Finite Group Databasebeta

← #6801017

Subgroups — #6801017

Order 512.

1625 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#36563C211yesmaximal, characteristic, complemented
2#34934C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#31804C214yesmaximal, complemented
5#31156C214yesmaximal, complemented
6#29998C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic, complemented
8#29121C212yesmaximal, complemented
9#28252C212yesmaximal, complemented
10#28252C211yesmaximal, characteristic, complemented
11#28186C212yesmaximal, complemented
12#26108C212yesmaximal, complemented
13#25295C212yesmaximal, complemented
14#24728C212yesmaximal
15Q8×C4⋊C8C211yesmaximal, characteristic, complemented
16#22146C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#476C2212yes—
20C2×#460C2214yes—
21C2×#460C2212yes—
22C2×#461C2212yes—
23C2×#461C2211yescharacteristic
24C2×C8×Q8C2212yes—
25C2×C4×SD16C2212yes—
26#2868C2211yescharacteristic
27#2859C2212yes—
28#2582C2212yes—
29#2567—21no—
30#2554—22no—
31#2552—21no—
32#2552C2211yescharacteristic
33#2550—22no—
34#2548C2211yescharacteristic
35#2541C2211yescharacteristic
36#2538C2211yescharacteristic
37#2512C2212yes—
38#2507—22no—
39#2490—21no—
40#2451C2214yes—
41#2445C2212yes—
42#2426—21no—
43#2392—22no—
44#2369—21no—
45#2355—21no—
46Q8×C4⋊C4C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2212yes—
50C4×#127C2211yescharacteristic
51C4×#121—21no—
52#2293C2212yes—
53C42×Q8C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2099C2214yes—
56#2088C2214yes—
57#2088C2214yes—
58#2065C2214yes—
59#2065C2214yes—
60#2055C2214yes—
61#2055C2214yes—
62#2055C2214yes—
63#2041C2214yes—
64#2023C2214yes—
65#1987C2212yes—
66#1981C2211yescharacteristic
67#1968C2214yes—
68#1957C2212yes—
69#1948C2211yescharacteristic
70#1948C2212yes—
71#1948C2212yes—
72#1939C2214yes—
73#1939C2212yes—
74#1934C2214yes—
75#1934C2214yes—
76#1933C2214yes—
77#1927C2212yes—
78#1880C2214yes—
79#1877C2212yes—
80#1865C2214yes—
81#1842C2212yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1803C2212yes—
85#1803C2214yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2211yescharacteristic
92C4×C4⋊C8C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96C22×SD16C2314yes—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#127C2312yes—
103#529—48no—
104C2×#126—22no—
105C2×#125—21no—
106C2×#125—22no—
107C2×#124—21no—
108C2×#123—22no—
109C2×C22⋊Q8—21no—
110C2×#121—21no—
111C2×#121—21no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C22×C42—21no—
127#476—44no—
128#461—24no—
129#461—28no—
130#460—48no—
131#460—44no—
132C8×Q8—44no—
133C4×SD16—44no—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2314yes—
154#398—24no—
155#398—22no—
156#398—24no—
157#397—22no—
158#396—22no—
159#396—22no—
160#395—24no—
161#394—22no—
162#394—22no—
163#394—22no—
164#392—22no—
165#391—22no—
166#391—24no—
167#391C2314yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—22no—
172#390—22no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#389—22no—
180#389—21no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—24no—
188#389C2314yes—
189#389C2314yes—
190#389C2312yes—
191#389C2312yes—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—24no—
196#386—22no—
197#384—22no—
198#384—21no—
199#384—21no—
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2311yescharacteristic
203#384C2312yes—
204#384C2312yes—
205#382—22no—
206#380—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C43C2312yes—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4C2412yes—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—28no—
252C2×SD16—42no—
253C2×SD16—44no—
254C2×SD16—44no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—22no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263C4⋊Q8#4114yes—
264#127—24no—
265#127—22no—
266#127—22no—
267#127—42no—
268#126—48no—
269#125—44no—
270#125—84no—
271#124—44no—
