Finite Group Databasebeta

← #6615693

Subgroups — #6615693

Order 512.

1795 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2C4×#547C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#33108C212yesmaximal, complemented
5#29206C211yesmaximal, characteristic, complemented
6#28283C211yesmaximal, characteristic, complemented
7#27625C212yesmaximal, complemented
8#27328C214yesmaximal, complemented
9#26786C214yesmaximal, complemented
10#25014C212yesmaximal, complemented
11#24651C212yesmaximal, complemented
12#24120C212yesmaximal
13#23062C212yesmaximal, complemented
14#22805C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17#21359C212yesmaximal, complemented
18C2×#547C2211yescharacteristic
19C2×C4×#41—21no—
20C2×#483C2214yes—
21C2×#475C2212yes—
22C2×#476C2212yes—
23C2×#461C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#431C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2571—21no—
29#2563—21no—
30#2538C2212yes—
31#2497—22no—
32#2492C2214yes—
33#2439—22no—
34#2369—21no—
35#2365C2211yescharacteristic
36#2362—21no—
37#2361C2212yes—
38#2359—22no—
39#2355C2212yes—
40#2354—21no—
41#2354C2211yescharacteristic
42D4×C4⋊C4C2211yescharacteristic
43#2344C2212yes—
44#2339—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8—21no—
48C4×#123—22no—
49C4×#121C2212yes—
50#2285C2212yes—
51C42×D4C2211yescharacteristic
52C42×D4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54#2077C2214yes—
55#2066C2214yes—
56#2058C2214yes—
57#2035C2214yes—
58#2034C2214yes—
59#2000C2214yes—
60#1990C2212yes—
61#1987C2212yes—
62#1981C2212yes—
63#1981C2211yescharacteristic
64#1968C2214yes—
65#1954C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1933C2214yes—
73#1927C2214yes—
74#1927C2214yes—
75#1927C2212yes—
76#1926C2214yes—
77#1923C2214yes—
78#1919C2214yes—
79#1877C2212yes—
80#1872C2214yes—
81#1853C2214yes—
82#1853C2214yes—
83#1844C2212yes—
84#1842C2211yescharacteristic
85#1803C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1797C2212yes—
90C4×#106C2214yes—
91C4×#106C2214yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×#41—21no—
96C22×SD16C2312yes—
97#547—24no—
98#547—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2314yes—
107C2×#121C2312yes—
108C4×#41—48no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—21no—
123C2×#117—22no—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C42—22no—
127C22×C42—21no—
128C22×C42C2312yes—
129#483—48no—
130#476—44no—
131#475—28no—
132#461—24no—
133C4×SD16—28no—
134#431—28no—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2314yes—
156#400—24no—
157#396—24no—
158#393—24no—
159#391—21no—
160#391—21no—
161#391—24no—
162#391C2312yes—
163#391C2314yes—
164#390—22no—
165#390C2312yes—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#388—24no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—24no—
182#387—22no—
183#386—22no—
184#386—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#382—22no—
194#382—22no—
195#381—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—21no—
200#380C2312yes—
201#380C2314yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226C2×#95—22no—
227C43C2311yescharacteristic
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2312yes—
237C2×#41—28no—
238C2×#41—44no—
239C2×#41—44no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8C2412yes—
243C22×D4—22no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2411yescharacteristic
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—42no—
264C4⋊Q8—42no—
265C4⋊Q8—44no—
266C4⋊Q8—82no—
267C4⋊Q8#4114yes—
268#123—216no—
269C22⋊Q8—28no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272#121—24no—
273#121—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—42no—
285C4×D4—41no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4SD1618yescomplemented
290C4×D4#4114yes—
291#117—44no—
292#117—48no—
293#117—84no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—24no—
348C2×C42—48no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#106—24no—
361#106—28no—
362#106—28no—
363#106—44no—
364#106—44no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—22no—
368C4⋊C8—42no—
369C4⋊C8#4114yescomplemented
370#103—24no—
371#103—24no—
372#103—28no—
373#103—24no—
374#103—44no—
375#102—24no—
376#102—44no—
377C22⋊C8—28no—
378C22⋊C8—44no—
379C4×C8—28no—
380C4×C8—24no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—21no—
400C24C22×D412yes—
401#41—432no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4C2×SD1614yes—
425C2×D4C2×#4112yes—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—81no—
459C22×C4—82no—
460C22×C4C2×SD1618yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2511yescharacteristic
471SD16—48no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8C2×#4114yes—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—81no—
509C4⋊C4—84no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4C2×SD1614yes—
513C4⋊C4C2×#4112yes—
514C4⋊C4C2×#4114yes—
515C4⋊C4H(5,𝔽2)12yes—
516C22⋊C4—28no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4C2×SD1618yes—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—22no—
535C42—21no—
536C42—41no—
537C42—44no—
538C42—42no—
539C42—44no—
540C42—42no—
541C42—44no—
542C42—42no—
543C42—44no—
544C42—84no—
545C42C2×SD1614yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C422−1+412yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C22×SD1614yes—
564C23C23×D411yescharacteristic
565Q8—48no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—164no—
571D4—28no—
572D4—22no—
573D4—22no—
574D4—416no—
575D4—41no—
576D4—41no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—216no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—416no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—88no—
628C2×C4—84no—
629C2×C4#52512yes—
630C2×C4#52512yes—
631C2×C4#54712yes—
632C2×C4#54712yes—
633C2×C4C22×SD1614yes—
634C2×C4C22×SD1612yes—
635C2×C4C22×SD1612yes—
636C2×C4C22×SD1612yes—
637C2×C4#57512yes—
638C2×C4#57712yes—
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C2×C4#58511yescharacteristic
645C8—24no—
646C8—44no—
647C8—42no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22C2×#52511yescharacteristic
663C22C2×#54711yescharacteristic
664C22#348811yescharacteristic
665C22C23×SD1611yescharacteristic
666C22C2×#57511yescharacteristic
667C22C2×#57711yescharacteristic
668C22#358611yescharacteristic
669C4—24no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—41no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—81no—
680C4—84no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4#310114yes—
685C4#333514yes—
686C4#347511yescharacteristic
687C4#348311yescharacteristic
688C4#351111yescharacteristic
689C4#351811yescharacteristic
690C2—44no—
691C2—82no—
692C2#3641111yescharacteristic
693C2C2×#310111yescharacteristic
694C2#6059911yescharacteristic
695C2C2×#333511yescharacteristic
696C2#6125711yescharacteristic
697C2#6194811yescharacteristic
698C2#6199711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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. #33197C2semidirect
4. #33108C2semidirect
5. #29206C2semidirect
6. #28283C2semidirect
7. #27625C2semidirect
8. #27328C2semidirect
9. #26786C2semidirect
10. #25014C2semidirect
11. #24651C2semidirect
13. #23062C2semidirect
14. #22805C2semidirect
15. #22795C2semidirect
16. C4×#461C2semidirect
17. #21359C2semidirect
289. C4×D4SD16semidirect
369. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#361942
C4×#5472
#331972
#331082
#292062
#282832
#276252
#273282
#267862
#250142
#246512
#241202
#230622
#228052
#227952
C4×#4612
#213592