Finite Group Databasebeta

← #6793017

Subgroups — #6793017

Order 512.

1681 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#31604C214yesmaximal, complemented
5#31552C214yesmaximal, complemented
6#29938C212yesmaximal, complemented
7#29218C211yesmaximal, characteristic, complemented
8#28283C211yesmaximal, characteristic, complemented
9#28141C212yesmaximal
10#26108C212yesmaximal, complemented
11#25341C212yesmaximal, complemented
12#24836C212yesmaximal, complemented
13#24813C212yesmaximal, complemented
14#24151C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21885C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#529—21no—
20C2×#475C2212yes—
21C2×#476C2212yes—
22C2×#461C2211yescharacteristic
23C2×C4×SD16C2212yes—
24C2×C8×D4C2212yes—
25C22×C4⋊C8C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2553—22no—
29#2552—21no—
30#2552C2211yescharacteristic
31#2547C2211yescharacteristic
32#2542—21no—
33#2542C2211yescharacteristic
34#2538C2212yes—
35#2537C2211yescharacteristic
36#2496C2212yes—
37#2489—21no—
38#2441—22no—
39#2429—22no—
40#2427—21no—
41#2423C2214yes—
42#2422C2212yes—
43#2395—22no—
44#2365—21no—
45#2354—21no—
46Q8×C4⋊C4C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C22⋊Q8—21no—
50C4×#121C2212yes—
51#2285C2212yes—
52C42×D4C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2100C2214yes—
55#2099C2214yes—
56#2088C2214yes—
57#2084C2214yes—
58#2077C2214yes—
59#2077C2214yes—
60#2066C2214yes—
61#2065C2214yes—
62#2035C2214yes—
63#2000C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1981C2212yes—
69#1981C2211yescharacteristic
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2211yescharacteristic
74#1939C2212yes—
75#1927C2212yes—
76#1893C2214yes—
77#1892C2214yes—
78#1892C2214yes—
79#1880C2214yes—
80#1877C2212yes—
81#1865C2214yes—
82#1850C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1807C2214yes—
86#1803C2214yes—
87#1803C2214yes—
88#1803C2212yes—
89#1802C2211yescharacteristic
90#1802C2212yes—
91#1802C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96C23×C8C2314yes—
97#533—28no—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104#529—48no—
105C2×#126—22no—
106C2×#125—21no—
107C2×#125—22no—
108C2×#124—21no—
109C2×#123—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2314yes—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2314yes—
121C2×C4×D4—22no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C2×#117—22no—
130C22×C42—21no—
131C22×C42C2312yes—
132#476—44no—
133#475—44no—
134#461—24no—
135C4×SD16—44no—
136C8×D4—44no—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2312yes—
140C2×C4⋊C8—44no—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#399—24no—
158#399—22no—
159#398—22no—
160#398—24no—
161#397—22no—
162#397—22no—
163#397—22no—
164#396—24no—
165#394—22no—
166#394—22no—
167#391—21no—
168#391—21no—
169#391—22no—
170#391C2312yes—
171#390—22no—
172#390C2312yes—
173#389—22no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—24no—
180#389C2312yes—
181#389C2314yes—
182#389C2312yes—
183#388—24no—
184#388—22no—
185#387—24no—
186#387—22no—
187#386—22no—
188#385—22no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199#380—22no—
200#380—22no—
201#380—21no—
202#380C2312yes—
203#380C2314yes—
204#380C2312yes—
205#380C2312yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—42no—
252C22×C8—44no—
253C22×C8—44no—
254C22×C8—44no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265C4⋊Q8#4114yes—
266#126—48no—
267#125—48no—
268#125—44no—
269#124—44no—
270#123—48no—
271C22⋊Q8—28no—
272C22⋊Q8—44no—
273C22⋊Q8—48no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—84no—
277#121—24no—
278#121—42no—
279#121—44no—
280C4×Q8—28no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×Q8—48no—
285C4×Q8—44no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—42no—
294C4×D4—41no—
295C4×D4—48no—
296C4×D4SD1618yescomplemented
297#117—48no—
298#117—48no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—44no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#106—42no—
365#106—44no—
366#106—44no—
367C4⋊C8—22no—
368C4⋊C8—22no—
369C4⋊C8—24no—
370C4⋊C8—24no—
371C4⋊C8—44no—
372C4⋊C8—42no—
373#103—24no—
374#103—42no—
375#103—44no—
376#103—42no—
377#103—44no—
378#102—42no—
379#102—44no—
380C22⋊C8—44no—
381C4×C8—24no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24C22×D412yes—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8C2×#4114yes—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—82no—
422C2×D4C2×SD1614yes—
423C2×D42−1+412yes—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—81no—
452C22×C4C2×SD1618yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2511yescharacteristic
463SD16—48no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—88no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4C2×SD1614yes—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—48no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—84no—
526C22⋊C4C2×SD1618yes—
527C42—24no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—41no—
536C42—48no—
537C42—44no—
538C42—42no—
539C42—42no—
540C42—42no—
541C42—44no—
542C42C2×SD1614yes—
543C42C2×#4112yes—
544C42H(5,𝔽2)12yes—
545C42H(5,𝔽2)12yes—
546C422−1+412yes—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23C22×SD1614yes—
561C23C23×D411yescharacteristic
562Q8—24no—
563Q8—24no—
564Q8—42no—
565Q8—42no—
566Q8—88no—
567Q8—84no—
568D4—28no—
569D4—22no—
570D4—22no—
571D4—41no—
572D4—41no—
573D4—88no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—216no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4#53712yes—
625C2×C4#53712yes—
626C2×C4#54612yes—
627C2×C4#54612yes—
628C2×C4C22×SD1612yes—
629C2×C4C22×SD1614yes—
630C2×C4C22×SD1612yes—
631C2×C4C22×SD1612yes—
632C2×C4#57512yes—
633C2×C4#57812yes—
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C2×C4#58511yescharacteristic
640C8—42no—
641C8—44no—
642C8—42no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22C2×#53711yescharacteristic
658C22C2×#54611yescharacteristic
659C22#348911yescharacteristic
660C22C23×SD1611yescharacteristic
661C22C2×#57511yescharacteristic
662C22C2×#57811yescharacteristic
663C22#358511yescharacteristic
664C4—22no—
665C4—22no—
666C4—24no—
667C4—22no—
668C4—41no—
669C4—42no—
670C4—42no—
671C4—44no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—81no—
677C4—82no—
678C4—84no—
679C4#324614yes—
680C4#329814yes—
681C4#351511yescharacteristic
682C4#352011yescharacteristic
683C4#354611yescharacteristic
684C4#355211yescharacteristic
685C2—44no—
686C2—82no—
687C2#3890811yescharacteristic
688C2C2×#324611yescharacteristic
689C2#6093011yescharacteristic
690C2C2×#329811yescharacteristic
691C2#6116011yescharacteristic
692C2#6209611yescharacteristic
693C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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. #36134C2semidirect
3. #33197C2semidirect
4. #31604C2semidirect
5. #31552C2semidirect
6. #29938C2semidirect
7. #29218C2semidirect
8. #28283C2semidirect
10. #26108C2semidirect
11. #25341C2semidirect
12. #24836C2semidirect
13. #24813C2semidirect
14. #24151C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21885C2semidirect
17. C4×#461C2semidirect
296. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#361342
#349322
#331972
#316042
#315522
#299382
#292182
#282832
#281412
#261082
#253412
#248362
#248132
#241512
D4×C4⋊C82
#218852
C4×#4612