Finite Group Databasebeta

← #6593466

Subgroups — #6593466

Order 512.

1775 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2C4×D4×Q8C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#32906C212yesmaximal, complemented
5#29141C211yesmaximal, characteristic, complemented
6#28255C211yesmaximal, characteristic, complemented
7#27630C212yesmaximal, complemented
8#27241C214yesmaximal, complemented
9#26743C214yesmaximal, complemented
10#24969C212yesmaximal
11#24660C212yesmaximal, complemented
12#24243C212yesmaximal
13#23102C212yesmaximal, complemented
14#22799C211yesmaximal, characteristic
15#22763C212yesmaximal, complemented
16C4×#462C211yesmaximal, characteristic, complemented
17#20740C212yesmaximal, complemented
18C2×D4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#475C2212yes—
21C2×#462C2212yes—
22C2×#462C2211yescharacteristic
23C2×#462C2212yes—
24C2×#451C2214yes—
25C2×C4×Q16C2212yes—
26C2×#432C2212yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2571—21no—
30#2563—21no—
31#2538C2212yes—
32#2497—22no—
33#2492C2214yes—
34#2439—22no—
35#2369—21no—
36#2365—21no—
37#2365C2211yescharacteristic
38#2360—22no—
39#2360C2212yes—
40#2355C2212yes—
41#2354—21no—
42Q8×C4⋊C4C2211yescharacteristic
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2211yescharacteristic
45Q8×C22⋊C4C2212yes—
46C4×C4⋊Q8—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C22⋊Q8—22no—
50C4×C22⋊Q8C2212yes—
51#2285C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4—21no—
54C42×D4C2211yescharacteristic
55#2077C2214yes—
56#2056C2214yes—
57#2036C2214yes—
58#2035C2214yes—
59#2024C2214yes—
60#2001C2214yes—
61#1988C2212yes—
62#1981C2211yescharacteristic
63#1971C2214yes—
64#1956C2212yes—
65#1954C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1947C2211yescharacteristic
69#1935C2214yes—
70#1928C2212yes—
71#1927C2214yes—
72#1924C2214yes—
73#1918C2214yes—
74#1878C2214yes—
75#1878C2212yes—
76#1878C2214yes—
77#1872C2214yes—
78#1842C2211yescharacteristic
79#1834C2212yes—
80#1823C2214yes—
81#1803C2211yescharacteristic
82#1803C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1797C2212yes—
89#1787C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93#1763C2214yes—
94C4×#103C2212yes—
95C23×Q8C2312yes—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98D4×Q8—24no—
99D4×Q8—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2314yes—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—28no—
112C2×C4×Q8—44no—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2312yes—
128C22×C42—22no—
129C22×C42C2312yes—
130#475—44no—
131#462—28no—
132#462—24no—
133#462—44no—
134#451—48no—
135C4×Q16—28no—
136#432—44no—
137C2×#107C2312yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#400—24no—
158#396—24no—
159#393—24no—
160#391—22no—
161#391—22no—
162#391—24no—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#391C2314yes—
167#390—22no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#388—24no—
181#387—22no—
182#387—24no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#382—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2312yes—
202#380C2314yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
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#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—24no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C22×Q8C2411yescharacteristic
250C22×D4C2×D411yescharacteristic
251C23×C4—22no—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C2×Q16—24no—
257C2×Q16—24no—
258C2×Q16—24no—
259C2×Q16—28no—
260C2×Q16—44no—
261C2×Q16—44no—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—28no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—42no—
270C4⋊Q8—42no—
271C4⋊Q8—44no—
272C4⋊Q8—82no—
273C4⋊Q8#4114yes—
274C22⋊Q8—216no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—24no—
282C4×Q8—22no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—48no—
289C4×Q8—42no—
290C4×Q8—42no—
291C4×Q8—41no—
292C4×Q8—48no—
293C4×Q8#4114yes—
