Finite Group Databasebeta

← #6848875

Subgroups — #6848875

Order 512.

1577 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#34929C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#32124C214yesmaximal, complemented
5#30929C214yesmaximal, complemented
6#29664C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#28255C211yesmaximal, characteristic, complemented
9#28151C212yesmaximal
10#27631C212yesmaximal, complemented
11#26022C212yesmaximal, complemented
12#25394C212yesmaximal
13#24760C212yesmaximal, complemented
14#24632C212yesmaximal, complemented
15#22650C211yesmaximal, characteristic
16#21894C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#475C2212yes—
20C2×#470C2214yes—
21C2×#462C2211yescharacteristic
22C2×#462C2212yes—
23C2×#462C2212yes—
24C2×C4×Q16C2212yes—
25C2×#436C2212yes—
26C2×#423C2212yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2571—21no—
30#2563—21no—
31#2556—22no—
32#2551C2211yescharacteristic
33#2546C2211yescharacteristic
34#2542C2211yescharacteristic
35#2538C2212yes—
36#2538C2211yescharacteristic
37#2506—21no—
38#2497—22no—
39#2492C2214yes—
40#2492C2212yes—
41#2452—21no—
42#2439—22no—
43#2429C2212yes—
44#2396—22no—
45#2369—21no—
46#2355C2212yes—
47#2354—21no—
48C4×C4⋊Q8—21no—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×C22⋊Q8C2212yes—
52#2293—21no—
53#2285C2212yes—
54C42×Q8C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2075C2214yes—
57#2066C2214yes—
58#2056C2214yes—
59#2056C2214yes—
60#2035C2214yes—
61#2024C2214yes—
62#2024C2214yes—
63#2016C2214yes—
64#2001C2214yes—
65#2001C2214yes—
66#1988C2212yes—
67#1988C2214yes—
68#1981C2214yes—
69#1981C2211yescharacteristic
70#1956C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2211yescharacteristic
74#1939C2212yes—
75#1928C2214yes—
76#1928C2212yes—
77#1919C2214yes—
78#1894C2214yes—
79#1894C2214yes—
80#1894C2214yes—
81#1878C2212yes—
82#1851C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1803C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×Q16C2314yes—
97C22×M16C2314yes—
98#533—28no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#125—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2314yes—
114C2×C4×Q8—22no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—22no—
125C22×C42C2312yes—
126#475—44no—
127#470—48no—
128#462—24no—
129#462—44no—
130#462—44no—
131C4×Q16—44no—
132#436—44no—
133C2×#107C2312yes—
134C2×#106C2314yes—
135#423—44no—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2312yes—
153#400—24no—
154#400—22no—
155#399—22no—
156#398—22no—
157#398—22no—
158#396—22no—
159#396—22no—
160#396—24no—
161#395—22no—
162#393—24no—
163#393—22no—
164#391—22no—
165#391—24no—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#391C2314yes—
170#390—22no—
171#390—21no—
172#390—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389C2312yes—
186#388—24no—
187#388—22no—
188#388—22no—
189#387—22no—
190#387—24no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#382—22no—
200#382—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227C43—21no—
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8C2412yes—
241C22×Q8C2412yes—
242C22×Q8C2411yescharacteristic
243C22×D4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C2×Q16—24no—
248C2×Q16—42no—
249C2×Q16—42no—
250C2×Q16—44no—
251C2×Q16—44no—
252C2×Q16—44no—
253C2×M16—44no—
254C2×M16—44no—
255C2×M16—44no—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—82no—
272#125—48no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—42no—
277C22⋊Q8—42no—
278C22⋊Q8—42no—
279C22⋊Q8—48no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—24no—
288C4×Q8—22no—
289C4×Q8—24no—
290C4×Q8—48no—
291C4×Q8—41no—
292C4×Q8#4114yes—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—22no—
296C4×D4—22no—
297#117—48no—
298#117—84no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
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⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—21no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—24no—
353C2×C42—21no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—42no—
365#106—28no—
366C4⋊C8—22no—
367C4⋊C8—22no—
368C4⋊C8—24no—
369C4⋊C8—24no—
370C4⋊C8—44no—
371C4⋊C8—42no—
372#103—24no—
373#103—24no—
374#103—44no—
375#103—42no—
376#103—42no—
377#103—44no—
378#103—44no—
379#103—44no—
380#103—44no—
381C22⋊C8—44no—
382C4×C8—24no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24C22×D412yes—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—82no—
423C2×Q8—82no—
424C2×Q8—82no—
425C2×Q8C2×#4112yes—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460Q16—48no—
461Q16—88no—
462M16—88no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—81no—
507C4⋊C4—84no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4#13714yes—
511C4⋊C4C2×#4112yes—
512C4⋊C4H(5,𝔽2)12yes—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—82no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C42—24no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—21no—
530C42—21no—
531C42—41no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42—44no—
536C42—42no—
537C42—42no—
538C42—81no—
539C42—84no—
540C42#13714yes—
541C42C2×#4112yes—
542C42H(5,𝔽2)12yes—
543C42H(5,𝔽2)12yes—
544C422−1+412yes—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23C23×D411yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—41no—
556Q8—41no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—88no—
566Q8—164no—
567D4—44no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4#53712yes—
619C2×C4#53712yes—
620C2×C4#54612yes—
621C2×C4#54612yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13714yes—
624C2×C4C2×#13712yes—
625C2×C4C2×#13712yes—
626C2×C4C2×#13712yes—
627C2×C4#57812yes—
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—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22C2×#53711yescharacteristic
645C22C2×#54611yescharacteristic
646C22#348811yescharacteristic
647C22C22×#13711yescharacteristic
648C22C22×#13711yescharacteristic
649C22C2×#57811yescharacteristic
650C22#358511yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—24no—
655C4—42no—
656C4—42no—
657C4—41no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—82no—
662C4—81no—
663C4—84no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4#325514yes—
668C4#332814yes—
669C4#350611yescharacteristic
670C4#351111yescharacteristic
671C4#354711yescharacteristic
672C4#354911yescharacteristic
673C2—82no—
674C2#3946911yescharacteristic
675C2C2×#325511yescharacteristic
676C2#6093011yescharacteristic
677C2C2×#332811yescharacteristic
678C2#6116211yescharacteristic
679C2#6196511yescharacteristic
680C2#6200011yescharacteristic
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. #33196C2semidirect
4. #32124C2semidirect
5. #30929C2semidirect
6. #29664C2semidirect
7. #29141C2semidirect
8. #28255C2semidirect
10. #27631C2semidirect
11. #26022C2semidirect
13. #24760C2semidirect
14. #24632C2semidirect
16. #21894C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349292
#331962
#321242
#309292
#296642
#291412
#282552
#281512
#276312
#260222
#253942
#247602
#246322
#226502
#218942
C4×#4622