Finite Group Databasebeta

← #6657543

Subgroups — #6657543

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#34971C211yesmaximal, characteristic
3#32957C212yesmaximal, complemented
4#32586C214yesmaximal, complemented
5#32469C212yesmaximal, complemented
6#32258C212yesmaximal, complemented
7#32006C214yesmaximal, complemented
8#31683C212yesmaximal, complemented
9#31262C212yesmaximal, complemented
10#29942C212yesmaximal, complemented
11#27630C212yesmaximal, complemented
12#24837C212yesmaximal, complemented
13#24711C212yesmaximal, complemented
14#24302C212yesmaximal, complemented
15#21851C211yesmaximal, characteristic
16C2×#536—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#477C2212yes—
21C2×#477C2212yes—
22C2×#432C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2621—21no—
27#2621—21no—
28#2599—21no—
29#2599—21no—
30#2581C2212yes—
31#2547—21no—
32#2547C2212yes—
33#2546—21no—
34#2546C2212yes—
35#2536C2212yes—
36#2486—21no—
37#2486—21no—
38#2486C2211yescharacteristic
39#2486C2212yes—
40#2486C2211yescharacteristic
41#2486C2212yes—
42#2484—21no—
43#2409C2211yescharacteristic
44#2409C2211yescharacteristic
45#2406—21no—
46#2399—21no—
47#2398—21no—
48#2310—21no—
49C4×C22⋊Q8C2212yes—
50C4×C22⋊Q8C2212yes—
51C4×#123—21no—
52C4×#123—21no—
53#2285C2211yescharacteristic
54#2283C2211yescharacteristic
55#2101C2214yes—
56#2087C2214yes—
57#2085C2214yes—
58#2077C2214yes—
59#2077C2214yes—
60#2075C2214yes—
61#2075C2214yes—
62#2068C2214yes—
63#2057C2214yes—
64#2036C2214yes—
65#2035C2214yes—
66#2024C2214yes—
67#1987C2212yes—
68#1987C2212yes—
69#1985C2212yes—
70#1955C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1944C2214yes—
74#1935C2214yes—
75#1927C2214yes—
76#1918C2214yes—
77#1892C2214yes—
78#1872C2214yes—
79#1869C2212yes—
80#1869C2212yes—
81#1863C2214yes—
82#1863C2214yes—
83#1852C2214yes—
84#1852C2214yes—
85#1808C2214yes—
86#1803C2212yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2212yes—
90#1787C2214yes—
91C22×#41D411yescharacteristic
92C23×Q8—21no—
93C23×Q8C2311yescharacteristic
94C23×C8C2312yes—
95#536—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×#124—22no—
104C2×#123—21no—
105C2×#123—21no—
106C2×#123—22no—
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C4×#41—42no—
113C4×#41—42no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—42no—
117C2×C4×Q8—42no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117D411yescharacteristic
130C2×#117D411yescharacteristic
131C2×#117D411yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#477—44no—
141#477—44no—
142#432—42no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8—42no—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2312yes—
158C2×#97C2312yes—
159#400—22no—
160#398—21no—
161#398—21no—
162#398—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#397—21no—
168#397—21no—
169#397—22no—
170#397—22no—
171#397—22no—
172#397—22no—
173#395—22no—
174#395—21no—
175#395—21no—
176#394—22no—
177#393—21no—
178#393—21no—
179#391—22no—
180#391—22no—
181#391—22no—
182#391—22no—
183#391C2312yes—
184#391C2312yes—
185#389—22no—
186#389—22no—
187#389C2312yes—
188#389C2314yes—
189#389C2312yes—
190#389C2314yes—
191#388—22no—
192#388—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#386—21no—
198#386—21no—
199#385—22no—
200#385—22no—
201#385—21no—
202#385—21no—
203#384C2312yes—
204#383—21no—
205#382—21no—
206#382C2312yes—
207#381—21no—
208#381—22no—
209#381—22no—
210#381—21no—
211#381—21no—
212#380—22no—
213#380—22no—
214#380—21no—
215#380C2312yes—
216#380C2314yes—
217#380C2311yescharacteristic
218#380C2314yes—
219#380C2311yescharacteristic
220#380C2312yes—
221#380C2312yes—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—21no—
230#376—21no—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C2×#41—24no—
242C2×#41—42no—
243C2×#41—42no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—24no—
248C22×Q8—41no—
249C22×Q8—41no—
250C22×Q8—42no—
251C22×Q8—44no—
252C22×Q8—44no—
253C22×Q8—44no—
254C22×Q8C2×D411yescharacteristic
255C22×Q8C2412yes—
256C22×D4C2×D411yescharacteristic
257C22×D4C2×D411yescharacteristic
258C22×D4C2×D411yescharacteristic
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C22×C8—44no—
