Finite Group Databasebeta

← #6640554

Subgroups — #6640554

Order 512.

1763 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35240C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic
3#32982C212yesmaximal, complemented
4#32269C212yesmaximal, complemented
5#32260C212yesmaximal, complemented
6#32037C212yesmaximal, complemented
7#32029C212yesmaximal, complemented
8#32022C214yesmaximal, complemented
9#32008C214yesmaximal, complemented
10#29894C212yesmaximal, complemented
11#27616C212yesmaximal, complemented
12#24559C212yesmaximal, complemented
13#24551C212yesmaximal, complemented
14#24063C212yesmaximal, complemented
15#21740C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#483C2212yes—
20C2×#483C2212yes—
21C2×#407C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2848C2212yes—
24#2848C2212yes—
25#2633—21no—
26#2633—21no—
27#2547C2211yescharacteristic
28#2547C2211yescharacteristic
29#2542C2211yescharacteristic
30#2542C2211yescharacteristic
31#2536C2212yes—
32#2536C2212yes—
33#2502—21no—
34#2487—21no—
35#2487—21no—
36#2487—21no—
37#2487—21no—
38#2487C2212yes—
39#2487C2212yes—
40#2487C2212yes—
41#2487C2212yes—
42#2482—21no—
43#2482—21no—
44#2435—21no—
45C2×#400—21no—
46C2×#399—21no—
47#2326—21no—
48#2326—21no—
49C2×#389C2212yes—
50C2×#386—21no—
51C2×#385—21no—
52#2283C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×#380C2212yes—
55#2101C2214yescomplemented
56#2085C2214yescomplemented
57#2077C2214yescomplemented
58#2077C2214yescomplemented
59#2077C2214yescomplemented
60#2075C2214yescomplemented
61#2058C2214yescomplemented
62#2043C2214yescomplemented
63#2036C2214yescomplemented
64#2036C2214yescomplemented
65#2035C2214yescomplemented
66#2035C2214yescomplemented
67#1921C2214yescomplemented
68#1920C2214yescomplemented
69#1894C2214yescomplemented
70#1892C2214yescomplemented
71#1872C2212yes—
72#1872C2212yes—
73#1872C2214yescomplemented
74#1872C2214yescomplemented
75#1870C2212yes—
76#1870C2212yes—
77#1869C2212yes—
78#1869C2212yes—
79#1865C2214yescomplemented
80#1865C2214yescomplemented
81#1853C2212yes—
82#1853C2214yescomplemented
83#1853C2214yescomplemented
84#1852C2212yes—
85#1823C2212yes—
86#1823C2212yes—
87#1817C2212yes—
88#1809C2214yescomplemented
89#1807C2214yescomplemented
90#1779C2212yes—
91C23×D4D411yescharacteristic
92C24×C4—21no—
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C22×M16C2312yes—
96C23×C8C2312yes—
97#533—42no—
98#533—42no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125—21no—
102C2×#124—21no—
103C2×#124—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—42no—
113C2×C4×D4—42no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—42no—
128C22×C4⋊C4—42no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—22no—
136C22×C22⋊C4D411yescharacteristic
137C22×C22⋊C4D411yescharacteristic
138C22×C42C2311yescharacteristic
139#483—44no—
140#483—44no—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149#407—42no—
150C2×C22⋊C8—42no—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157#400—22no—
158#400—22no—
159#400—22no—
160#400—22no—
161#399—21no—
162#399—21no—
163#399—21no—
164#399—21no—
165#399—22no—
166#399—22no—
167#399—22no—
168#399—22no—
169#398—22no—
170#398—22no—
171#397—21no—
172#397—21no—
173#397—21no—
174#397—21no—
175#394—22no—
176#394—22no—
177#391—21no—
178#391—21no—
179#390—21no—
180#390—21no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#384—21no—
204#384—21no—
205#383—21no—
206#382C2312yes—
207#381—21no—
208#380—22no—
209#380—22no—
210#380—21no—
211#380—21no—
212#380C2312yes—
213#380C2314yes—
214#380C2312yes—
215#380C2314yes—
216#380C2312yes—
217#380C2312yes—
218#380C2312yes—
219#380C2312yes—
220#380C2312yes—
221#379—22no—
222#379—22no—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—21no—
226#376—21no—
227C2×#95—21no—
228C2×#95—22no—
229C2×#95—22no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#336C2314yes—
239#336C2314yes—
240C25C2×D411yescharacteristic
241C25C2×D411yescharacteristic
242C22×Q8—21no—
243C22×Q8C2412yes—
244C22×D4—24no—
245C22×D4—42no—
246C22×D4—42no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2×D411yescharacteristic
249C22×D4C2×D411yescharacteristic
250C22×D4C2×D411yescharacteristic
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—44no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4—84no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C23×C4C2411yescharacteristic
275C23×C4C2411yescharacteristic
276C2×M16—44no—
277C2×M16—42no—
278C22×C8—44no—
279C22×C8—42no—
280C22×C8C2414yes—
281C22×C8C2414yes—
282C4⋊Q8—41no—
283C4⋊Q8—41no—
284C4⋊Q8—41no—
285C4⋊Q8—41no—
286#125—44no—
287#124—44no—
288#124—44no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—44no—
292C22⋊Q8—44no—
