Finite Group Databasebeta

← #6623901

Subgroups — #6623901

Order 512.

1697 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2C4×#556C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic, complemented
4#33025C212yesmaximal, complemented
5#28512C211yesmaximal, characteristic, complemented
6#28383C212yesmaximal
7#28252C211yesmaximal, characteristic, complemented
8#28203C212yesmaximal, complemented
9#27510C214yesmaximal, complemented
10#27090C214yesmaximal, complemented
11#25059C212yesmaximal, complemented
12#24185C212yesmaximal, complemented
13#23100C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22813C212yesmaximal, complemented
16#21615C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#556—21no—
19C2×#556C2211yescharacteristic
20C2×#475C2212yes—
21C2×#475C2212yes—
22C2×#474C2212yes—
23C2×#460C2212yes—
24C2×#461C2212yes—
25C2×#459C2211yescharacteristic
26C2×C4×D8C2212yes—
27#2868C2211yescharacteristic
28#2858C2212yes—
29#2547—21no—
30#2545—22no—
31#2538—21no—
32#2538C2212yes—
33#2512—22no—
34#2512C2214yes—
35#2373—22no—
36#2367C2212yes—
37#2365C2211yescharacteristic
38#2362—21no—
39#2356C2212yes—
40#2355C2211yescharacteristic
41#2354—21no—
42#2354C2211yescharacteristic
43#2354C2212yes—
44#2347—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#126—22no—
49C4×#124C2212yes—
50C4×C22⋊Q8—21no—
51C42×Q8C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54C4×#117—21no—
55#2096C2214yes—
56#2092C2214yes—
57#2085C2214yes—
58#2055C2214yes—
59#2042C2214yes—
60#2012C2214yes—
61#1988C2212yes—
62#1984C2212yes—
63#1982C2211yescharacteristic
64#1973C2214yes—
65#1972C2214yes—
66#1956C2212yes—
67#1947C2212yes—
68#1947C2211yescharacteristic
69#1947C2212yes—
70#1928C2214yes—
71#1926C2214yes—
72#1926C2214yes—
73#1926C2214yes—
74#1925C2212yes—
75#1920C2214yes—
76#1917C2214yes—
77#1880C2214yes—
78#1880C2214yes—
79#1879C2212yes—
80#1863C2214yes—
81#1844C2212yes—
82#1842C2211yescharacteristic
83#1831C2212yes—
84#1803C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#102C2212yes—
95C22×SD16C2314yes—
96C22×D8C2312yes—
97#556—24no—
98#556—24no—
99#556—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#126—22no—
106C2×#126—24no—
107C2×#124—22no—
108C2×#124—22no—
109C2×#124C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—21no—
126C2×#117—22no—
127C2×#117—21no—
128C22×C42—21no—
129#475—28no—
130#475—44no—
131#474—44no—
132#461—28no—
133#460—44no—
134#459—24no—
135C4×D8—28no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#397—24no—
157#396—24no—
158#396—24no—
159#391—21no—
160#391—22no—
161#391C2312yes—
162#391C2312yes—
163#390—21no—
164#390—21no—
165#390—22no—
166#390—22no—
167#390C2312yes—
168#390C2314yes—
169#390C2314yes—
170#390C2312yes—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389C2314yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—24no—
182#388—24no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#383—22no—
195#382C2312yes—
196#380—21no—
197#380—22no—
198#380—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225#376—22no—
226C43—22no—
227C43C2311yescharacteristic
228C43C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—22no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2412yes—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C2×SD16—28no—
250C2×SD16—44no—
251C2×SD16—44no—
252C2×D8—24no—
253C2×D8—24no—
254C2×D8—42no—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263C4⋊Q8#4114yes—
264#127—22no—
265#127—22no—
266#126—216no—
267#126—416no—
268#124—24no—
269#124—24no—
270#124—48no—
271#124—84no—
272C22⋊Q8—28no—
273C22⋊Q8—48no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×Q8—48no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—41no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4#4114yes—
295#117—28no—
296#117—44no—
297#117—48no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—24no—
363#107—44no—
364#106—28no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—42no—
370C4⋊C8#4114yes—
371#103—28no—
372#103—28no—
373#103—44no—
374#102—24no—
375#102—24no—
376#102—24no—
377#102—24no—
378#102—44no—
379#102—44no—
380#97—44no—
381C4×C8—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—22no—
394#95—24no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8C2×#4114yes—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4C2×#4112yes—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460SD16—88no—
461D8—48no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8C2×#4114yes—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4C2×#4112yes—
503C4⋊C42−1+412yes—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—82no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—28no—
519C42—24no—
520C42—22no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—41no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42—44no—
534C42—48no—
535C42—44no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—82no—
540C42AGL(1,ℤ/8)14yes—
541C42C2×#4112yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C42H(5,𝔽2)12yes—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23C23×D411yescharacteristic
560Q8—24no—
561Q8—24no—
562Q8—44no—
563Q8—42no—
564Q8—42no—
565Q8—88no—
566D4—22no—
567D4—22no—
568D4—42no—
569D4—41no—
570D4—42no—
571D4—42no—
572D4—41no—
573D4—42no—
574D4—84no—
575D4—84no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4—84no—
622C2×C4#52512yes—
623C2×C4#52512yes—
624C2×C4#55012yes—
625C2×C4#55012yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4#57512yes—
630C2×C4#57712yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4#58511yescharacteristic
637C8—24no—
638C8—42no—
639C8—44no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22C2×#52511yescharacteristic
655C22C2×#55011yescharacteristic
656C22#348911yescharacteristic
657C22C22×AGL(1,ℤ/8)11yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57711yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—41no—
672C4—82no—
673C4—81no—
674C4—84no—
675C4—82no—
676C4#311514yes—
677C4#339714yes—
678C4#347511yescharacteristic
679C4#348211yescharacteristic
680C4#351111yescharacteristic
681C4#351411yescharacteristic
682C2—82no—
683C2—162no—
684C2#3650411yescharacteristic
685C2C2×#311511yescharacteristic
686C2#6059911yescharacteristic
687C2C2×#339711yescharacteristic
688C2#6140311yescharacteristic
689C2#6210711yescharacteristic
690C2#6214511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36594C2semidirect
2. C4×#556C2semidirect
3. #33195C2semidirect
4. #33025C2semidirect
5. #28512C2semidirect
7. #28252C2semidirect
8. #28203C2semidirect
9. #27510C2semidirect
10. #27090C2semidirect
11. #25059C2semidirect
12. #24185C2semidirect
13. #23100C2semidirect
14. #22825C2semidirect
15. #22813C2semidirect
16. #21615C2semidirect

Maximal subgroups

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

subgroupindex
#365942
C4×#5562
#331952
#330252
#285122
#283832
#282522
#282032
#275102
#270902
#250592
#241852
#231002
#228252
#228132
#216152
C4×#4592