Finite Group Databasebeta

← #626621

Subgroups — #626621

Order 512.

3694 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C212yesmaximal, complemented
2C2×#2325C211yesmaximal, characteristic
3C22×#391C211yesmaximal, characteristic, complemented
4#15883C214yesmaximal, complemented
5#15853C214yesmaximal, complemented
6#15279C214yesmaximal, complemented
7#15275C214yesmaximal, complemented
8#15014C211yesmaximal, characteristic, complemented
9#14913C211yesmaximal, characteristic
10#14821C211yesmaximal, characteristic, complemented
11#14719C211yesmaximal, characteristic
12#14696C212yesmaximal, complemented
13#14347C211yesmaximal, characteristic, complemented
14#14259C212yesmaximal, complemented
15#13960C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2409—216no—
23C2×#398C2212yes—
24C2×#395C2214yes—
25C2×#394C2212yes—
26C2×#393C2214yes—
27#2325—28no—
28C2×#391—28no—
29C2×#391C2214yes—
30C2×#391C2214yes—
31C2×#391C2214yes—
32C2×#388C2212yes—
33C2×#388C2214yes—
34C2×#387C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1445C2212yes—
51#1444C2212yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2212yes—
68#1437C2214yes—
69#1436C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2212yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2212yes—
78C23×Q8—24no—
79C23×Q8—24no—
80C23×Q8—44no—
81C23×Q8—44no—
82C23×Q8D418yescomplemented
83C23×Q8C2314yes—
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—48no—
96C2×C22⋊Q8—48no—
97C2×#121—28no—
98C2×#121—24no—
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C2×C4×D4—28no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42—24no—
125C22×C42—44no—
126C22×C42C2312yes—
127C22×C42C2314yes—
128#398—44no—
129#395—216no—
130#394—44no—
131#393—216no—
132#391—216no—
133#391—216no—
134#391—216no—
135#388—28no—
136#388—48no—
137#387—28no—
138#386—28no—
139#386—28no—
140#386—28no—
141#386—28no—
142#386—28no—
143#386—28no—
144#385—28no—
145#385—28no—
146#385—28no—
147#380—24no—
148#380—28no—
149C2×#95—24no—
150C2×#95—24no—
151C2×#95—24no—
152C2×#95D418yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2412yes—
180C25C2411yescharacteristic
181C22×Q8—232no—
182C22×Q8—28no—
183C22×Q8—28no—
184C22×Q8—28no—
185C22×Q8—28no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—432no—
189C22×Q8—416no—
190C22×Q8—416no—
191C22×Q8—432no—
192C22×Q8C2×D4116yes—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4#4118yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4#4114yes—
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C22⋊Q8—464no—
254#121—232no—
255C4×D4—432no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—416no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—24no—
316C2×C42—28no—
317C2×C42—44no—
318C2×C42—416no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—42no—
336#95—42no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—42no—
341#95—42no—
342#95—44no—
343#95C2×D418yes—
344#95C2×D418yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24C22×D412yes—
360C24C22×D412yes—
361C24C2×#4114yes—
362C24C2×#4112yes—
363C24C2×#4114yes—
364C242−1+414yes—
365C24C2511yescharacteristic
366C2×Q8—232no—
367C2×Q8—232no—
368C2×Q8—416no—
369C2×Q8—416no—
370C2×Q8—416no—
371C2×Q8—416no—
372C2×Q8—416no—
373C2×Q8—4128no—
374C2×D4—216no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—28no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—88no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—216no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—44no—
464C22×C4—416no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4—88no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4#121116yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C42−1+414yes—
491C4⋊C4—28no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—416no—
495C4⋊C4—416no—
496C4⋊C4—48no—
497C4⋊C4—416no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C22⋊C4—216no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—88no—
511C42—416no—
512C42—48no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—42no—
531C23—44no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23C2×#12114yes—
549C23#52514yes—
550C23#53614yes—
551C23#54818yes—
552C23#55218yes—
553C23C23×D411yescharacteristic
554C23C23×D411yescharacteristic
555C23C22×#4111yescharacteristic
556C23C22×#4112yes—
557C23C2×H(5,𝔽2)12yes—
558C23C2×H(5,𝔽2)12yes—
559C23C2×2−1+412yes—
560C23C2×2−1+412yes—
561C23#58512yes—
562Q8—464no—
563D4—432no—
564D4—88no—
565C2×C4—232no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—416no—
593C2×C4—416no—
594C2×C4—416no—
595C2×C4—416no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—816no—
639C2×C4—88no—
640C2×C4—88no—
641C2×C4C2×#12118yes—
642C2×C4#53418yes—
643C22—24no—
644C22—216no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—216no—
651C22—44no—
652C22—48no—
653C22—44no—
654C22—44no—
655C22—48no—
656C22—84no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—84no—
662C22#253618yes—
663C22C22×#12111yescharacteristic
664C22C2×#52511yescharacteristic
665C22C2×#53412yes—
666C22C2×#53611yescharacteristic
667C22#346112yes—
668C22C2×#54812yes—
669C22C2×#55212yes—
670C22#348411yescharacteristic
671C22#348411yescharacteristic
672C22#348511yescharacteristic
673C22#348611yescharacteristic
674C22#349012yes—
675C22#349712yes—
676C22#351414yes—
677C22#352014yes—
678C22#354114yes—
679C22#355214yes—
680C4—28no—
681C4—28no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—84no—
692C4—84no—
693C2—28no—
694C2—84no—
695C2C2×#253611yescharacteristic
696C2#3498111yescharacteristic
697C2#3515311yescharacteristic
698C2#3535711yescharacteristic
699C2#3545111yescharacteristic
700C2#3671611yescharacteristic
701C2#3723311yescharacteristic
702C2#3822114yes—
703C2#3916614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2409C2semidirect
3. C22×#391C2semidirect
4. #15883C2semidirect
5. #15853C2semidirect
6. #15279C2semidirect
7. #15275C2semidirect
8. #15014C2semidirect
10. #14821C2semidirect
12. #14696C2semidirect
13. #14347C2semidirect
14. #14259C2semidirect
15. #13960C2semidirect
82. C23×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#23252
C22×#3912
#158832
#158532
#152792
#152752
#150142
#149132
#148212
#147192
#146962
#143472
#142592
#139602