Finite Group Databasebeta

← #6485947

Subgroups — #6485947

Order 512.

1736 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#33070C214yesmaximal, complemented
4#30171C214yesmaximal, complemented
5#29397C214yesmaximal, complemented
6#29207C211yesmaximal, characteristic, complemented
7#29143C211yesmaximal, characteristic, complemented
8#28718C211yesmaximal, characteristic, complemented
9#28688C211yesmaximal, characteristic, complemented
10#27877C211yesmaximal, characteristic, complemented
11#27875C211yesmaximal, characteristic, complemented
12#27633C214yesmaximal, complemented
13#25673C211yesmaximal, characteristic
14#25671C211yesmaximal, characteristic
15#22248C211yesmaximal, characteristic, complemented
16#22246C211yesmaximal, characteristic, complemented
17#22220C211yesmaximal, characteristic, complemented
18#22219C211yesmaximal, characteristic, complemented
19#20777C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×C4×#41—21no—
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#486C2211yescharacteristic
26#2864C2212yes—
27#2863—21no—
28#2863—21no—
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2855—21no—
34#2855—21no—
35#2851C2212yes—
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2211yescharacteristic
40C4×#127C2211yescharacteristic
41C4×#127C2211yescharacteristic
42C4×#125C2212yes—
43C4×#124—21no—
44C4×#124—21no—
45C4×C22⋊Q8—21no—
46C4×C22⋊Q8—21no—
47C4×#121—21no—
48C4×#121—21no—
49C4×#121C2212yes—
50C4×#121C2212yes—
51#2297—21no—
52#2295—21no—
53#2295—21no—
54#2293—21no—
55#2293C2212yes—
56#2285C2211yescharacteristic
57#2283C2212yes—
58C42×D4—21no—
59C42×D4C2212yes—
60C2×C43C2211yescharacteristic
61#2085C2214yes—
62#2077C2214yes—
63#2075C2214yes—
64#2073C2214yes—
65#2059C2214yes—
66#2034C2214yes—
67#2033C2214yes—
68#2025C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1949C2214yes—
78#1949C2214yes—
79#1926C2214yes—
80#1925C2214yes—
81#1921C2214yes—
82#1921C2214yes—
83#1920C2214yes—
84#1920C2214yes—
85#1919C2214yes—
86#1918C2214yes—
87#1864C2214yes—
88#1864C2214yes—
89#1862C2214yes—
90#1862C2214yes—
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1839C2212yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1831C2212yes—
97#1817C2212yes—
98#1804C2214yes—
99#1804C2214yes—
100#1797C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105#1744C2211yescharacteristic
106#1744C2211yescharacteristic
107C22×#41—21no—
108#532—24no—
109#532—24no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#125C2312yes—
115C2×#124—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×#121—21no—
119C2×#121C2312yes—
120C4×#41—48no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×D4—22no—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4C2312yes—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×#117—21no—
136C2×#117—22no—
137C22×C42—22no—
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42D411yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#500—24no—
146#498—24no—
147#486—24no—
148#486—24no—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165#391—21no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#386—22no—
210#386—22no—
211#385—22no—
212#385—22no—
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#383—22no—
218#382—21no—
219#382C2312yes—
220#382C2312yes—
221#382C2312yes—
222#381—22no—
223#380—21no—
224#380—21no—
225#380—21no—
226#380C2312yes—
227#380C2312yes—
228#380C2312yes—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4C2312yes—
237C4×C4⋊C4C2312yes—
238C4×C4⋊C4C2312yes—
239C4×C4⋊C4C2312yes—
240C4×C4⋊C4C2312yes—
241C4×C22⋊C4—22no—
242C4×C22⋊C4—22no—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—22no—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248#376—22no—
249C43D411yescharacteristic
250C43D411yescharacteristic
251C43D411yescharacteristic
252C43D411yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C2×#41—28no—
274C2×#41—44no—
275C2×#41—44no—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×D4—22no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C23×C4—22no—
284C23×C4—22no—
285C23×C4—22no—
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2×D411yescharacteristic
292C23×C4C2411yescharacteristic
