Finite Group Databasebeta

← #474920

Subgroups — #474920

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2088C211yesmaximal, characteristic
2C2×#2056C211yesmaximal, characteristic
3C2×#1933C211yesmaximal, characteristic
4C2×#1933C211yesmaximal, characteristic
5#15237C211yesmaximal, characteristic
6#15184C211yesmaximal, characteristic
7#11373C211yesmaximal, characteristic
8#11372C211yesmaximal, characteristic
9#11122C211yesmaximal, characteristic
10#11122C211yesmaximal, characteristic
11#11052C211yesmaximal, characteristic
12#11050C211yesmaximal, characteristic
13#11024C211yesmaximal, characteristic
14#11024C211yesmaximal, characteristic
15#10834C211yesmaximal, characteristic
16C23×Q16C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#2088—24no—
38#2056—24no—
39#1933—24no—
40#1933—24no—
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53#1443—22no—
54#1441—22no—
55#1441—22no—
56#1441—22no—
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C23×Q8C2311yescharacteristic
60C23×Q8C2311yescharacteristic
61C22×Q16—22no—
62C22×Q16—22no—
63C22×Q16—22no—
64C22×Q16—41no—
65C22×Q16—41no—
66C22×Q16—41no—
67C22×Q16—41no—
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#106—22no—
86C2×#106—41no—
87C2×#106—41no—
88C2×C4⋊C8—41no—
89C2×C4⋊C8—41no—
90C2×C4⋊C8D414yes—
91C2×#103—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—41no—
104C2×#103—41no—
105C2×#103—41no—
106C2×#103—41no—
107C2×#103—41no—
108C2×#103—41no—
109C2×C4×C8—41no—
110C2×C4×C8—41no—
111C2×C4×C8D414yes—
112#391—22no—
113#391—22no—
114#391—22no—
115#391—22no—
116#391—22no—
117#391—22no—
118#391—41no—
119#391—41no—
120#391—41no—
121#391—41no—
122#391—41no—
123#391—41no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#384—22no—
129#384—41no—
130#382—22no—
131#382—41no—
132C4×C4⋊C4—41no—
133C4×C4⋊C4—41no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95—22no—
147C2×#95D412yes—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#336—41no—
157#336—41no—
158#336—41no—
159#336—41no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—21no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8—21no—
186C22×Q8#4112yes—
187C22×Q8#4112yes—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4C2×D412yes—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2411yescharacteristic
201C2×Q16—44no—
202C2×Q16—44no—
203C2×Q16—44no—
204C2×Q16—44no—
205C2×Q16—44no—
206C2×Q16—44no—
207C2×Q16—44no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8C2×D412yes—
233C22×C8C2×D412yes—
234C22×C8C2×D412yes—
235C22×C8C2×D412yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4#4112yes—
281C2×C4⋊C4#4112yes—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300#106—44no—
301C4⋊C8—44no—
302#103—28no—
303#103—28no—
304#103—44no—
305#103—44no—
306#103—44no—
307#103—44no—
308C4×C8—44no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—42no—
319#95—42no—
320#95—42no—
321#95—44no—
322#95—82no—
323C24C22×D411yescharacteristic
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4#12712yes—
445C22×C4#12712yes—
446C22×C4#13514yes—
447C22×C4#13714yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4H(5,𝔽2)11yescharacteristic
455C22×C4H(5,𝔽2)11yescharacteristic
456C22×C4H(5,𝔽2)11yescharacteristic
457C22×C4H(5,𝔽2)11yescharacteristic
458C22×C4H(5,𝔽2)11yescharacteristic
459C22×C42−1+411yescharacteristic
460C22×C42−1+411yescharacteristic
461C22×C42−1+411yescharacteristic
462Q16—416no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—48no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—88no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C23C2×#12711yescharacteristic
532C23#53411yescharacteristic
533C23#53711yescharacteristic
534C23#54611yescharacteristic
535C23#54611yescharacteristic
536C23#54811yescharacteristic
537C23#54811yescharacteristic
538C23C2×#13512yes—
539C23C2×#13712yes—
540C23#57712yes—
541C23#57812yes—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—81no—
551Q8—81no—
552Q8—81no—
553Q8—81no—
554Q8—81no—
555Q8—81no—
556Q8—81no—
557Q8—81no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4#49714yes—
626C2×C4#52511yescharacteristic
627C2×C4#52511yescharacteristic
628C2×C4#53411yescharacteristic
629C2×C4#53411yescharacteristic
630C2×C4#53711yescharacteristic
631C2×C4#53711yescharacteristic
632C2×C4#53711yescharacteristic
633C2×C4#53711yescharacteristic
634C2×C4#53811yescharacteristic
635C2×C4#53811yescharacteristic
636C2×C4#53911yescharacteristic
637C2×C4#53911yescharacteristic
638C2×C4#54011yescharacteristic
639C2×C4#54011yescharacteristic
640C2×C4#54211yescharacteristic
641C2×C4#54211yescharacteristic
642C2×C4#54611yescharacteristic
643C2×C4#54611yescharacteristic
644C2×C4#54811yescharacteristic
645C2×C4#54811yescharacteristic
646C2×C4#55111yescharacteristic
647C2×C4#55211yescharacteristic
648C2×C4#55211yescharacteristic
649C2×C4#55511yescharacteristic
650C2×C4#56011yescharacteristic
651C2×C4#56311yescharacteristic
652C2×C4#56311yescharacteristic
653C2×C4#56311yescharacteristic
654C8—42no—
655C8—42no—
656C8—42no—
657C8—42no—
658C22#254011yescharacteristic
659C22#254011yescharacteristic
660C22#265611yescharacteristic
661C22#269911yescharacteristic
662C22#270211yescharacteristic
663C22#270211yescharacteristic
664C22#272811yescharacteristic
665C22#314711yescharacteristic
666C22C2×#49711yescharacteristic
667C22#315411yescharacteristic
668C22#315611yescharacteristic
669C22#319012yes—
670C22#320112yes—
671C22#324812yes—
672C22#325812yes—
673C22#329612yes—
674C22#330012yes—
675C22#332812yes—
676C22#333112yes—
677C22#336412yes—
678C22#336512yes—
679C22#336912yes—
680C22#337412yes—
681C4—24no—
682C4—42no—
683C4—44no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—161no—
690C4—161no—
691C4—161no—
692C4—161no—
693C4#260911yescharacteristic
694C4#263011yescharacteristic
695C4#263011yescharacteristic
696C4#269711yescharacteristic
697C4#270711yescharacteristic
698C4#270711yescharacteristic
699C4#273111yescharacteristic
700C4#282311yescharacteristic
701C2#1701511yescharacteristic
702C2#2840711yescharacteristic
703C2#2840811yescharacteristic
704C2#2841711yescharacteristic
705C2#2841911yescharacteristic
706C2#3024411yescharacteristic
707C2#3025211yescharacteristic
708C2#3172412yes—
709C2#3181212yes—
710C2#3181312yes—
711C2#3255312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20882
C2×#20562
C2×#19332
#152372
#151842
#113732
#113722
#111222
#110522
#110502
#110242
#108342