Finite Group Databasebeta

← #6491909

Subgroups — #6491909

Order 512.

1696 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#35275C211yesmaximal, characteristic, complemented
2#34127C211yesmaximal, characteristic
3#33197C212yesmaximal, complemented
4#33072C212yesmaximal, complemented
5#29997C212yesmaximal, complemented
6#29983C212yesmaximal, complemented
7#26106C212yesmaximal, complemented
8#26088C212yesmaximal, complemented
9#26044C212yesmaximal, complemented
10#26026C212yesmaximal, complemented
11#26007C214yesmaximal, complemented
12#25978C214yesmaximal
13#21981C211yesmaximal, characteristic, complemented
14#21974C211yesmaximal, characteristic, complemented
15#21964C211yesmaximal, characteristic, complemented
16#20737C211yesmaximal, characteristic
17#20648C211yesmaximal, characteristic
18C2×#544—21no—
19C2×C4×#41C2211yescharacteristic
20C2×#476C2214yes—
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#417C2212yes—
24#2868C2212yes—
25#2859C2212yes—
26#2660—22no—
27#2552—21no—
28#2552C2212yes—
29#2551—21no—
30#2551C2212yes—
31#2440—22no—
32#2440C2214yes—
33#2355C2211yescharacteristic
34#2355C2211yescharacteristic
35#2352—21no—
36#2352—21no—
37D4×C4⋊C4C2211yescharacteristic
38D4×C4⋊C4C2211yescharacteristic
39#2347C2212yes—
40#2343—22no—
41#2339C2212yes—
42#2317—21no—
43#2295C2212yes—
44#2290—22no—
45#2285C2211yescharacteristic
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2283—21no—
49C42×D4C2212yes—
50C4×#117C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52#2099C2214yes—
53#2096C2214yes—
54#2088C2214yes—
55#2086C2214yes—
56#2068C2214yes—
57#2066C2214yes—
58#2058C2214yes—
59#2024C2214yes—
60#1987C2214yes—
61#1948C2212yes—
62#1948C2212yes—
63#1948C2212yes—
64#1948C2214yes—
65#1948C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1939C2212yes—
69#1939C2212yes—
70#1921C2214yes—
71#1919C2214yes—
72#1864C2214yes—
73#1862C2214yes—
74#1853C2214yes—
75#1851C2214yes—
76#1844C2212yes—
77#1842C2212yes—
78#1834C2212yes—
79#1831C2212yes—
80#1825C2212yes—
81#1809C2214yes—
82#1803C2211yescharacteristic
83#1803C2211yescharacteristic
84#1803C2212yes—
85#1803C2212yes—
86#1802C2211yescharacteristic
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1802C2214yes—
90#1802C2212yes—
91#1788C2212yes—
92C4×#103C2212yes—
93C4×#102C2212yes—
94#1736C2212yes—
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97#544—48no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8C2314yes—
102C2×#125—22no—
103C2×#125—22no—
104C2×C22⋊Q8—24no—
105C4×#41—24no—
106C4×#41Q818yescomplemented
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×#117—21no—
116C2×#117—21no—
117C2×#117—21no—
118C2×#117D411yescharacteristic
119C2×#117D411yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C42—21no—
127C22×C42—22no—
128C22×C42C2311yescharacteristic
129#476—48no—
130#431—24no—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135#423—24no—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142#417—44no—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C4×C8C2312yes—
151#398—24no—
152#398—24no—
153#395—24no—
154#395—24no—
155#391C2312yes—
156#391C2312yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2314yes—
163#389C2314yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#386—22no—
172#386—22no—
173#385—24no—
174#385—22no—
175#385—22no—
176#384—22no—
177#384—22no—
178#384—21no—
179#384—21no—
180#384C2312yes—
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2311yescharacteristic
184#384C2314yes—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#383—22no—
188#382—21no—
189#382—21no—
190#382—22no—
191#382—21no—
192#382—21no—
193#381—22no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380C2312yes—
199#380C2312yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376—22no—
220#376D412yes—
221C2×#95—22no—
222C43C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2314yes—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—41no—
239C2×#41—41no—
240C2×#41C2×Q814yes—
241C22×Q8—24no—
242C22×Q8C2411yescharacteristic
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×M16—24no—
255C2×M16—42no—
256C2×M16—42no—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8—48no—
263C4⋊Q8—48no—
264C4⋊Q8—84no—
265#125—48no—
266#125—48no—
267C22⋊Q8—416no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8C2×Q814yes—
