Finite Group Databasebeta

← #4930261

Subgroups — #4930261

Order 512.

2406 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2570C211yesmaximal, characteristic, complemented
3#15024C212yesmaximal, complemented
4#15024C212yesmaximal, complemented
5#15004C214yesmaximal, complemented
6#14982C212yesmaximal
7#14982C212yesmaximal
8#14956C214yesmaximal, complemented
9#14282C211yesmaximal, characteristic, complemented
10#14155C211yesmaximal, characteristic, complemented
11#13994C212yesmaximal, complemented
12#13993C212yesmaximal, complemented
13#13905C211yesmaximal, characteristic
14#13876C212yesmaximal, complemented
15#13875C212yesmaximal, complemented
16#13847C211yesmaximal, characteristic
17#13750C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2570—28no—
21#2563—28no—
22C2×#400C2212yes—
23C2×#398C2212yes—
24C2×#393C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#390C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×#376C2211yescharacteristic
50#1446C2214yes—
51#1445C2211yescharacteristic
52#1445C2211yescharacteristic
53#1445C2212yes—
54#1445C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1445C2212yes—
58#1443C2214yes—
59#1443C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2211yescharacteristic
69#1438C2211yescharacteristic
70#1438C2212yes—
71#1437C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2214yes—
81#1434C2214yes—
82C4×#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2314yes—
121C22×C42C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125#400—28no—
126#398—28no—
127#393—216no—
128#391—44no—
129#391—44no—
130#390—28no—
131#389—24no—
132#389—24no—
133#388—48no—
134#388—48no—
135#387—28no—
136#387—28no—
137#384—24no—
138#382—24no—
139#382—28no—
140#382—24no—
141#380—28no—
142#380—28no—
143#380—28no—
144#380—28no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4D418yes—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—28no—
152#376—24no—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C25C2412yes—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—42no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—42no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C4⋊Q8—416no—
221C4×D4—416no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4#4118yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C2×D418yes—
302C2×C42C2×D414yes—
303C2×C42#4118yes—
304C2×C42#4114yes—
305C2×C42#4114yes—
306#95—22no—
307#95—22no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95C2×D418yes—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C242−1+414yes—
352C24C2511yescharacteristic
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—84no—
357C2×Q8—84no—
358C2×Q8—84no—
359C2×Q8—84no—
360C2×D4—28no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—44no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4118yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C4⋊C4—216no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C22⋊C4—216no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C42—216no—
488C42—48no—
489C42—44no—
490C42—44no—
491C42—84no—
492C42—88no—
493C42—88no—
494C42#124116yes—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23C23×D411yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4111yescharacteristic
533C23C22×#4112yes—
534C23C22×#4112yes—
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×2−1+412yes—
538C23#58512yes—
539C23#58511yescharacteristic
540C23#58511yescharacteristic
541Q8—816no—
542D4—416no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—816no—
629C2×C4C2×#12414yes—
630C2×C4C2×#12414yes—
631C2×C4C2×#12414yes—
632C2×C4#52918yes—
633C2×C4#52918yes—
634C2×C4#53214yes—
635C2×C4#53214yes—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22C22×#12411yescharacteristic
652C22C2×#52912yes—
653C22C2×#52912yes—
654C22C2×#53211yescharacteristic
655C22C2×#53211yescharacteristic
656C22#348511yescharacteristic
657C22#348611yescharacteristic
658C22#348811yescharacteristic
659C22#348811yescharacteristic
660C22#349212yes—
661C22#349312yes—
662C22#349412yes—
663C22#349512yes—
664C22#349512yes—
665C22#349512yes—
666C22#350611yescharacteristic
667C22#350611yescharacteristic
668C22#350811yescharacteristic
669C22#351011yescharacteristic
670C22#351412yes—
671C22#351612yes—
672C22#352012yes—
673C22#352412yes—
674C4—24no—
675C4—24no—
676C4—42no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—42no—
681C4—48no—
682C4—44no—
683C4—82no—
684C4—82no—
685C4—81no—
686C4—82no—
687C4—81no—
688C4—84no—
689C4#286118yes—
690C2—84no—
691C2#3555312yes—
692C2#3555611yescharacteristic
693C2#3564411yescharacteristic
694C2#3565512yes—
695C2#3577012yes—
696C2#3582812yes—
697C2#3590911yescharacteristic
698C2#3592611yescharacteristic
699C2#3605111yescharacteristic
700C2#3606211yescharacteristic
701C2C2×#286111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2570C2semidirect
3. #15024C2semidirect
4. #15024C2semidirect
5. #15004C2semidirect
8. #14956C2semidirect
9. #14282C2semidirect
10. #14155C2semidirect
11. #13994C2semidirect
12. #13993C2semidirect
14. #13876C2semidirect
15. #13875C2semidirect
17. #13750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#25702
#150242
#150042
#149822
#149562
#142822
#141552
#139942
#139932
#139052
#138762
#138752
#138472
#137502