Finite Group Databasebeta

← #6793185

Subgroups — #6793185

Order 512.

1727 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33162C211yesmaximal, characteristic, complemented
4#31488C214yesmaximal, complemented
5#30597C214yesmaximal, complemented
6#29922C212yesmaximal, complemented
7#29287C211yesmaximal, characteristic, complemented
8#28748C211yesmaximal, characteristic, complemented
9#28600C212yesmaximal, complemented
10#26528C212yesmaximal, complemented
11#25385C212yesmaximal
12#25320C212yesmaximal, complemented
13#24751C212yesmaximal, complemented
14#24151C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21885C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#529—21no—
20C2×#487C2212yes—
21C2×#476C2211yescharacteristic
22C2×#465C2214yes—
23C2×#461C2212yes—
24C2×#461C2212yes—
25C2×#461C2212yes—
26C2×C8×D4C2212yes—
27C22×C4⋊C8C2212yes—
28#2868C2211yescharacteristic
29#2857C2212yes—
30#2553—22no—
31#2552—21no—
32#2552C2211yescharacteristic
33#2547C2211yescharacteristic
34#2542C2211yescharacteristic
35#2542C2212yes—
36#2538—21no—
37#2537C2211yescharacteristic
38#2496C2212yes—
39#2489—21no—
40#2441—22no—
41#2429C2214yes—
42#2427—21no—
43#2423—22no—
44#2422C2212yes—
45#2395—22no—
46#2365C2212yes—
47#2354—21no—
48Q8×C4⋊C4—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×C22⋊Q8—21no—
52C4×#121C2212yes—
53#2285C2212yes—
54C42×D4C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2096C2214yes—
57#2088C2214yes—
58#2072C2214yes—
59#2066C2214yes—
60#2055C2214yes—
61#2043C2214yes—
62#2035C2214yes—
63#2035C2214yes—
64#2035C2214yes—
65#2000C2214yes—
66#1988C2212yes—
67#1987C2212yes—
68#1987C2211yescharacteristic
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1956C2212yes—
75#1948C2212yes—
76#1948C2212yes—
77#1948C2211yescharacteristic
78#1939C2212yes—
79#1920C2212yes—
80#1892C2214yes—
81#1892C2214yes—
82#1892C2214yes—
83#1880C2214yes—
84#1865C2214yes—
85#1864C2212yes—
86#1850C2214yes—
87#1842C2211yescharacteristic
88#1803C2214yes—
89#1803C2214yes—
90#1802C2211yescharacteristic
91#1797C2212yes—
92C4×#106C2212yes—
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
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—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2314yes—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
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#487—44no—
133#476—24no—
134#465—48no—
135#461—44no—
136#461—44no—
137#461—44no—
138C8×D4—44no—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8—44no—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×C22⋊C8C2314yes—
159C2×C4×C8C2312yes—
160#400—22no—
161#399—24no—
162#399—22no—
163#398—22no—
164#398—24no—
165#397—22no—
166#397—22no—
167#397—22no—
168#396—24no—
169#394—22no—
170#394—22no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391C2314yes—
175#390—22no—
176#390C2312yes—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389—24no—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#388—24no—
188#388—22no—
189#387—24no—
190#387—22no—
191#386—22no—
192#385—22no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#382—22no—
203#380—22no—
204#380—22no—
205#380—22no—
206#380—21no—
207#380C2312yes—
208#380C2312yes—
209#380C2312yes—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—21no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8C2414yes—
242C22×D4—22no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C2×SD16—28no—
252C2×SD16—44no—
253C2×SD16—44no—
254C22×C8—44no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—24no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8#4114yes—
270#126—48no—
271#125—48no—
272#125—44no—
273#124—44no—
274#123—48no—
275C22⋊Q8—28no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—48no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282#121—24no—
283#121—42no—
284#121—44no—
285C4×Q8—28no—
286C4×Q8—44no—
287C4×Q8—48no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—42no—
298C4×D4—41no—
299C4×D4—48no—
300C4×D4SD1618yescomplemented
301#117—48no—
302#117—48no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—44no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2412yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#106—24no—
369#106—42no—
370#106—44no—
371C4⋊C8—22no—
372C4⋊C8—22no—
373C4⋊C8—24no—
374C4⋊C8—24no—
375C4⋊C8—44no—
376C4⋊C8—42no—
377#103—44no—
378#103—44no—
379#103—44no—
380#103—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384#102—24no—
385#102—24no—
386C22⋊C8—44no—
387C4×C8—24no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24—21no—
405C24C22×D412yes—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—82no—
428C2×D4C2×SD1614yes—
429C2×D4H(5,𝔽2)12yes—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—81no—
458C22×C4C2×SD1618yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2511yescharacteristic
469SD16—88no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—88no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C4⋊C4—84no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4C2×SD1614yes—
518C4⋊C4C2×#4114yes—
519C22⋊C4—24no—
520C22⋊C4—42no—
521C22⋊C4—48no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—84no—
532C22⋊C4C2×SD1618yes—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—21no—
540C42—21no—
541C42—41no—
542C42—48no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—44no—
547C42—44no—
548C42C2×SD1614yes—
549C42C2×#4112yes—
550C42H(5,𝔽2)12yes—
551C42H(5,𝔽2)12yes—
552C422−1+412yes—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23C22×SD1614yes—
567C23C23×D411yescharacteristic
568Q8—48no—
569Q8—44no—
570Q8—44no—
571Q8—44no—
572Q8—44no—
573Q8—84no—
574D4—28no—
575D4—22no—
576D4—22no—
577D4—41no—
578D4—41no—
579D4—88no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—216no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4#53712yes—
631C2×C4#53712yes—
632C2×C4#54612yes—
633C2×C4#54612yes—
634C2×C4C22×SD1612yes—
635C2×C4C22×SD1614yes—
636C2×C4C22×SD1612yes—
637C2×C4C22×SD1612yes—
638C2×C4#57512yes—
639C2×C4#57812yes—
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×2−1+411yescharacteristic
645C2×C4#58511yescharacteristic
646C8—42no—
647C8—42no—
648C8—44no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22C2×#53711yescharacteristic
664C22C2×#54611yescharacteristic
665C22#348811yescharacteristic
666C22C23×SD1611yescharacteristic
667C22C2×#57511yescharacteristic
668C22C2×#57811yescharacteristic
669C22#358611yescharacteristic
670C4—24no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—41no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—82no—
679C4—81no—
680C4—81no—
681C4—82no—
682C4—81no—
683C4—82no—
684C4—84no—
685C4#324614yes—
686C4#329814yes—
687C4#351711yescharacteristic
688C4#352211yescharacteristic
689C4#354311yescharacteristic
690C4#355411yescharacteristic
691C2—44no—
692C2—82no—
693C2#3890911yescharacteristic
694C2C2×#324611yescharacteristic
695C2#6092511yescharacteristic
696C2C2×#329811yescharacteristic
697C2#6115911yescharacteristic
698C2#6194811yescharacteristic
699C2#6198511yescharacteristic
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. #31488C2semidirect
5. #30597C2semidirect
6. #29922C2semidirect
7. #29287C2semidirect
8. #28748C2semidirect
9. #28600C2semidirect
10. #26528C2semidirect
12. #25320C2semidirect
13. #24751C2semidirect
14. #24151C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21885C2semidirect
17. C4×#476C2semidirect
300. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#361342
#349322
#331622
#314882
#305972
#299222
#292872
#287482
#286002
#265282
#253852
#253202
#247512
#241512
D4×C4⋊C82
#218852
C4×#4762