Finite Group Databasebeta

← #6631920

Subgroups — #6631920

Order 512.

1443 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2#36638C211yesmaximal, characteristic
3#32852C212yesmaximal, complemented
4#32837C211yesmaximal, characteristic, complemented
5#32837C211yesmaximal, characteristic, complemented
6#32834C211yesmaximal, characteristic, complemented
7#32834C211yesmaximal, characteristic, complemented
8#32825C212yesmaximal, complemented
9#32789C212yesmaximal, complemented
10#32775C212yesmaximal
11#32744C212yesmaximal, complemented
12#32742C212yesmaximal
13#23841C212yesmaximal, complemented
14#23821C212yesmaximal, complemented
15#23783C212yesmaximal, complemented
16#23781C212yesmaximal, complemented
17#23763C212yesmaximal, complemented
18#23761C212yesmaximal
19#20778C211yesmaximal, characteristic, complemented
20C2×#533—21no—
21#2868C2211yescharacteristic
22#2868C2211yescharacteristic
23#2866—21no—
24#2865—21no—
25#2864C2211yescharacteristic
26#2863C2211yescharacteristic
27#2862—21no—
28#2861C2211yescharacteristic
29#2860C2211yescharacteristic
30#2859—21no—
31#2859C2212yes—
32#2857—21no—
33#2857—21no—
34#2857C2212yes—
35#2857C2212yes—
36#2855C2211yescharacteristic
37#2854C2211yescharacteristic
38#2853—21no—
39#2852—21no—
40#2851C2211yescharacteristic
41#2850C2211yescharacteristic
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8—21no—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#125C2211yescharacteristic
47C4×#126—21no—
48C4×#124C2211yescharacteristic
49C4×C22⋊Q8C2211yescharacteristic
50C4×#123—21no—
51C4×#121C2211yescharacteristic
52#2298—21no—
53#2295C2211yescharacteristic
54#2293C2211yescharacteristic
55#2290—21no—
56#2285—21no—
57#2285C2212yes—
58C2×C43C2211yescharacteristic
59#2101C2214yes—
60#2099C2212yes—
61#2099C2212yes—
62#2099C2212yes—
63#2099C2212yes—
64#2096C2212yes—
65#2096C2212yes—
66#2093C2212yes—
67#2093C2212yes—
68#2088C2214yes—
69#2087C2214yes—
70#2085C2214yes—
71#2084C2212yes—
72#2084C2212yes—
73#2083C2212yes—
74#2083C2212yes—
75#2077C2214yes—
76#2076C2214yes—
77#2072C2212yes—
78#2072C2212yes—
79#2071C2212yes—
80#2071C2212yes—
81#2068C2214yes—
82#2065C2214yes—
83#2059C2214yes—
84#2058C2212yes—
85#2058C2212yes—
86#2057C2212yes—
87#2057C2212yes—
88#2042C2214yes—
89#2036C2214yes—
90#2034C2214yes—
91#2024C2212yes—
92#2024C2212yes—
93#2023C2212yes—
94#2023C2212yes—
95#1839C2211yescharacteristic
96#1839C2211yescharacteristic
97#1831C2212yes—
98#1817C2212yes—
99#1797C2212yes—
100#1776C2214yes—
101#1776C2214yes—
102#1767C2212yes—
103#1763C2214yes—
104#1762C2214yes—
105#1749C2212yes—
106#1745C2212yes—
107#533—48no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2312yes—
114C2×#126—21no—
115C2×#125—22no—
116C2×#125C2311yescharacteristic
117C2×#124C2311yescharacteristic
118C2×#123—21no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8C2311yescharacteristic
122C2×#121C2311yescharacteristic
123C2×C4×Q8—22no—
124C2×C4×Q8—22no—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4—21no—
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C2×#117—21no—
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2314yes—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391C2311yescharacteristic
153#391C2311yescharacteristic
154#391C2311yescharacteristic
155#391C2311yescharacteristic
156#390C2311yescharacteristic
157#390C2311yescharacteristic
158#390C2311yescharacteristic
159#390C2311yescharacteristic
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389C2311yescharacteristic
167#389C2311yescharacteristic
168#389C2314yes—
169#389C2314yes—
170#389C2311yescharacteristic
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#389C2311yescharacteristic
174#389C2311yescharacteristic
175#389C2311yescharacteristic
176#388—21no—
177#388—21no—
178#388—21no—
179#388—21no—
180#388—21no—
181#388—21no—
182#388—21no—
183#388—21no—
184#388—21no—
185#388—21no—
186#388—21no—
187#388—21no—
188#388—21no—
189#388—21no—
190#388—21no—
191#388—21no—
192#387—21no—
193#387—21no—
194#387—21no—
195#387—21no—
196#387—21no—
197#387—21no—
198#387—21no—
199#387—21no—
200#387—21no—
201#387—21no—
202#387—21no—
203#387—21no—
204#387—21no—
205#387—21no—
206#387—21no—