293C22⋊Q8—44no—
294C22⋊Q8—44no—
295C22⋊Q8—44no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—44no—
299C4×D4—42no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—42no—
303C4×D4—41no—
304C4×D4—41no—
305C4×D4—41no—
306C4×D4—41no—
307C4×D4—41no—
308C4×D4—41no—
309C4×D4—41no—
310C4×D4—41no—
311#117—44no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—84no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—42no—
363C2×C22⋊C4—42no—
364C2×C22⋊C4—44no—
365C2×C22⋊C4—84no—
366C2×C22⋊C4C2×D411yescharacteristic
367C2×C22⋊C4C2×D411yescharacteristic
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—44no—
384#106—44no—
385#106—44no—
386#106—44no—
387#103—44no—
388#103—44no—
389#103—44no—
390#103—44no—
391C22⋊C8—24no—
392C22⋊C8—24no—
393C22⋊C8—42no—
394C22⋊C8—44no—
395C22⋊C8—44no—
396C22⋊C8—42no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—21no—
407#95—21no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—24no—
417C24—22no—
418C24—24no—
419C24—42no—
420C24—42no—
421C24—41no—
422C24C22×D411yescharacteristic
423C24C22×D411yescharacteristic
424C24C22×D411yescharacteristic
425C24C22×D411yescharacteristic
426C24C22×D411yescharacteristic
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—41no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—416no—
446C2×D4#13714yes—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—41no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—84no—
494C22×C4—84no—
495C22×C4—84no—
496C22×C4—84no—
497C22×C4—816no—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4H(5,𝔽2)12yes—
508C22×C4H(5,𝔽2)12yes—
509C22×C4C2511yescharacteristic
510M16—84no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—84no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—84no—
562C22⋊C4—84no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C42—22no—
568C42—22no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—44no—
574C42—42no—
575C42—42no—
576C23—22no—
577C23—21no—
578C23—21no—
579C23—28no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—21no—
593C23—22no—
594C23—21no—
595C23—21no—
596C23—22no—
597C23—48no—
598C23—48no—
599C23—44no—
600C23—48no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—82no—
605C23—84no—
606C23#53412yes—
607C23C2×#13712yes—
608C23C2×#13712yes—
609C23#57512yes—
610C23#57714yes—
611C23C23×D411yescharacteristic
612Q8—44no—
613Q8—44no—
614Q8—84no—
615D4—44no—
616D4—44no—
617D4—44no—
618D4—44no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
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—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4—88no—
674C2×C4—88no—
675C2×C4—88no—
676C2×C4—88no—
677C2×C4#53412yes—
678C2×C4C2×#13712yes—
679C2×C4C2×H(5,𝔽2)11yescharacteristic
680C2×C4C2×H(5,𝔽2)11yescharacteristic
681C2×C4#58514yes—
682C8—44no—
683C8—44no—
684C22—24no—
685C22—24no—
686C22—22no—
687C22—22no—
688C22—24no—
689C22—24no—
690C22—24no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—416no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—42no—
705C22—42no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—88no—
710C22#320014yes—
711C22C2×#53411yescharacteristic
712C22#348512yes—
713C22C22×#13711yescharacteristic
714C22C2×#57511yescharacteristic
715C22C2×#57712yes—
716C4—22no—
717C4—22no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—82no—
726C4—82no—
727C4—84no—
728C4—84no—
729C4—82no—
730C4—82no—
731C4#353312yes—
732C4#353312yes—
733C2—24no—
734C2—42no—
735C2—42no—
736C2—42no—
737C2—81no—
738C2#3695511yescharacteristic
739C2C2×#320011yescharacteristic
740C2#6075111yescharacteristic
741C2#6173512yes—
742C2#6173612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32982C2semidirect
4. #32269C2semidirect
5. #32260C2semidirect
6. #32037C2semidirect
7. #32029C2semidirect
8. #32022C2semidirect
9. #32008C2semidirect
10. #29894C2semidirect
11. #27616C2semidirect
12. #24559C2semidirect
13. #24551C2semidirect
14. #24063C2semidirect
55. #2101C22semidirect
56. #2085C22semidirect
57. #2077C22semidirect
58. #2077C22semidirect
59. #2077C22semidirect
60. #2075C22semidirect
61. #2058C22semidirect
62. #2043C22semidirect
63. #2036C22semidirect
64. #2036C22semidirect
65. #2035C22semidirect
66. #2035C22semidirect
67. #1921C22semidirect
68. #1920C22semidirect
69. #1894C22semidirect
70. #1892C22semidirect
73. #1872C22semidirect
74. #1872C22semidirect
79. #1865C22semidirect
80. #1865C22semidirect
82. #1853C22semidirect
83. #1853C22semidirect
88. #1809C22semidirect
89. #1807C22semidirect

Maximal subgroups

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

subgroupindex
#352402
C2×#24872
#329822
#322692
#322602
#320372
#320292
#320222
#320082
#298942
#276162
#245592
#245512
#240632
#217402