Finite Group Databasebeta

← #6793401

Subgroups — #6793401

Order 512.

1749 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic
2#34945C211yesmaximal, characteristic
3#33163C211yesmaximal, characteristic, complemented
4#31312C214yesmaximal, complemented
5#31006C214yesmaximal, complemented
6#29926C212yesmaximal, complemented
7#28748C211yesmaximal, characteristic, complemented
8#28417C212yesmaximal, complemented
9#28283C211yesmaximal, characteristic, complemented
10#27757C212yesmaximal, complemented
11#25884C212yesmaximal
12#24660C212yesmaximal, complemented
13#24243C212yesmaximal
14#24212C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic
16#21811C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×D4×Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C2×#487C2212yes—
21C2×#475C2212yes—
22C2×#462C2211yescharacteristic
23C2×#462C2212yes—
24C2×#462C2212yes—
25C2×#462C2212yes—
26C2×#451C2214yes—
27C2×C4×Q16C2212yes—
28C2×#432C2212yes—
29#2868C2211yescharacteristic
30#2857C2212yes—
31#2573—22no—
32#2571C2211yescharacteristic
33#2568C2211yescharacteristic
34#2551—21no—
35#2551—21no—
36#2547C2211yescharacteristic
37#2546C2212yes—
38#2538C2211yescharacteristic
39#2514—21no—
40#2494C2212yes—
41#2447—21no—
42#2440—22no—
43#2440—22no—
44#2434C2214yes—
45#2421C2212yes—
46#2401—22no—
47D4×C4⋊C4—21no—
48D4×C4⋊C4—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×#124—21no—
52C4×C22⋊Q8C2212yes—
53#2295—21no—
54#2283C2212yes—
55C42×Q8C2211yescharacteristic
56C42×D4C2211yescharacteristic
57#2097C2214yes—
58#2074C2214yes—
59#2066C2214yes—
60#2043C2214yes—
61#2043C2214yes—
62#2036C2214yes—
63#2035C2214yes—
64#2035C2214yes—
65#2016C2214yes—
66#2001C2214yes—
67#1981C2212yes—
68#1981C2211yescharacteristic
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1971C2214yes—
75#1960C2214yes—
76#1960C2214yes—
77#1956C2212yes—
78#1954C2212yes—
79#1948C2212yes—
80#1947C2211yescharacteristic
81#1935C2214yes—
82#1928C2212yes—
83#1924C2214yes—
84#1878C2212yes—
85#1842C2211yescharacteristic
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1797C2212yes—
89#1787C2214yes—
90#1787C2214yes—
91#1787C2214yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C23×Q8C2314yes—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98D4×Q8—44no—
99D4×Q8—44no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—28no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×#124—21no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2314yes—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4—22no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C42C2312yes—
127#487—44no—
128#475—44no—
129#462—24no—
130#462—44no—
131#462—44no—
132#462—44no—
133#451—48no—
134C4×Q16—44no—
135#432—44no—
136C2×#107C2312yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×#97C2314yes—
159C2×C4×C8C2312yes—
160#401—22no—
161#400—22no—
162#400—22no—
163#400—22no—
164#398—24no—
165#397—22no—
166#397—22no—
167#397—22no—
168#396—22no—
169#395—24no—
170#395—24no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391C2312yes—
176#391C2312yes—
177#391C2314yes—
178#390—21no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—22no—
187#389—22no—
188#389C2312yes—
189#389C2312yes—
190#388—22no—
191#388—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#385—22no—
196#385—24no—
197#385—24no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384—21no—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#383—22no—
205#382—21no—
206#382C2312yes—
207#382C2312yes—
208#381—22no—
209#380—22no—
210#380—22no—
211#380C2312yes—
212#380C2312yes—
213#380C2312yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4D412yes—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4C2312yes—
233#376—22no—
234C43—21no—
235C43C2311yescharacteristic
236#340C2314yes—
237#340C2312yes—
238#340C2312yes—
239#336C2314yes—
240#336C2312yes—
241C22×Q8—21no—
242C22×Q8—24no—
243C22×Q8—28no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8—44no—
249C22×Q8C2414yes—
250C22×Q8C2412yes—
251C22×Q8C2411yescharacteristic
252C22×D4C2×D411yescharacteristic