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8#4114yes—
303#127—22no—
304#127#4114yes—
305#125—44no—
306#124—28no—
307C22⋊Q8—44no—
308C22⋊Q8—44no—
309#121—28no—
310#121—24no—
311#121—24no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×Q8—44no—
316C4×D4—28no—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—28no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4—44no—
323C4×D4—44no—
324C4×D4—48no—
325C4×D4—48no—
326C4×D4—44no—
327#117—48no—
328#117—44no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—24no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—42no—
377C2×C42—42no—
378C2×C42—42no—
379C2×C42—48no—
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×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408#107—28no—
409#107—44no—
410#106—28no—
411#106—44no—
412C4⋊C8—24no—
413C4⋊C8—24no—
414#102—28no—
415#102—28no—
416#102—44no—
417#102—44no—
418C22⋊C8—42no—
419C22⋊C8—42no—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8—22no—
423C4×C8—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444C24—22no—
445C24C22×D411yescharacteristic
446#41—432no—
447C2×Q8—24no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×D4—28no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4C2×#4114yes—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—28no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4AGL(1,ℤ/8)14yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C2511yescharacteristic
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—44no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4C2×#4114yes—
562C22⋊C4—28no—
563C22⋊C4—28no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—21no—
589C42—21no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—41no—
594C42—41no—
595C42—41no—
596C42—41no—
597C42—41no—
598C42—41no—
599C42—44no—
600C42—44no—
601C42—44no—
602C42—44no—
603C42—44no—
604C42#13514yes—
605C42C2×#4112yes—
606C42C2×#4112yes—
607C422−1+414yes—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—24no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23C2×AGL(1,ℤ/8)12yes—
622C23C23×D411yescharacteristic
623Q8—48no—
624Q8—84no—
625Q8—84no—
626D4—28no—
627D4—416no—
628D4—48no—
629D4—48no—
630D4—44no—
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—28no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—24no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—416no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—88no—
690C2×C4—88no—
691C2×C4#52512yes—
692C2×C4#52512yes—
693C2×C4#53612yes—
694C2×C4#53612yes—
695C2×C4C2×#13512yes—
696C2×C4C2×#13512yes—
697C2×C4C2×#13512yes—
698C2×C4C2×AGL(1,ℤ/8)12yes—
699C2×C4C2×AGL(1,ℤ/8)12yes—
700C2×C4#57512yes—
701C2×C4#57512yes—
702C2×C4C22×#4111yescharacteristic
703C2×C4C22×#4111yescharacteristic
704C2×C4C2×2−1+412yes—
705C2×C4#58512yes—
706C8—42no—
707C8—42no—
708C8—42no—
709C8—42no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22C2×#52511yescharacteristic
725C22C2×#53611yescharacteristic
726C22#344911yescharacteristic
727C22C22×#13511yescharacteristic
728C22C22×AGL(1,ℤ/8)11yescharacteristic
729C22C2×#57511yescharacteristic
730C22C2×#57511yescharacteristic
731C4—22no—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—44no—
741C4—41no—
742C4—41no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4#310214yes—
750C4#311414yes—
751C4#351512yes—
752C4#351812yes—
753C2—42no—
754C2—84no—
755C2#3520011yescharacteristic
756C2C2×#310211yescharacteristic
757C2C2×#311411yescharacteristic
758C2#6084611yescharacteristic
759C2#6086411yescharacteristic
760C2#6094611yescharacteristic
761C2#6097211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #36635C2semidirect
3. #33070C2semidirect
4. #30171C2semidirect
5. #29397C2semidirect
6. #29207C2semidirect
7. #29143C2semidirect
8. #28718C2semidirect
9. #28688C2semidirect
10. #27877C2semidirect
11. #27875C2semidirect
12. #27633C2semidirect
15. #22248C2semidirect
16. #22246C2semidirect
17. #22220C2semidirect
18. #22219C2semidirect
19. #20777C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C4×#5322
#330702
#301712
#293972
#292072
#291432
#287182
#286882
#278772
#278752
#276332
#256732
#256712
#222482
#222462
#222202
#222192
#207772