Finite Group Databasebeta

← #6593446

Subgroups — #6593446

Order 512.

1859 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2C4×#550C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#32907C212yesmaximal, complemented
5#29139C211yesmaximal, characteristic, complemented
6#28252C211yesmaximal, characteristic, complemented
7#27627C212yesmaximal, complemented
8#27233C214yesmaximal, complemented
9#26731C214yesmaximal, complemented
10#24972C212yesmaximal, complemented
11#24651C212yesmaximal, complemented
12#24242C212yesmaximal
13#23095C212yesmaximal, complemented
14#22795C211yesmaximal, characteristic, complemented
15#22771C212yesmaximal, complemented
16C4×#462C211yesmaximal, characteristic, complemented
17#20737C212yesmaximal, complemented
18C2×#550C2211yescharacteristic
19C2×C4×#41C2212yes—
20C2×#475C2212yes—
21C2×#462C2211yescharacteristic
22C2×#460C2212yes—
23C2×#461C2212yes—
24C2×#452C2214yes—
25C2×C4×Q16C2212yes—
26C2×#431C2212yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2571—21no—
30#2563—21no—
31#2538C2212yes—
32#2497—22no—
33#2492C2214yes—
34#2439—22no—
35#2369—21no—
36#2366—21no—
37#2366C2211yescharacteristic
38#2363—22no—
39#2363C2212yes—
40#2355C2212yes—
41#2354—21no—
42#2354—21no—
43#2354C2211yescharacteristic
44Q8×C4⋊C4C2211yescharacteristic
45#2341C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127—21no—
48C4×#127C2211yescharacteristic
49C4×#121—22no—
50C4×#121C2212yes—
51#2285C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4—21no—
54C42×D4C2211yescharacteristic
55#2077C2214yes—
56#2055C2214yes—
57#2035C2214yes—
58#2033C2214yes—
59#2023C2214yes—
60#2000C2214yes—
61#1988C2212yes—
62#1981C2211yescharacteristic
63#1968C2214yes—
64#1957C2212yes—
65#1954C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1947C2211yescharacteristic
69#1933C2214yes—
70#1928C2212yes—
71#1926C2214yes—
72#1923C2214yes—
73#1921C2214yes—
74#1878C2212yes—
75#1878C2214yes—
76#1877C2214yes—
77#1873C2214yes—
78#1842C2211yescharacteristic
79#1834C2212yes—
80#1825C2214yes—
81#1803C2211yescharacteristic
82#1803C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1797C2212yes—
89#1788C2214yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2214yes—
93C4×#103C2212yes—
94C4×#102C2214yes—
95C22×#41C2312yes—
96C22×Q16C2312yes—
97C22×SD16C2314yes—
98#550—24no—
99#550—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127—21no—
105C2×#127C2311yescharacteristic
106C2×C22⋊Q8C2314yes—
107C2×#121—22no—
108C2×#121C2312yes—
109C4×#41—28no—
110C4×#41—44no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C2×#117—22no—
130C2×#117C2312yes—
131C22×C42—22no—
132C22×C42C2312yes—
133#475—44no—
134#462—24no—
135#461—44no—
136#460—28no—
137#452—48no—
138C4×Q16—28no—
139#431—44no—
140C2×#107C2312yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C4×C8C2312yes—
159C2×C4×C8C2314yes—
160#400—24no—
161#396—24no—
162#393—24no—
163#391—24no—
164#391C2312yes—
165#391C2314yes—
166#390—21no—
167#390—22no—
168#390—21no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#389—22no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#388—24no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—24no—
188#386—22no—
189#386—22no—
190#386—22no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199#382—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4C2312yes—
228C43—21no—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—42no—
244C2×#41—42no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C2×D411yescharacteristic
253C22×D4C2412yes—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2412yes—
260C2×Q16—24no—
261C2×Q16—24no—
262C2×Q16—24no—
263C2×SD16—28no—
264C2×SD16—44no—
265C2×SD16—44no—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—42no—
272C4⋊Q8—42no—
273C4⋊Q8—44no—
274C4⋊Q8—82no—
275#127—28no—
276#127—22no—
277#127#4114yes—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280#121—216no—
281#121—24no—
282#121—24no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—41no—
290C4×Q8—42no—
291C4×Q8#4114yes—
292C4×Q8#4114yes—
293C4×D4—28no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—42no—
302C4×D4—48no—
303C4×D4—48no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4—82no—
307C4×D4—84no—
308#117—24no—
309#117—48no—
310#117—42no—
311#117—84no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
