Finite Group Databasebeta

← #6480639

Subgroups — #6480639

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#33068C212yesmaximal, complemented
4#33066C212yesmaximal, complemented
5#30093C212yesmaximal, complemented
6#30077C212yesmaximal, complemented
7#29582C214yesmaximal, complemented
8#29320C211yesmaximal, characteristic, complemented
9#29319C211yesmaximal, characteristic, complemented
10#29130C211yesmaximal, characteristic, complemented
11#29130C211yesmaximal, characteristic, complemented
12#27679C212yesmaximal, complemented
13#27646C212yesmaximal, complemented
14#27612C212yesmaximal, complemented
15#24062C212yesmaximal
16#22086C214yesmaximal, complemented
17#20778C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×C4×#41—21no—
20C2×#482C2214yes—
21#2864C2212yes—
22#2864C2212yes—
23#2859C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2851—21no—
28#2851—21no—
29#2851—21no—
30#2851—21no—
31C4×C4⋊Q8C2211yescharacteristic
32C4×C4⋊Q8C2211yescharacteristic
33C4×#127C2211yescharacteristic
34C4×#127C2211yescharacteristic
35C4×#125—21no—
36C4×#125—21no—
37C4×#125—21no—
38C4×#125—21no—
39C4×#124—21no—
40C4×#124—21no—
41C4×#121—21no—
42C4×#121—21no—
43C4×#121C2212yes—
44C4×#121C2212yes—
45#2297—21no—
46#2296C2212yes—
47#2296C2212yes—
48#2293—21no—
49#2293C2212yes—
50#2285C2211yescharacteristic
51#2285C2211yescharacteristic
52#2285C2211yescharacteristic
53C42×D4—21no—
54C42×D4C2212yes—
55C2×C43C2211yescharacteristic
56#2101C2214yes—
57#2100C2212yes—
58#2100C2212yes—
59#2092C2214yes—
60#2085C2214yes—
61#2084C2212yes—
62#2083C2212yes—
63#2073C2214yes—
64#2058C2212yes—
65#2057C2212yes—
66#2041C2212yes—
67#2041C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1944C2214yes—
73#1943C2214yes—
74#1936C2214yes—
75#1936C2214yes—
76#1920C2214yes—
77#1920C2214yes—
78#1918C2214yes—
79#1918C2214yes—
80#1875C2214yes—
81#1864C2214yes—
82#1864C2214yes—
83#1862C2214yes—
84#1862C2214yes—
85#1854C2214yes—
86#1839C2212yes—
87#1831C2212yes—
88#1825C2214yes—
89#1817C2212yes—
90#1802C2212yes—
91#1802C2212yes—
92#1802C2212yes—
93#1802C2212yes—
94#1797C2212yes—
95#1767C2211yescharacteristic
96#1767C2211yescharacteristic
97#1749C2212yes—
98#1745C2212yes—
99C22×#41—21no—
100#532—28no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#125—21no—
104C2×#125—21no—
105C2×#125—21no—
106C2×#125—21no—
107C2×#124—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C4×#41—48no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C2×#117—22no—
125C2×#117C2312yes—
126C2×#117C2312yes—
127C22×C42—22no—
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#482—216no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2312yes—
148C2×#97C2312yes—
149#391—21no—
150#391—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390C2312yes—
156#390C2312yes—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#386—22no—
190#386—22no—
191#385—22no—
192#385—22no—
193#385—22no—
194#385—22no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#382—21no—
202#382—21no—
203#382—21no—
204#382—21no—
205#382—21no—
206#382C2312yes—
207#382C2312yes—
208#380—21no—
209#380—21no—
210#380—21no—
211#380—21no—
212#380—21no—
213#380C2312yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C43D411yescharacteristic
235C43D411yescharacteristic
236C43D411yescharacteristic
237C43D411yescharacteristic
238C43C2311yescharacteristic
239C43C2311yescharacteristic
240C43C2311yescharacteristic
241C43C2311yescharacteristic
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#336C2314yes—
251#336C2312yes—
252#336C2312yes—
253#336C2314yes—
254#336C2314yes—
255C2×#41—28no—
256C2×#41—44no—
257C2×#41—44no—
258C22×Q8—21no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4—21no—
