Finite Group Databasebeta

← #6793186

Subgroups — #6793186

Order 512.

1629 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34943C211yesmaximal, characteristic
3#33162C211yesmaximal, characteristic, complemented
4#31497C214yesmaximal, complemented
5#30601C214yesmaximal, complemented
6#29926C212yesmaximal, complemented
7#29287C211yesmaximal, characteristic, complemented
8#28748C211yesmaximal, characteristic, complemented
9#28598C212yesmaximal, complemented
10#26527C212yesmaximal, complemented
11#25385C212yesmaximal
12#25319C212yesmaximal, complemented
13#24751C212yesmaximal, complemented
14#24153C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21889C212yesmaximal, complemented
17C4×#474C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#487C2212yes—
20C2×#474C2211yescharacteristic
21C2×#465C2214yes—
22C2×#461C2212yes—
23C2×#461C2212yes—
24C2×#461C2212yes—
25C2×C8×D4C2212yes—
26C22×C4⋊C8C2212yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2572—22no—
30#2571C2211yescharacteristic
31#2563C2211yescharacteristic
32#2563C2211yescharacteristic
33#2552—21no—
34#2542C2212yes—
35#2538—21no—
36#2537C2211yescharacteristic
37#2495—21no—
38#2495C2212yes—
39#2441—22no—
40#2429C2214yes—
41#2423—22no—
42#2420—21no—
43#2420C2212yes—
44#2398—22no—
45#2365C2212yes—
46#2354—21no—
47Q8×C4⋊C4—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#121—21no—
51C4×#121C2212yes—
52#2283—21no—
53#2283C2212yes—
54C42×D4C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2097C2214yes—
57#2088C2214yes—
58#2074C2214yes—
59#2066C2214yes—
60#2055C2214yes—
61#2043C2214yes—
62#2036C2214yes—
63#2036C2214yes—
64#2035C2214yes—
65#2000C2214yes—
66#1988C2212yes—
67#1988C2211yescharacteristic
68#1987C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1956C2212yes—
75#1948C2212yes—
76#1948C2212yes—
77#1947C2211yescharacteristic
78#1939C2212yes—
79#1918C2212yes—
80#1892C2214yes—
81#1892C2214yes—
82#1892C2214yes—
83#1880C2214yes—
84#1863C2214yes—
85#1862C2212yes—
86#1850C2214yes—
87#1842C2211yescharacteristic
88#1805C2214yes—
89#1805C2214yes—
90#1802C2211yescharacteristic
91#1797C2212yes—
92C4×#107C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×SD16C2314yes—
96C23×C8C2314yes—
97#533—28no—
98#533—44no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2314yes—
107C2×C22⋊Q8C2314yes—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2314yes—
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—
122C22×C42—21no—
123C22×C42C2312yes—
124#487—44no—
125#474—24no—
126#465—48no—
127#461—44no—
128#461—44no—
129#461—44no—
130C8×D4—44no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8—44no—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152#400—22no—
153#400—22no—
154#399—24no—
155#398—24no—
156#397—22no—
157#397—22no—
158#396—24no—
159#394—22no—
160#394—22no—
161#393—22no—
162#393—22no—
163#391—21no—
164#391—21no—
165#391C2314yes—
166#390—21no—
167#390—22no—
168#390C2312yes—
169#389—21no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—24no—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—24no—
181#388—22no—
182#387—24no—
183#386—22no—
184#386—22no—
185#384—22no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#383—22no—
193#383—22no—
194#382—21no—
195#382—21no—
196#382C2312yes—
197#382C2312yes—
198#381—22no—
199#381—22no—
200#380—22no—
201#380—22no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380C2312yes—
206#380C2312yes—
207#380C2312yes—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227#376—22no—
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8C2414yes—
241C22×D4—22no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C2×SD16—28no—
251C2×SD16—44no—
252C2×SD16—44no—
253C22×C8—44no—
254C22×C8—44no—
255C22×C8—44no—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—24no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8#4114yes—
269#125—48no—
270C22⋊Q8—28no—
271C22⋊Q8—44no—
272C22⋊Q8—48no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275#121—24no—
276#121—42no—
277#121—44no—
278C4×Q8—28no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×Q8—44no—
282C4×Q8—82no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—41no—
293C4×D4—84no—
294#117—48no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—24no—
357#107—42no—
358#106—44no—
359C4⋊C8—22no—
360C4⋊C8—22no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363C4⋊C8—44no—
364C4⋊C8—42no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#102—24no—
373#102—24no—
374C22⋊C8—44no—
375C4×C8—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C24C22×D412yes—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—81no—
409C2×Q8—81no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—82no—
420C2×D4#13714yes—
421C2×D4H(5,𝔽2)12yes—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
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—81no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2511yescharacteristic
461SD16—88no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—88no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—82no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—84no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4C2×#4114yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—48no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—84no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—41no—
530C42—48no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—44no—
536C42#13514yes—
537C42C2×#4112yes—
538C42H(5,𝔽2)12yes—
539C42H(5,𝔽2)12yes—
540C422−1+412yes—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C2×#13714yes—
555C23C23×D411yescharacteristic
556Q8—48no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560Q8—44no—
561Q8—84no—
562D4—28no—
563D4—22no—
564D4—22no—
565D4—41no—
566D4—41no—
567D4—88no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—48no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4#53712yes—
620C2×C4#53712yes—
621C2×C4#54612yes—
622C2×C4#54612yes—
623C2×C4C2×#13512yes—
624C2×C4C2×#13512yes—
625C2×C4C2×#13512yes—
626C2×C4C2×#13712yes—
627C2×C4#57712yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C2×C4#58511yescharacteristic
634C8—42no—
635C8—42no—
636C8—44no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22C2×#53711yescharacteristic
652C22C2×#54611yescharacteristic
653C22#348811yescharacteristic
654C22C22×#13511yescharacteristic
655C22C22×#13711yescharacteristic
656C22C2×#57711yescharacteristic
657C22#358611yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—41no—
662C4—42no—
663C4—44no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—81no—
668C4—81no—
669C4—82no—
670C4—84no—
671C4—82no—
672C4—81no—
673C4#324814yes—
674C4#330314yes—
675C4#351711yescharacteristic
676C4#352211yescharacteristic
677C4#354311yescharacteristic
678C4#355411yescharacteristic
679C2—44no—
680C2—82no—
681C2#3890911yescharacteristic
682C2C2×#324811yescharacteristic
683C2#6092311yescharacteristic
684C2C2×#330311yescharacteristic
685C2#6115711yescharacteristic
686C2#6195011yescharacteristic
687C2#6198011yescharacteristic
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. #33162C2semidirect
4. #31497C2semidirect
5. #30601C2semidirect
6. #29926C2semidirect
7. #29287C2semidirect
8. #28748C2semidirect
9. #28598C2semidirect
10. #26527C2semidirect
12. #25319C2semidirect
13. #24751C2semidirect
14. #24153C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21889C2semidirect
17. C4×#474C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#349432
#331622
#314972
#306012
#299262
#292872
#287482
#285982
#265272
#253852
#253192
#247512
#241532
D4×C4⋊C82
#218892
C4×#4742