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—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—21no—
365C2×C42—48no—
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D412yes—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—24no—
377#106—44no—
378C4⋊C8—24no—
379C4⋊C8—22no—
380C4⋊C8—22no—
381C4⋊C8—24no—
382C4⋊C8—42no—
383#103—24no—
384#103—28no—
385#103—24no—
386#103—24no—
387#103—42no—
388#103—44no—
389#102—28no—
390#102—44no—
391#102—44no—
392C22⋊C8—44no—
393C22⋊C8—44no—
394C4×C8—24no—
395C4×C8—44no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95C2×D412yes—
408#95C2×D412yes—
409#95C2×D412yes—
410C24—22no—
411C24C22×D412yes—
412#41—48no—
413#41—48no—
414#41—48no—
415#41—48no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8C2×#4112yes—
429C2×Q8C2×#4112yes—
430C2×Q8H(5,𝔽2)12yes—
431C2×D4—28no—
432C2×D4—28no—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4C2×#4114yes—
452C22×C4—21no—
453C22×C4—28no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—82no—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C2511yescharacteristic
491Q16—48no—
492SD16—88no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—81no—
528C4⋊C4—82no—
529C4⋊C4—84no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4#13714yes—
533C4⋊C4C2×#4112yes—
534C4⋊C4C2×#4112yes—
535C22⋊C4—28no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—84no—
547C22⋊C4—82no—
548C22⋊C4—84no—
549C42—24no—
550C42—24no—
551C42—24no—
552C42—24no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—21no—
558C42—21no—
559C42—41no—
560C42—44no—
561C42—44no—
562C42—42no—
563C42—44no—
564C42—42no—
565C42—42no—
566C42—81no—
567C42—84no—
568C42#13514yes—
569C42C2×#4112yes—
570C42C2×#4112yes—
571C42C2×#4112yes—
572C42H(5,𝔽2)12yes—
573C23—22no—
574C23—22no—
575C23—24no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23C23×D411yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—22no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—164no—
596Q8#55514yes—
597D4—24no—
598D4—24no—
599D4—416no—
600D4—44no—
601D4—48no—
602D4—42no—
603D4—42no—
604D4—84no—
605D4—88no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—48no—
629C2×C4—416no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—48no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—88no—
655C2×C4—88no—
656C2×C4#52512yes—
657C2×C4#52512yes—
658C2×C4#54612yes—
659C2×C4#54612yes—
660C2×C4C2×#13512yes—
661C2×C4C2×#13512yes—
662C2×C4C2×#13512yes—
663C2×C4C2×#13714yes—
664C2×C4C2×#13712yes—
665C2×C4#57712yes—
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C2×H(5,𝔽2)11yescharacteristic
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4#58511yescharacteristic
672C8—42no—
673C8—42no—
674C8—44no—
675C22—42no—
676C22—42no—
677C22—44no—
678C22—42no—
679C22—42no—
680C22—44no—
681C22—42no—
682C22—44no—
683C22—42no—
684C22—42no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22C2×#52511yescharacteristic
690C22C2×#54611yescharacteristic
691C22#348811yescharacteristic
692C22C22×#13511yescharacteristic
693C22C22×#13711yescharacteristic
694C22C2×#57711yescharacteristic
695C22#358511yescharacteristic
696C4—22no—
697C4—24no—
698C4—22no—
699C4—22no—
700C4—24no—
701C4—41no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—84no—
707C4—81no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4#310714yes—
712C4#330414yes—
713C4C2×#55511yescharacteristic
714C4#347511yescharacteristic
715C4#350611yescharacteristic
716C4#351911yescharacteristic
717C2—82no—
718C2—84no—
719C2#3616011yescharacteristic
720C2C2×#310711yescharacteristic
721C2#6059711yescharacteristic
722C2C2×#330411yescharacteristic
723C2#6116211yescharacteristic
724C2#6195311yescharacteristic
725C2#6200211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #33196C2semidirect
4. #32907C2semidirect
5. #29139C2semidirect
6. #28252C2semidirect
7. #27627C2semidirect
8. #27233C2semidirect
9. #26731C2semidirect
10. #24972C2semidirect
11. #24651C2semidirect
13. #23095C2semidirect
14. #22795C2semidirect
15. #22771C2semidirect
16. C4×#462C2semidirect
17. #20737C2semidirect

Maximal subgroups

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

subgroupindex
#361942
C4×#5502
#331962
#329072
#291392
#282522
#276272
#272332
#267312
#249722
#246512
#242422
#230952
#227952
#227712
C4×#4622
#207372