262C22×D4C2412yes—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2411yescharacteristic
273C22×C8C2414yes—
274C22×C8C2414yes—
275C4⋊Q8—24no—
276#127—24no—
277#125—44no—
278#125—44no—
279#125—44no—
280#125—44no—
281#124—28no—
282#121—28no—
283#121—24no—
284#121#4118yes—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×D4—28no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—42no—
293C4×D4—42no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—48no—
299C4×D4—48no—
300#117—28no—
301#117—48no—
302#117—42no—
303#117—42no—
304#117—44no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42—41no—
358C2×C42—41no—
359C2×C42—48no—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388#107—28no—
389#107—44no—
390#106—28no—
391#106—44no—
392C4⋊C8—24no—
393C4⋊C8—24no—
394#102—28no—
395#102—28no—
396#102—44no—
397#102—44no—
398C22⋊C8—28no—
399#97—42no—
400#97—42no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419C24—22no—
420C24C22×D411yescharacteristic
421#41—432no—
422C2×Q8—24no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—28no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C2×#4114yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—28no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4114yes—
491C22×C4C2511yescharacteristic
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4C2×#4114yes—
533C22⋊C4—28no—
534C22⋊C4—28no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—44no—
573C42—44no—
574C42—44no—
575C42—44no—
576C42—44no—
577C42AGL(1,ℤ/8)14yes—
578C422−1+414yes—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—24no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23#52512yes—
593C23C23×D411yescharacteristic
594Q8—48no—
595D4—24no—
596D4—24no—
597D4—48no—
598D4—48no—
599D4—416no—
600D4—42no—
601D4—42no—
602C2×C4—28no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—21no—
618C2×C4—24no—
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—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—416no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—88no—
663C2×C4—88no—
664C2×C4#52512yes—
665C2×C4#53612yes—
666C2×C4#53612yes—
667C2×C4C2×AGL(1,ℤ/8)12yes—
668C2×C4C2×AGL(1,ℤ/8)12yes—
669C2×C4C2×AGL(1,ℤ/8)12yes—
670C2×C4#57512yes—
671C2×C4#57512yes—
672C2×C4#57512yes—
673C2×C4C22×#4112yes—
674C2×C4C2×2−1+412yes—
675C2×C4#58511yescharacteristic
676C2×C4#58511yescharacteristic
677C8—44no—
678C8—44no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22C2×#52511yescharacteristic
694C22C2×#53611yescharacteristic
695C22#344911yescharacteristic
696C22C22×AGL(1,ℤ/8)11yescharacteristic
697C22C2×#57511yescharacteristic
698C22C2×#57511yescharacteristic
699C22C2×#57511yescharacteristic
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—44no—
711C4—41no—
712C4—41no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4#323514yes—
720C4#351512yes—
721C4#351512yes—
722C2—42no—
723C2—84no—
724C2#3513811yescharacteristic
725C2#6054611yescharacteristic
726C2#6055811yescharacteristic
727C2C2×#323511yescharacteristic
728C2#6086211yescharacteristic
729C2#6095111yescharacteristic
730C2#6096711yescharacteristic
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. #36640C2semidirect
3. #33068C2semidirect
4. #33066C2semidirect
5. #30093C2semidirect
6. #30077C2semidirect
7. #29582C2semidirect
8. #29320C2semidirect
9. #29319C2semidirect
10. #29130C2semidirect
11. #29130C2semidirect
12. #27679C2semidirect
13. #27646C2semidirect
14. #27612C2semidirect
16. #22086C2semidirect
17. #20778C2semidirect

Maximal subgroups

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

subgroupindex
#366402
C4×#5322
#330682
#330662
#300932
#300772
#295822
#293202
#293192
#291302
#276792
#276462
#276122
#240622
#220862
#207782