294C4×Q8#4114yes—
295C4×D4—28no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—44no—
301#117—84no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—48no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—24no—
357C2×C42—21no—
358C2×C42—48no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#107—24no—
370#106—44no—
371C4⋊C8—24no—
372C4⋊C8—22no—
373C4⋊C8—22no—
374C4⋊C8—24no—
375C4⋊C8—42no—
376#103—24no—
377#103—28no—
378#103—28no—
379#103—24no—
380#103—24no—
381#103—42no—
382#103—44no—
383#103—44no—
384#103—44no—
385C22⋊C8—44no—
386C22⋊C8—44no—
387#97—44no—
388C4×C8—24no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407C24C22×D412yes—
408C2×Q8—28no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—48no—
426C2×Q8—48no—
427C2×Q8—48no—
428C2×Q8—48no—
429C2×Q8—44no—
430C2×Q8—82no—
431C2×Q8—82no—
432C2×Q8C2×#4114yes—
433C2×Q8C2×#4112yes—
434C2×Q8C2×#4112yes—
435C2×Q8H(5,𝔽2)12yes—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4#13714yes—
439C22×C4—21no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—82no—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2511yescharacteristic
476Q16—48no—
477Q16—88no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C4⋊C4—28no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—24no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—84no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4C2×#4112yes—
525C4⋊C4C2×#4112yes—
526C22⋊C4—28no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—44no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C42—24no—
534C42—24no—
535C42—24no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—21no—
542C42—21no—
543C42—41no—
544C42—44no—
545C42—44no—
546C42—42no—
547C42—44no—
548C42—44no—
549C42—42no—
550C42—42no—
551C42—81no—
552C42—84no—
553C42#13714yes—
554C42C2×#4112yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C42H(5,𝔽2)12yes—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23C2×#13714yes—
565C23C23×D411yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—24no—
570Q8—24no—
571Q8—416no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—164no—
579Q8#55514yes—
580D4—28no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—48no—
606C2×C4—416no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—88no—
634C2×C4#52512yes—
635C2×C4#52512yes—
636C2×C4#54612yes—
637C2×C4#54612yes—
638C2×C4C2×#13712yes—
639C2×C4C2×#13712yes—
640C2×C4C2×#13712yes—
641C2×C4C2×#13712yes—
642C2×C4#57812yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4#58511yescharacteristic
649C8—42no—
650C8—42no—
651C8—44no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22C2×#52511yescharacteristic
660C22C2×#54611yescharacteristic
661C22#348811yescharacteristic
662C22C22×#13711yescharacteristic
663C22C22×#13711yescharacteristic
664C22C2×#57811yescharacteristic
665C22#358511yescharacteristic
666C4—22no—
667C4—24no—
668C4—22no—
669C4—22no—
670C4—41no—
671C4—42no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—84no—
678C4—81no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#311314yes—
683C4#332814yes—
684C4C2×#55511yescharacteristic
685C4#347511yescharacteristic
686C4#350611yescharacteristic
687C4#351911yescharacteristic
688C2—44no—
689C2#3616011yescharacteristic
690C2C2×#311311yescharacteristic
691C2#6060011yescharacteristic
692C2C2×#332811yescharacteristic
693C2#6116211yescharacteristic
694C2#6196511yescharacteristic
695C2#6198411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #33196C2semidirect
4. #32906C2semidirect
5. #29141C2semidirect
6. #28255C2semidirect
7. #27630C2semidirect
8. #27241C2semidirect
9. #26743C2semidirect
11. #24660C2semidirect
13. #23102C2semidirect
15. #22763C2semidirect
16. C4×#462C2semidirect
17. #20740C2semidirect

Maximal subgroups

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

subgroupindex
#361942
C4×D4×Q82
#331962
#329062
#291412
#282552
#276302
#272412
#267432
#249692
#246602
#242432
#231022
#227992
#227632
C4×#4622
#207402