207#387—21no—
208#386—21no—
209#386—21no—
210#386—21no—
211#386—21no—
212#385—21no—
213#385—21no—
214#385—21no—
215#385—21no—
216#384—21no—
217#384—21no—
218#384—21no—
219#384—21no—
220#384C2312yes—
221#384C2312yes—
222#384C2312yes—
223#384C2312yes—
224#383—21no—
225#382—22no—
226#382—22no—
227#382C2311yescharacteristic
228#382C2311yescharacteristic
229#381—21no—
230#380—22no—
231#380—22no—
232#380—22no—
233#380C2311yescharacteristic
234#380C2314yes—
235#380C2311yescharacteristic
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2311yescharacteristic
245C4×C4⋊C4C2314yes—
246C4×C22⋊C4—21no—
247C4×C22⋊C4—21no—
248C4×C22⋊C4—21no—
249C4×C22⋊C4—21no—
250C4×C22⋊C4—21no—
251#376—21no—
252C43D411yescharacteristic
253C43D411yescharacteristic
254C43D411yescharacteristic
255C43D411yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2314yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—22no—
276C22×Q8—22no—
277C22×Q8C2411yescharacteristic
278C22×D4—21no—
279C22×D4C2411yescharacteristic
280C23×C4—21no—
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2411yescharacteristic
288C22×C8C2414yes—
289C22×C8C2414yes—
290C4⋊Q8—44no—
291C4⋊Q8—44no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296#126—44no—
297#125—42no—
298#125—48no—
299#124—42no—
300#123—44no—
301C22⋊Q8—42no—
302C22⋊Q8—48no—
303C22⋊Q8—48no—
304#121—42no—
305C4×Q8—24no—
306C4×Q8—48no—
307C4×Q8—48no—
308C4×D4—24no—
309C4×D4—44no—
310#117—44no—
311#117—48no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2414yes—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—42no—
368C2×C42—42no—
369C2×C42—42no—
370C2×C42—42no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399#107—44no—
400#107—44no—
401#106—44no—
402#106—44no—
403C4⋊C8—44no—
404#103—44no—
405#103—44no—
406#102—44no—
407#102—44no—
408C22⋊C8—44no—
409#97—28no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—24no—
436#95—21no—
437#95—21no—
438C24—21no—
439C24C22×D411yescharacteristic
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4#13714yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C2511yescharacteristic
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—48no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42—41no—
584C42—42no—
585C42—48no—
586C42—42no—
587C42—42no—
588C42AGL(1,ℤ/8)14yes—
589C422−1+414yes—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23C2×#13712yes—
604C23C23×D411yescharacteristic
605Q8—44no—
606Q8—88no—
607Q8—88no—
608D4—44no—
609D4—84no—
610C2×C4—28no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—84no—
675C2×C4#53612yes—
676C2×C4#53612yes—
677C2×C4C2×AGL(1,ℤ/8)12yes—
678C2×C4C2×AGL(1,ℤ/8)12yes—
679C2×C4C2×AGL(1,ℤ/8)12yes—
680C2×C4C2×#13712yes—
681C2×C4C2×#13712yes—
682C2×C4#57512yes—
683C2×C4#57512yes—
684C2×C4C2×2−1+412yes—
685C2×C4#58512yes—
686C2×C4#58512yes—
687C8—44no—
688C8—44no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22—81no—
700C22—81no—
701C22—81no—
702C22—81no—
703C22C2×#53611yescharacteristic
704C22#344911yescharacteristic
705C22#344911yescharacteristic
706C22C22×AGL(1,ℤ/8)11yescharacteristic
707C22C22×#13711yescharacteristic
708C22C2×#57511yescharacteristic
709C22C2×#57511yescharacteristic
710C4—22no—
711C4—22no—
712C4—22no—
713C4—22no—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—81no—
724C4—81no—
725C4—81no—
726C4—81no—
727C4—84no—
728C4—84no—
729C4—81no—
730C4—81no—
731C4—81no—
732C4#353114yes—
733C2—42no—
734C2—161no—
735C2#3669911yescharacteristic
736C2#6086011yescharacteristic
737C2#6086611yescharacteristic
738C2#6096011yescharacteristic
739C2#6096311yescharacteristic
740C2#6096511yescharacteristic
741C2#6096511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #32852C2semidirect
4. #32837C2semidirect
5. #32837C2semidirect
6. #32834C2semidirect
7. #32834C2semidirect
8. #32825C2semidirect
9. #32789C2semidirect
11. #32744C2semidirect
13. #23841C2semidirect
14. #23821C2semidirect
15. #23783C2semidirect
16. #23781C2semidirect
17. #23763C2semidirect
19. #20778C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#366382
#328522
#328372
#328342
#328252
#327892
#327752
#327442
#327422
#238412
#238212
#237832
#237812
#237632
#237612
#207782