253C23×C4—24no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C2×Q16—24no—
258C2×Q16—24no—
259C2×Q16—28no—
260C2×Q16—42no—
261C2×Q16—44no—
262C2×Q16—44no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—28no—
267C4⋊Q8—24no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—48no—
275C4⋊Q8—48no—
276C4⋊Q8—44no—
277C4⋊Q8—42no—
278C4⋊Q8—42no—
279C4⋊Q8—48no—
280C4⋊Q8#4114yes—
281#124—44no—
282C22⋊Q8—24no—
283C22⋊Q8—416no—
284C22⋊Q8—42no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—41no—
295C4×Q8—44no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—48no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—48no—
325C2×C4⋊C4—48no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—42no—
364#106—44no—
365C4⋊C8—22no—
366C4⋊C8—22no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8—42no—
370#103—24no—
371#103—42no—
372#103—44no—
373#103—42no—
374#103—42no—
375#103—44no—
376#103—44no—
377#103—44no—
378#103—44no—
379#103—44no—
380#103—44no—
381#103—44no—
382#103—44no—
383C22⋊C8—44no—
384C22⋊C8—44no—
385#97—44no—
386C4×C8—24no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24C22×D412yes—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—24no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—48no—
427C2×Q8—48no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—84no—
432C2×Q8—88no—
433C2×Q8—88no—
434C2×Q8—88no—
435C2×Q8—88no—
436C2×Q8C2×#4114yes—
437C2×Q8H(5,𝔽2)12yes—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4#13714yes—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—84no—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C2511yescharacteristic
475Q16—48no—
476Q16—88no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—81no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—44no—
528C22⋊C4—48no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—82no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—21no—
540C42—41no—
541C42—42no—
542C42—44no—
543C42—42no—
544C42—42no—
545C42—44no—
546C42—42no—
547C42—42no—
548C42—81no—
549C42#13714yes—
550C42C2×#4112yes—
551C42H(5,𝔽2)12yes—
552C42H(5,𝔽2)12yes—
553C42H(5,𝔽2)12yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23C2×#13714yes—
561C23C23×D411yescharacteristic
562Q8—24no—
563Q8—22no—
564Q8—24no—
565Q8—22no—
566Q8—41no—
567Q8—42no—
568Q8—41no—
569Q8—42no—
570Q8—44no—
571Q8—44no—
572Q8—44no—
573Q8—44no—
574Q8—816no—
575Q8—84no—
576D4—28no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—816no—
629C2×C4#53512yes—
630C2×C4#53512yes—
631C2×C4#54612yes—
632C2×C4#54612yes—
633C2×C4C2×#13712yes—
634C2×C4C2×#13712yes—
635C2×C4C2×#13712yes—
636C2×C4C2×#13712yes—
637C2×C4#57812yes—
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4#58511yescharacteristic
644C8—42no—
645C8—44no—
646C8—42no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22C2×#53511yescharacteristic
655C22C2×#54611yescharacteristic
656C22#348811yescharacteristic
657C22C22×#13711yescharacteristic
658C22C22×#13711yescharacteristic
659C22C2×#57811yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—41no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—42no—
670C4—82no—
671C4—81no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4—81no—
676C4—84no—
677C4#321814yes—
678C4#332814yes—
679C4#351411yescharacteristic
680C4#351911yescharacteristic
681C4#354311yescharacteristic
682C4#354611yescharacteristic
683C2—44no—
684C2#3891011yescharacteristic
685C2C2×#321811yescharacteristic
686C2#6083711yescharacteristic
687C2C2×#332811yescharacteristic
688C2#6115711yescharacteristic
689C2#6196411yescharacteristic
690C2#6198411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #33163C2semidirect
4. #31312C2semidirect
5. #31006C2semidirect
6. #29926C2semidirect
7. #28748C2semidirect
8. #28417C2semidirect
9. #28283C2semidirect
10. #27757C2semidirect
12. #24660C2semidirect
14. #24212C2semidirect
16. #21811C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#349452
#331632
#313122
#310062
#299262
#287482
#284172
#282832
#277572
#258842
#246602
#242432
#242122
#227992
#218112
C4×#4622