Finite Group Databasebeta

← #6463934

Subgroups — #6463934

Order 512.

1742 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2C4×#536C211yesmaximal, characteristic
3#29138C214yesmaximal, complemented
4#29137C214yesmaximal, complemented
5#29132C212yesmaximal, complemented
6#29131C212yesmaximal, complemented
7#29123C212yesmaximal, complemented
8#29122C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23512C212yesmaximal
11#21090C212yesmaximal, complemented
12#21087C212yesmaximal
13#20648C211yesmaximal, characteristic
14#20647C211yesmaximal, characteristic, complemented
15#20645C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic, complemented
17#20431C211yesmaximal, characteristic, complemented
18C2×#536C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#417C2212yes—
24C2×#419C2212yes—
25C2×#416C2212yes—
26C22×#103C2212yes—
27C2×#405C2212yes—
28#2859C2212yes—
29#2859C2212yes—
30#2647C2214yes—
31#2571—21no—
32#2571—21no—
33#2552—21no—
34#2552—21no—
35#2442—22no—
36#2442—22no—
37#2327C2212yes—
38#2326C2212yes—
39#2323—21no—
40#2323—21no—
41#2322—22no—
42#2317C2212yes—
43#2317C2211yescharacteristic
44#2316C2212yes—
45#2311C2211yescharacteristic
46#2310C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#124C2212yes—
49C4×C22⋊Q8—21no—
50C4×C22⋊Q8—21no—
51C4×#123—22no—
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2101C2214yes—
56#2101C2214yes—
57#2100C2214yes—
58#2099C2214yes—
59#2043C2214yes—
60#2043C2214yes—
61#2042C2214yes—
62#2042C2214yes—
63#1936C2214yes—
64#1935C2214yes—
65#1935C2214yes—
66#1935C2214yes—
67#1934C2214yes—
68#1933C2214yes—
69#1933C2214yes—
70#1933C2214yes—
71#1805C2212yes—
72#1805C2214yes—
73#1804C2212yes—
74#1803C2212yes—
75#1803C2212yes—
76#1803C2214yes—
77#1803C2211yescharacteristic
78#1803C2211yescharacteristic
79#1803C2212yes—
80#1803C2212yes—
81#1803C2212yes—
82#1803C2212yes—
83#1802C2211yescharacteristic
84#1802C2211yescharacteristic
85#1802C2212yes—
86#1802C2212yes—
87#1768C2214yes—
88#1767C2212yes—
89C4×C4⋊C8C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93C2×#340C2212yes—
94#1729C2212yes—
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#536—24no—
100#536—24no—
101C2×C4⋊Q8C2312yes—
102C2×#125—22no—
103C2×#125—22no—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C4×#41—24no—
109C4×#41—24no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—24no—
113C2×C4×Q8—24no—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C42—21no—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135#423—24no—
136C2×C4⋊C8—24no—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142#419—28no—
143#417—44no—
144#416—28no—
145C2×#103—44no—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156#405—44no—
157C2×#97C2314yes—
158C2×C4×C8C2314yes—
159#400—24no—
160#400—24no—
161#398—24no—
162#398—24no—
163#391—21no—
164#391—21no—
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#390C2312yes—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389C2314yes—
180#389C2314yes—
181#389C2314yes—
182#389C2314yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#386—22no—
192#385—22no—
193#385—24no—
194#385—22no—
195#385—22no—
196#384—22no—
197#384—21no—
198#384—21no—
199#384C2312yes—
200#384C2312yes—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2312yes—
204#382—22no—
205#382—22no—
206#382—22no—
207#382C2312yes—
208#382C2312yes—
209#381—22no—
210#380—21no—
211#380—21no—
212#380—21no—
213#380—21no—
214#380—24no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D411yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227#376C2312yes—
228C43C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41#4114yes—
247C22×Q8—21no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8#4114yes—
254C22×Q8C2412yes—
255C22×Q8C2411yescharacteristic
256C22×D4—21no—
257C22×D4C2411yescharacteristic
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C4⋊Q8—24no—
277C4⋊Q8—24no—
278#125—84no—
279#125—84no—
280#124—24no—
281#124—24no—
282#123—216no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—44no—
300C4×D4—44no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—24no—
