Finite Group Databasebeta

← #6800726

Subgroups — #6800726

Order 512.

1815 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#127C211yesmaximal, characteristic, complemented
2#34945C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic, complemented
4#32448C214yesmaximal, complemented
5#31654C214yesmaximal, complemented
6#29984C212yesmaximal, complemented
7#29148C212yesmaximal, complemented
8#28717C211yesmaximal, characteristic, complemented
9#28414C212yesmaximal, complemented
10#28252C211yesmaximal, characteristic, complemented
11#28158C212yesmaximal
12#27736C212yesmaximal, complemented
13#25929C212yesmaximal, complemented
14#24326C212yesmaximal, complemented
15#22825C211yesmaximal, characteristic, complemented
16#22153C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic
18C2×D4×Q8—21no—
19C2×D4×Q8—21no—
20C2×D4×Q8C2212yes—
21C2×#487C2212yes—
22C2×#474C2212yes—
23C2×#460C2212yes—
24C2×#461C2212yes—
25C2×#459C2211yescharacteristic
26C2×C4×D8C2212yes—
27#2868C2212yes—
28#2868C2211yescharacteristic
29#2573—22no—
30#2571C2211yescharacteristic
31#2568C2211yescharacteristic
32#2547C2211yescharacteristic
33#2542—22no—
34#2542—22no—
35#2542—21no—
36#2542—21no—
37#2542C2212yes—
38#2542C2214yes—
39#2538C2211yescharacteristic
40#2514C2212yes—
41#2494—21no—
42#2447C2212yes—
43#2421—21no—
44#2401—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#127C2212yes—
48C4×#124C2212yes—
49C4×C22⋊Q8—21no—
50#2295C2212yes—
51#2283—21no—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2100C2214yes—
55#2095C2214yes—
56#2073C2214yes—
57#2072C2214yes—
58#2057C2214yes—
59#2055C2214yes—
60#2024C2214yes—
61#2024C2214yes—
62#2023C2214yes—
63#2012C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1973C2214yes—
69#1959C2214yes—
70#1947C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1936C2214yes—
74#1925C2212yes—
75#1917C2214yes—
76#1879C2214yes—
77#1879C2212yes—
78#1877C2214yes—
79#1877C2214yes—
80#1862C2214yes—
81#1842C2211yescharacteristic
82#1836C2214yes—
83#1831C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2211yescharacteristic
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C23×Q8—22no—
96C22×SD16C2314yes—
97C22×D8C2312yes—
98D4×Q8—44no—
99D4×Q8—44no—
100D4×Q8—48no—
101D4×Q8—48no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×#127C2311yescharacteristic
110C2×#124C2312yes—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2314yes—
131C22×C42—21no—
132#487—44no—
133#474—44no—
134#461—44no—
135#460—28no—
136#459—24no—
137C4×D8—44no—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#401—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#399—24no—
161#399—24no—
162#399—24no—
163#397—22no—
164#397—22no—
165#397—22no—
166#396—22no—
167#391—21no—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#390—22no—
173#390—21no—
174#390—21no—
175#390—22no—
176#390C2312yes—
177#390C2314yes—
178#390C2312yes—
179#389—21no—
180#389—21no—
181#389—21no—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#385—22no—
190#384—22no—
191#384—21no—
192#384—21no—
193#384—22no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2314yes—
199#383—22no—
200#382—21no—
201#382—21no—
202#382C2312yes—
203#381—22no—
204#380—21no—
205#380—21no—
206#380—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224#376—22no—
225C43C2311yescharacteristic
226C43C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C22×Q8—24no—
239C22×Q8—22no—
240C22×Q8—48no—
241C22×Q8—48no—
242C22×Q8—48no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2412yes—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2412yes—
248C22×D4C2411yescharacteristic
249C23×C4—24no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C2×SD16—28no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×D8—24no—
258C2×D8—24no—
259C2×D8—42no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—48no—
270C4⋊Q8—44no—
271C4⋊Q8—48no—
272C4⋊Q8—44no—
273#127—22no—
274#127#4114yes—
275#124—24no—
276#124—42no—
277C22⋊Q8—48no—
