Finite Group Databasebeta

← #401824

Subgroups — #401824

Order 512.

1650 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#2072C211yesmaximal, characteristic, complemented
4C2×#2035C212yesmaximal
5#14686C211yesmaximal, characteristic
6#11207C211yesmaximal, characteristic, complemented
7#11199C212yesmaximal
8#11050C211yesmaximal, characteristic, complemented
9#10908C212yesmaximal, complemented
10#10789C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#106C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19C4×C22⋊Q8—24no—
20C4×C22⋊Q8—24no—
21C2×#391C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#387—21no—
24C2×#385—21no—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28C2×#380—21no—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34#2084—24no—
35#2072—24no—
36#2035—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C4⋊#95—21no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8D414yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—24no—
63C2×C4×Q8—42no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—24no—
84C22×C42—42no—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—42no—
101#391—22no—
102#391D414yes—
103#389—22no—
104#389D414yes—
105#387—28no—
106#385—28no—
107#384—22no—
108#384—22no—
109#384—42no—
110#380—22no—
111#380—22no—
112#380—24no—
113#380—24no—
114#380—44no—
115#380—44no—
116C4×C4⋊C4—28no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C22⋊C4—28no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—42no—
133#336—22no—
134#336—22no—
135#336—24no—
136C25C2×D411yescharacteristic
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8C2×D412yes—
145C22×Q8C2×D412yes—
146C22×Q8#4112yes—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4SD1614yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4112yes—
182C23×C4C2411yescharacteristic
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8#4114yes—
197C22⋊Q8—28no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8—28no—
201C4×Q8—48no—
202C4×Q8—48no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4#4112yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—48no—
271C2×C42—48no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42#4114yes—
281#106—48no—
282#103—48no—
283#103—48no—
284C22⋊C8—48no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—44no—
291#95—44no—
292#95—44no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24#12112yes—
306C24C2×SD1612yes—
307C24AGL(1,ℤ/8)12yes—
308C24C22×D411yescharacteristic
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8#12114yes—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—28no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
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—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4#12012yes—
422C22×C4#12012yes—
423C22×C4#12112yes—
424C22×C4#12312yes—
425C22×C4#12312yes—
426C22×C4C2×SD1612yes—
427C22×C4C2×SD1612yes—
428C22×C4#13512yes—
429C22×C4#13712yes—
430C22×C4#13712yes—
431C22×C4#13712yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C42−1+412yes—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C4⋊C4—216no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—48no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4#12114yes—
479C22⋊C4—216no—
480C22⋊C4—48no—
481C22⋊C4—88no—
482C42—24no—
483C42—24no—
484C42—24no—
485C42—24no—
486C42—44no—
487C42—44no—
488C42—48no—
489C42—48no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23#46014yes—
525C23C2×#12011yescharacteristic
526C23C2×#12111yescharacteristic
527C23C2×#12311yescharacteristic
528C23#52511yescharacteristic
529C23#53211yescharacteristic
530C23#53411yescharacteristic
531C23#53611yescharacteristic
532C23C22×SD1611yescharacteristic
533C23C2×#13511yescharacteristic
534C23C2×AGL(1,ℤ/8)11yescharacteristic
535C23C2×#13711yescharacteristic
536C23C2×#13711yescharacteristic
537C23C2×#13711yescharacteristic
538C23#57511yescharacteristic
539C23#57511yescharacteristic
540Q8—28no—
541Q8—44no—
542Q8—44no—
543Q8—44no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—82no—
611C2×C4—82no—
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—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4#44814yes—
625C2×C4#45214yes—
626C2×C4#48314yes—
627C2×C4C2×#12111yescharacteristic
628C2×C4C2×#12111yescharacteristic
629C2×C4#52511yescharacteristic
630C2×C4#52511yescharacteristic
631C2×C4#52912yes—
632C2×C4#52912yes—
633C2×C4#53912yes—
634C2×C4#54212yes—
635C2×C4#54612yes—
636C2×C4#54612yes—
637C2×C4#54612yes—
638C2×C4#54712yes—
639C2×C4#54712yes—
640C2×C4#54812yes—
641C2×C4#55212yes—
642C2×C4#55312yes—
643C8—44no—
644C8—82no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22#239511yescharacteristic
661C22#240511yescharacteristic
662C22#243211yescharacteristic
663C22#248611yescharacteristic
664C22#254311yescharacteristic
665C22#263811yescharacteristic
666C22#284711yescharacteristic
667C22C2×#44811yescharacteristic
668C22C2×#45211yescharacteristic
669C22#300611yescharacteristic
670C22#300711yescharacteristic
671C22C2×#46011yescharacteristic
672C22#303511yescharacteristic
673C22#303711yescharacteristic
674C22#303911yescharacteristic
675C22C2×#48311yescharacteristic
676C22#309111yescharacteristic
677C22#309211yescharacteristic
678C22#309511yescharacteristic
679C22#310311yescharacteristic
680C22#310711yescharacteristic
681C22#311211yescharacteristic
682C22#311611yescharacteristic
683C22#316711yescharacteristic
684C22#317211yescharacteristic
685C22#318011yescharacteristic
686C22#318511yescharacteristic
687C22#318711yescharacteristic
688C22#318811yescharacteristic
689C22#319511yescharacteristic
690C22#319611yescharacteristic
691C22#322911yescharacteristic
692C22#323111yescharacteristic
693C22#323811yescharacteristic
694C22#324011yescharacteristic
695C4—24no—
696C4—24no—
697C4—24no—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—82no—
703C4—82no—
704C4—162no—
705C4—162no—
706C4#242512yes—
707C4#242712yes—
708C4#246012yes—
709C4#246812yes—
710C2—44no—
711C2#1482411yescharacteristic
712C2#2375611yescharacteristic
713C2#2376211yescharacteristic
714C2#2400311yescharacteristic
715C2#2400711yescharacteristic
716C2#2567211yescharacteristic
717C2#2567911yescharacteristic
718C2#2764711yescharacteristic
719C2#2765211yescharacteristic
720C2#2857411yescharacteristic
721C2#2858111yescharacteristic
722C2#3016811yescharacteristic
723C2#3017611yescharacteristic
724C2#3290311yescharacteristic
725C2#3290711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
3. C2×#2072C2semidirect
6. #11207C2semidirect
8. #11050C2semidirect
9. #10908C2semidirect
10. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C2×#20842
C2×#20722
C2×#20352
#146862
#112072
#111992
#110502
#109082
#107892
#107122