Finite Group Databasebeta

← #762896

Subgroups — #762896

Order 512.

2766 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#17058C211yesmaximal, characteristic, complemented
4#17057C211yesmaximal, characteristic, complemented
5#16363C211yesmaximal, characteristic
6#16362C211yesmaximal, characteristic
7#15122C212yesmaximal, complemented
8#15120C212yesmaximal, complemented
9#15017C212yesmaximal, complemented
10#15016C212yesmaximal, complemented
11#14838C214yesmaximal, complemented
12#14249C212yesmaximal, complemented
13#14231C212yesmaximal, complemented
14#14173C214yesmaximal, complemented
15#13959C212yesmaximal, complemented
16#13952C212yesmaximal, complemented
17#13753C211yesmaximal, characteristic, complemented
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C42C2211yescharacteristic
22C2×#399C2212yes—
23C2×#397C2212yes—
24C2×#394C2214yes—
25C4×#121—28no—
26C2×#390C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#388C2212yes—
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#386C2212yes—
32#2283—28no—
33C2×#383C2212yes—
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#381C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×#376C2212yes—
47C2×#376C2212yes—
48#1445C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1444C2212yes—
53#1444C2212yes—
54#1444C2212yes—
55#1443C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1438C2212yes—
67#1438C2212yes—
68#1438C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2214yes—
76#1434C2212yes—
77#1434C2214yes—
78#1434C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#121—24no—
92C2×#121—28no—
93C2×C4×D4—28no—
94C2×C4×D4—24no—
95C2×C4×D4—28no—
96C2×C4×D4—48no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C42—28no—
121C22×C42D418yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#399—28no—
131#397—28no—
132#394—216no—
133#390—24no—
134#389—24no—
135#388—28no—
136#388—48no—
137#387—48no—
138#386—28no—
139#383—28no—
140#382—24no—
141#382—24no—
142#381—28no—
143#380—24no—
144#380—24no—
145#380—24no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C22⋊C4—28no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—28no—
152#376—28no—
153#376—28no—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—44no—
192C22×D4—42no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—41no—
202C23×C4—41no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226#121—232no—
227C4×D4—432no—
228C4×D4—416no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4#4118yes—
284C2×C22⋊C4#4118yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—232no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42#4118yes—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—21no—
338C24—22no—
339C24—21no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24C22×D412yes—
351C24C2×#4114yes—
352C24C2×#4114yes—
353C24C2511yescharacteristic
354C2×D4—28no—
355C2×D4—216no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—44no—
359C2×D4—416no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—44no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C2×#4118yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4H(5,𝔽2)14yes—
456C4⋊C4—28no—
457C4⋊C4—416no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—88no—
466C22⋊C4—216no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C42—28no—
480C42—28no—
481C42—28no—
482C42—416no—
483C42—48no—
484C42—48no—
485C42—48no—
486C42#123116yes—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—28no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23#53214yes—
527C23C23×D411yescharacteristic
528C23C22×#4112yes—
529C23C22×#4112yes—
530C23C22×#4112yes—
531C23C2×H(5,𝔽2)12yes—
532C23#58512yes—
533C23#58512yes—
534C23#58511yescharacteristic
535C23#58511yescharacteristic
536D4—416no—
537D4—432no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—416no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
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—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—816no—
612C2×C4C2×#12314yes—
613C2×C4C2×#12314yes—
614C2×C4C2×#12314yes—
615C2×C4#52514yes—
616C2×C4#53514yes—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—48no—
624C22—48no—
625C22—42no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—48no—
630C22—44no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—84no—
635C22—84no—
636C22—82no—
637C22C22×#12311yescharacteristic
638C22C2×#52511yescharacteristic
639C22C2×#53211yescharacteristic
640C22C2×#53511yescharacteristic
641C22#344911yescharacteristic
642C22#344911yescharacteristic
643C22#344911yescharacteristic
644C22#347514yes—
645C22#347514yes—
646C22#348114yes—
647C22#349112yes—
648C22#349312yes—
649C22#349412yes—
650C22#349412yes—
651C22#351012yes—
652C22#351212yes—
653C22#356214yes—
654C4—24no—
655C4—24no—
656C4—24no—
657C4—24no—
658C4—24no—
659C4—44no—
660C4—48no—
661C4—42no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—84no—
669C2—44no—
670C2—84no—
671C2#3480911yescharacteristic
672C2#3481011yescharacteristic
673C2#3496511yescharacteristic
674C2#3511911yescharacteristic
675C2#3526211yescharacteristic
676C2#3534211yescharacteristic
677C2#3755911yescharacteristic
678C2#5934914yes—
679C2#5935614yes—
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
2. C2×#2283C2semidirect
3. #17058C2semidirect
4. #17057C2semidirect
7. #15122C2semidirect
8. #15120C2semidirect
9. #15017C2semidirect
10. #15016C2semidirect
11. #14838C2semidirect
12. #14249C2semidirect
13. #14231C2semidirect
14. #14173C2semidirect
15. #13959C2semidirect
16. #13952C2semidirect
17. #13753C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#22832
#170582
#170572
#163632
#163622
#151222
#151202
#150172
#150162
#148382
#142492
#142312
#141732
#139592
#139522
#137532