278C22⋊Q8—416no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C22⋊Q8—416no—
282C22⋊Q8—416no—
283C22⋊Q8—44no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—41no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—48no—
301C4×D4—44no—
302C4×D4—48no—
303C4×D4—44no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—44no—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—42no—
365#107—44no—
366C4⋊C8—22no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8—22no—
370C4⋊C8—42no—
371#103—44no—
372#103—44no—
373#103—44no—
374#102—24no—
375#102—28no—
376#102—42no—
377#102—42no—
378#102—42no—
379#102—44no—
380#97—44no—
381C4×C8—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—21no—
401C24—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—44no—
413C2×Q8—48no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—832no—
417C2×Q8AGL(1,ℤ/8)14yes—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4C2×#4114yes—
435C2×D42−1+412yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—42no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—81no—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2511yescharacteristic
469SD16—88no—
470D8—48no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—82no—
528C42—24no—
529C42—22no—
530C42—24no—
531C42—22no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—22no—
537C42—22no—
538C42—21no—
539C42—41no—
540C42—42no—
541C42—44no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42AGL(1,ℤ/8)14yes—
546C42C2×#4112yes—
547C42H(5,𝔽2)12yes—
548C42H(5,𝔽2)12yes—
549C422−1+412yes—
550C23—24no—
551C23—22no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23C23×D411yescharacteristic
565Q8—28no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—816no—
571Q8—88no—
572D4—24no—
573D4—22no—
574D4—24no—
575D4—22no—
576D4—48no—
577D4—48no—
578D4—41no—
579D4—41no—
580D4—42no—
581D4—42no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—84no—
624C2×C4—816no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—88no—
628C2×C4#53512yes—
629C2×C4#53512yes—
630C2×C4#54812yes—
631C2×C4#54812yes—
632C2×C4C2×AGL(1,ℤ/8)14yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4#57812yes—
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×2−1+411yescharacteristic
642C2×C4C2×2−1+411yescharacteristic
643C2×C4#58511yescharacteristic
644C8—44no—
645C8—42no—
646C8—42no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—82no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—82no—
658C22—81no—
659C22—82no—
660C22—82no—
661C22C2×#53511yescharacteristic
662C22C2×#54811yescharacteristic
663C22#348911yescharacteristic
664C22C22×AGL(1,ℤ/8)11yescharacteristic
665C22C22×AGL(1,ℤ/8)11yescharacteristic
666C22C2×#57811yescharacteristic
667C22#358611yescharacteristic
668C4—24no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—41no—
675C4—42no—
676C4—81no—
677C4—84no—
678C4—82no—
679C4—81no—
680C4—82no—
681C4—81no—
682C4—82no—
683C4#321714yes—
684C4#336714yes—
685C4#351411yescharacteristic
686C4#352011yescharacteristic
687C4#354711yescharacteristic
688C4#355211yescharacteristic
689C2—84no—
690C2—82no—
691C2#3912411yescharacteristic
692C2C2×#321711yescharacteristic
693C2#6083711yescharacteristic
694C2C2×#336711yescharacteristic
695C2#6131711yescharacteristic
696C2#6210711yescharacteristic
697C2#6214811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
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. Q8×#127C2semidirect
2. #34945C2semidirect
3. #33195C2semidirect
4. #32448C2semidirect
5. #31654C2semidirect
6. #29984C2semidirect
7. #29148C2semidirect
8. #28717C2semidirect
9. #28414C2semidirect
10. #28252C2semidirect
12. #27736C2semidirect
13. #25929C2semidirect
14. #24326C2semidirect
15. #22825C2semidirect
16. #22153C2semidirect

Maximal subgroups

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

subgroupindex
Q8×#1272
#349452
#331952
#324482
#316542
#299842
#291482
#287172
#284142
#282522
#281582
#277362
#259292
#243262
#228252
#221532
C4×#4592