272#123—48no—
273C22⋊Q8—44no—
274#121—44no—
275#121—44no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—42no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×Q8SD1618yescomplemented
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—24no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—24no—
291C4×D4—41no—
292C4×D4—44no—
293C4×D4—48no—
294C4×D4—84no—
295C4×D4—82no—
296#117—48no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—44no—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#106—42no—
362#106—44no—
363C4⋊C8—22no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—22no—
367C4⋊C8—44no—
368C4⋊C8—44no—
369C4⋊C8—42no—
370#103—28no—
371#103—24no—
372#103—42no—
373#103—42no—
374#102—28no—
375#102—44no—
376#102—42no—
377#102—44no—
378#102—44no—
379C4×C8—24no—
380C4×C8—44no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8C2×SD1614yes—
408C2×Q8C2×#4114yes—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—81no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—81no—
432C2×D4H(5,𝔽2)12yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—81no—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2511yescharacteristic
463SD16—48no—
464SD16—88no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4C2×SD1614yes—
510C4⋊C4C2×SD1618yes—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—22no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—21no—
530C42—22no—
531C42—22no—
532C42—21no—
533C42—22no—
534C42—41no—
535C42—42no—
536C42—42no—
537C42—44no—
538C42—42no—
539C42—42no—
540C42—42no—
541C42—84no—
542C42C2×SD1614yes—
543C42C2×SD1618yes—
544C42C2×#4112yes—
545C42H(5,𝔽2)12yes—
546C422−1+412yes—
547C422−1+412yes—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23C23×D411yescharacteristic
563Q8—28no—
564Q8—24no—
565Q8—24no—
566Q8—42no—
567Q8—42no—
568Q8—164no—
569D4—22no—
570D4—22no—
571D4—41no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—41no—
576D4—42no—
577D4—88no—
578D4—84no—
579D4—88no—
580D4—84no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—88no—
624C2×C4—84no—
625C2×C4#53712yes—
626C2×C4#53712yes—
627C2×C4#54812yes—
628C2×C4#54812yes—
629C2×C4C22×SD1612yes—
630C2×C4C22×SD1614yes—
631C2×C4C22×SD1614yes—
632C2×C4C22×SD1612yes—
633C2×C4C22×SD1612yes—
634C2×C4#57512yes—
635C2×C4#57612yes—
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C2×C4C2×2−1+411yescharacteristic
641C2×C4#58511yescharacteristic
642C8—42no—
643C8—44no—
644C8—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22C2×#53711yescharacteristic
660C22C2×#54811yescharacteristic
661C22#348811yescharacteristic
662C22C23×SD1611yescharacteristic
663C22C2×#57511yescharacteristic
664C22C2×#57611yescharacteristic
665C22#358611yescharacteristic
666C4—22no—
667C4—22no—
668C4—24no—
669C4—22no—
670C4—24no—
671C4—42no—
672C4—42no—
673C4—41no—
674C4—44no—
675C4—82no—
676C4—84no—
677C4—81no—
678C4—82no—
679C4—81no—
680C4—81no—
681C4#324614yes—
682C4#336214yes—
683C4#351511yescharacteristic
684C4#351911yescharacteristic
685C4#354411yescharacteristic
686C4#354711yescharacteristic
687C2—82no—
688C2—162no—
689C2#3912611yescharacteristic
690C2C2×#324611yescharacteristic
691C2#6092811yescharacteristic
692C2C2×#336211yescharacteristic
693C2#6132011yescharacteristic
694C2#6194811yescharacteristic
695C2#6199511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36563C2semidirect
3. #33197C2semidirect
4. #31804C2semidirect
5. #31156C2semidirect
6. #29998C2semidirect
7. #29139C2semidirect
8. #29121C2semidirect
9. #28252C2semidirect
10. #28252C2semidirect
11. #28186C2semidirect
12. #26108C2semidirect
13. #25295C2semidirect
15. Q8×C4⋊C8C2semidirect
16. #22146C2semidirect
17. C4×#461C2semidirect
283. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#365632
#349342
#331972
#318042
#311562
#299982
#291392
#291212
#282522
#281862
#261082
#252952
#247282
Q8×C4⋊C82
#221462
C4×#4612