308#117—24no—
309#117—44no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—82no—
338C2×C4⋊C4—82no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—44no—
366C2×C42—44no—
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42#4114yes—
374C2×C42#4114yes—
375C2×C42C2412yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C4⋊C8—22no—
380C4⋊C8—22no—
381C4⋊C8—24no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384#103—24no—
385#103—24no—
386#103—28no—
387#103—28no—
388#103—44no—
389#103—44no—
390#103—44no—
391#102—24no—
392#102—24no—
393#102—44no—
394#97—44no—
395C4×C8—44no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95C2×D412yes—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414C24—21no—
415C24C22×D411yescharacteristic
416#41—28no—
417#41—28no—
418#41—44no—
419#41—44no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—28no—
424C2×Q8—28no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8C2×#4114yes—
438C2×Q8C2×#4112yes—
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4C2×#4112yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4#12414yes—
479C22×C4#12414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C42−1+412yes—
495C22×C4C2511yescharacteristic
496M16—48no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—22no—
506C2×C8—48no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C22⋊C4—28no—
546C22⋊C4—28no—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—44no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C42—24no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—42no—
562C42—42no—
563C42—44no—
564C42—42no—
565C42—42no—
566C42—42no—
567C42—44no—
568C42—44no—
569C42#12414yes—
570C42#12414yes—
571C42C2×#4112yes—
572C42C2×#4112yes—
573C42C2×#4112yes—
574C42C2×#4112yes—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—22no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23#52512yes—
587C23C2×#12412yes—
588C23#53712yes—
589C23C23×D411yescharacteristic
590Q8—24no—
591Q8—22no—
592Q8—24no—
593Q8—22no—
594Q8—48no—
595Q8—42no—
596Q8—41no—
597Q8—42no—
598Q8—41no—
599D4—22no—
600D4—22no—
601D4—48no—
602D4—41no—
603D4—41no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—28no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—164no—
654C2×C4#52512yes—
655C2×C4C2×#12412yes—
656C2×C4C2×#12412yes—
657C2×C4C2×#12412yes—
658C2×C4C2×#12412yes—
659C2×C4C2×#12412yes—
660C2×C4#53712yes—
661C2×C4#55214yes—
662C2×C4#55214yes—
663C2×C4#55314yes—
664C2×C4#55314yes—
665C2×C4#57712yes—
666C2×C4#57812yes—
667C2×C4C22×#4111yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×2−1+411yescharacteristic
673C8—42no—
674C8—42no—
675C8—44no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22#256214yes—
690C22C2×#52511yescharacteristic
691C22C22×#12411yescharacteristic
692C22C2×#53711yescharacteristic
693C22C2×#57711yescharacteristic
694C22C2×#57811yescharacteristic
695C22#358511yescharacteristic
696C22#358611yescharacteristic
697C4—22no—
698C4—22no—
699C4—22no—
700C4—42no—
701C4—44no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—82no—
707C4—82no—
708C4—81no—
709C4—82no—
710C4—81no—
711C4—84no—
712C4#256214yes—
713C4#313714yes—
714C4C2×#55211yescharacteristic
715C4C2×#55211yescharacteristic
716C4C2×#55311yescharacteristic
717C4C2×#55311yescharacteristic
718C2—24no—
719C2—82no—
720C2C2×#256211yescharacteristic
721C2#6059911yescharacteristic
722C2#6060011yescharacteristic
723C2C2×#313711yescharacteristic
724C2#6062111yescharacteristic
725C2#6092411yescharacteristic
726C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #29138C2semidirect
4. #29137C2semidirect
5. #29132C2semidirect
6. #29131C2semidirect
7. #29123C2semidirect
8. #29122C2semidirect
9. #23749C2semidirect
11. #21090C2semidirect
14. #20647C2semidirect
16. C2×#1803C2semidirect
17. #20431C2semidirect

Maximal subgroups

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

subgroupindex
#352862
C4×#5362
#291382
#291372
#291322
#291312
#291232
#291222
#237492
#235122
#210902
#210872
#206482
#206472
#206452
C2×#18032
#204312