271C4×Q8C2×Q814yes—
272C4×D4—28no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—42no—
279C4×D4—42no—
280C4×D4C2×Q814yes—
281C4×D4C2×Q814yes—
282#117—22no—
283#117—24no—
284#117—24no—
285#117—28no—
286#117—44no—
287#117—82no—
288#117C2×Q814yes—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—28no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—84no—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×Q814yes—
353C2×C42#4114yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—44no—
359#106—44no—
360#106—44no—
361#106—44no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364C4⋊C8—22no—
365C4⋊C8—22no—
366C4⋊C8—44no—
367C4⋊C8—44no—
368#103—42no—
369#103—42no—
370#103—44no—
371#102—28no—
372#102—24no—
373#102—24no—
374C22⋊C8—24no—
375C4×C8—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—21no—
395C24C22×D411yescharacteristic
396#41—28no—
397#41—44no—
398#41—44no—
399#41—44no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—48no—
406C2×Q8—84no—
407C2×Q8—84no—
408C2×Q8C22×Q812yes—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4C22×Q812yes—
416C2×D4H(5,𝔽2)12yes—
417C2×D42−1+412yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4C22⋊Q814yes—
453C22×C4C22⋊Q814yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×Q812yes—
464C22×C4C2×#4112yes—
465C22×C4C2511yescharacteristic
466M16—84no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4AGL(1,ℤ/8)14yes—
513C4⋊C4#13714yes—
514C4⋊C4C22×Q812yes—
515C4⋊C4C22×Q812yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C42—24no—
526C42—24no—
527C42—24no—
528C42—24no—
529C42—24no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—44no—
537C42—44no—
538C42—82no—
539C42—82no—
540C42C22⋊Q814yes—
541C42C22⋊Q814yes—
542C42C22×Q812yes—
543C42C22×Q812yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C23—22no—
547C23—22no—
548C23—21no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×C22⋊Q812yes—
560C23C23×D411yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—41no—
564Q8—41no—
565Q8—168no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4—48no—
570D4#54414yescomplemented
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—48no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—88no—
621C2×C4—88no—
622C2×C4—164no—
623C2×C4C2×C22⋊Q812yes—
624C2×C4C2×C22⋊Q812yes—
625C2×C4C2×C22⋊Q812yes—
626C2×C4C2×C22⋊Q812yes—
627C2×C4C2×C22⋊Q812yes—
628C2×C4C2×AGL(1,ℤ/8)14yes—
629C2×C4C2×AGL(1,ℤ/8)12yes—
630C2×C4C2×#13714yes—
631C2×C4C2×#13712yes—
632C2×C4#57712yes—
633C2×C4#57712yes—
634C2×C4C23×Q811yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C2×C4#58511yescharacteristic
639C2×C4#58511yescharacteristic
640C8—42no—
641C8—44no—
642C8—42no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—24no—
647C22—42no—
648C22—41no—
649C22—41no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—41no—
655C22—41no—
656C22—42no—
657C22C22×C22⋊Q811yescharacteristic
658C22#348811yescharacteristic
659C22#348911yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57711yescharacteristic
663C22C2×#57711yescharacteristic
664C4—24no—
665C4—22no—
666C4—22no—
667C4—24no—
668C4—22no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—82no—
675C4—84no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#307914yes—
680C4#308414yes—
681C4C2×#54411yescharacteristic
682C4#345811yescharacteristic
683C4#349411yescharacteristic
684C4#349511yescharacteristic
685C2—42no—
686C2—44no—
687C2#3525311yescharacteristic
688C2C2×#307911yescharacteristic
689C2C2×#308411yescharacteristic
690C2#6198011yescharacteristic
691C2#6199911yescharacteristic
692C2#6212811yescharacteristic
693C2#6214711yescharacteristic
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. #35275C2semidirect
3. #33197C2semidirect
4. #33072C2semidirect
5. #29997C2semidirect
6. #29983C2semidirect
7. #26106C2semidirect
8. #26088C2semidirect
9. #26044C2semidirect
10. #26026C2semidirect
11. #26007C2semidirect
13. #21981C2semidirect
14. #21974C2semidirect
15. #21964C2semidirect
106. C4×#41Q8semidirect
570. D4#544semidirect

Maximal subgroups

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

subgroupindex
#352752
#341272
#331972
#330722
#299972
#299832
#261062
#260882
#260442
#260262
#260072
#259782
#219812
#219742
#219642
#207372
#206482