272C22×C8—42no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275C4⋊Q8—24no—
276C4⋊Q8—24no—
277C4⋊Q8—42no—
278C4⋊Q8—42no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—48no—
282#124—48no—
283#123—44no—
284#123—48no—
285#123—48no—
286#123—44no—
287C22⋊Q8—24no—
288C22⋊Q8—24no—
289C22⋊Q8—42no—
290C22⋊Q8—42no—
291C22⋊Q8—48no—
292C22⋊Q8—48no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—42no—
296C4×Q8—41no—
297C4×Q8—41no—
298C4×Q8—42no—
299C4×Q8—42no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×D4—42no—
303C4×D4—42no—
304C4×D4—42no—
305C4×D4—41no—
306C4×D4—41no—
307C4×D4—41no—
308C4×D4—41no—
309C4×D4—41no—
310C4×D4—41no—
311#117—42no—
312#117—42no—
313#117—42no—
314#117—44no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—41no—
330C2×C4⋊C4—41no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2414yes—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4C2×D411yescharacteristic
356C2×C22⋊C4C2×D411yescharacteristic
357C2×C22⋊C4C2×D411yescharacteristic
358C2×C22⋊C4C2×D411yescharacteristic
359C2×C22⋊C4C2×D411yescharacteristic
360C2×C22⋊C4C2×D411yescharacteristic
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—44no—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382#107—44no—
383#107—44no—
384#107—44no—
385#106—44no—
386C4⋊C8—24no—
387C4⋊C8—24no—
388C4⋊C8—42no—
389#103—28no—
390#103—44no—
391#103—44no—
392#103—44no—
393C22⋊C8—44no—
394C22⋊C8—44no—
395C22⋊C8—42no—
396#97—42no—
397#95—21no—
398#95—21no—
399#95—22no—
400#95—22no—
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—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416C24C22×D411yescharacteristic
417C24C22×D411yescharacteristic
418C24C22×D411yescharacteristic
419#41—416no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—41no—
427C2×Q8—41no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—48no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—816no—
440C2×Q82−1+414yes—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4AGL(1,ℤ/8)14yes—
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—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—44no—
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×C4C22×D411yescharacteristic
502C22×C4C2511yescharacteristic
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—84no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—42no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—42no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—44no—
568C42—44no—
569C42—44no—
570C42—44no—
571C42H(5,𝔽2)12yes—
572C42H(5,𝔽2)12yes—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23C2×AGL(1,ℤ/8)12yes—
586C23C2×AGL(1,ℤ/8)12yes—
587C23#57512yes—
588C23C23×D411yescharacteristic
589Q8—24no—
590Q8—24no—
591Q8—44no—
592Q8—48no—
593Q8—42no—
594Q8—42no—
595Q8—88no—
596Q8—88no—
597D4—44no—
598D4—44no—
599D4—44no—
600C2×C4—28no—
601C2×C4—21no—
602C2×C4—21no—
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—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—84no—
657C2×C4—88no—
658C2×C4#53412yes—
659C2×C4#53412yes—
660C2×C4C2×AGL(1,ℤ/8)12yes—
661C2×C4#57814yes—
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×2−1+412yes—
665C2×C4#58512yes—
666C8—44no—
667C8—44no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22C2×#53411yescharacteristic
684C22#348912yes—
685C22C22×AGL(1,ℤ/8)11yescharacteristic
686C22C2×#57511yescharacteristic
687C22C2×#57812yes—
688C4—22no—
689C4—22no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—44no—
699C4—82no—
700C4—84no—
701C4—82no—
702C4—82no—
703C4—84no—
704C4#320214yes—
705C4#353811yescharacteristic
706C4#354011yescharacteristic
707C4#354112yes—
708C2—42no—
709C2—42no—
710C2—42no—
711C2#3737411yescharacteristic
712C2C2×#320211yescharacteristic
713C2#6075011yescharacteristic
714C2#6213012yes—
715C2#6213212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32957C2semidirect
4. #32586C2semidirect
5. #32469C2semidirect
6. #32258C2semidirect
7. #32006C2semidirect
8. #31683C2semidirect
9. #31262C2semidirect
10. #29942C2semidirect
11. #27630C2semidirect
12. #24837C2semidirect
13. #24711C2semidirect
14. #24302C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#349712
#329572
#325862
#324692
#322582
#320062
#316832
#312622
#299422
#276302
#248372
#247112
#243022
#218512