22#330812yes—
662C22#330812yes—
663C22#331212yes—
664C22#331312yes—
665C22#332712yes—
666C22#332812yes—
667C4—28no—
668C4—24no—
669C4—42no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—81no—
674C4—82no—
675C4—82no—
676C4—162no—
677C4—162no—
678C4C2×#39711yescharacteristic
679C4#241212yes—
680C4#248012yes—
681C4#248211yescharacteristic
682C4#256111yescharacteristic
683C4#261711yescharacteristic
684C2—44no—
685C2#1482011yescharacteristic
686C2#2404111yescharacteristic
687C2#2405111yescharacteristic
688C2#2452812yes—
689C2#2454012yes—
690C2#2588411yescharacteristic
691C2#2589611yescharacteristic
692C2#2752212yes—
693C2#2753512yes—
694C2#3290311yescharacteristic
695C2#3291511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2072C2semidirect
3. C2×#1803C2semidirect
7. #11190C2semidirect
8. #11077C2semidirect
10. #10789C2semidirect
11. #10789C2semidirect
546. Q8#397semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊C42
C2×#20722
C2×#18032
C2×#17872
#151502
#111992
#111902
#110772
#110422
#107892