Finite Group Databasebeta

← #6566752

Subgroups — #6566752

Order 512.

1919 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34310C211yesmaximal, characteristic
3#30113C212yesmaximal, complemented
4#29980C211yesmaximal, characteristic, complemented
5#28508C211yesmaximal, characteristic, complemented
6#28311C212yesmaximal, complemented
7#28160C211yesmaximal, characteristic, complemented
8#27582C212yesmaximal, complemented
9#27245C214yesmaximal, complemented
10#26803C214yesmaximal, complemented
11#25016C212yesmaximal, complemented
12#24240C212yesmaximal
13#23761C212yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23101C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22811C212yesmaximal, complemented
18C2×D42C2211yescharacteristic
19C2×#518—21no—
20C2×#484C2214yescomplemented
21C2×#474C2214yescomplemented
22C2×#460C2212yes—
23C2×#450C2214yes—
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic, complemented
26#2857C2212yescomplemented
27#2571—21no—
28#2563—21no—
29#2538C2212yes—
30#2497—22no—
31#2492C2214yescomplemented
32#2439—22no—
33#2369—21no—
34#2366C2211yescharacteristic
35#2363—21no—
36#2361C2212yes—
37#2358—22no—
38#2355C2212yescomplemented
39#2354—21no—
40#2354C2211yescharacteristic
41D4×C4⋊C4C2211yescharacteristic
42D4×C22⋊C4—21no—
43D4×C22⋊C4C2212yes—
44#2324—21no—
45#2321—22no—
46C4×C4⋊Q8C2211yescharacteristic, complemented
47C4×#127C2211yescharacteristic
48C4×#121C2212yes—
49#2293C2211yescharacteristic
50C2×#390C2212yes—
51#2285C2212yescomplemented
52C42×D4C2211yescharacteristic
53#2099C2212yescomplemented
54#2093C2212yes—
55#2093C2212yescomplemented
56#2093C2212yescomplemented
57#2084C2211yescharacteristic, complemented
58#2076C2214yescomplemented
59#2071C2211yescharacteristic
60#2071C2212yescomplemented
61#2066C2212yescomplemented
62#2057C2212yescomplemented
63#2055C2212yes—
64#2043C2211yescharacteristic, complemented
65#2036C2214yescomplemented
66#2034C2214yes—
67#2023C2212yes—
68#2013C2211yescharacteristic
69#2013C2212yescomplemented
70#2000C2214yes—
71#1973C2214yes—
72#1973C2214yescomplemented
73#1957C2212yes—
74#1954C2212yes—
75#1947C2212yescomplemented
76#1928C2214yescomplemented
77#1925C2214yes—
78#1925C2214yescomplemented
79#1924C2214yescomplemented
80#1877C2214yescomplemented
81#1877C2214yescomplemented
82#1852C2214yescomplemented
83#1845C2212yescomplemented
84#1831C2211yescharacteristic, complemented
85#1817C2212yescomplemented
86#1803C2212yescomplemented
87#1802C2212yes—
88#1786C2214yes—
89#1776C2214yescomplemented
90#1770C2212yescomplemented
91C4×C4⋊C8C2211yescharacteristic, complemented
92#1762C2214yescomplemented
93C4×#103C2212yescomplemented
94#1749C2212yescomplemented
95C23×D4—21no—
96C23×D4C2312yes—
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99D42—24no—
100D42—24no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×C22⋊Q8C2314yes—
107C2×#121—21no—
108C2×#121C2312yes—
109C2×#120—22no—
110#518—48no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C2×#117—21no—
126C2×#117—21no—
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—22no—
129C22×C22⋊C4—22no—
130C22×C22⋊C4—22no—
131C22×C42C2312yes—
132#484—48no—
133#474—48no—
134#460—28no—
135#450—48no—
136C4×SD16—28no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#400—24no—
157#396—24no—
158#393—24no—
159#391—24no—
160#391C2312yes—
161#391C2314yes—
162#390—22no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—22no—
167#390—22no—
168#390C2312yes—
169#390C2311yescharacteristic
170#390C2311yescharacteristic
171#390C2312yes—
172#390C2312yes—
173#389—22no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389C2311yescharacteristic
180#389C2311yescharacteristic
181#388—24no—
182#388—21no—
183#388—22no—
184#388—21no—
185#388—22no—
186#388—22no—
187#387—24no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382—22no—
202#382—22no—
203#380—22no—
204#380—22no—
205#380—22no—
206#380—21no—
207#380C2312yes—
208#380C2314yes—
209#379—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225C43C2311yescharacteristic
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2314yes—
235#336C2311yescharacteristic
236C25—22no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×D4—28no—
240C22×D4—22no—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4—44no—
249C22×D4—44no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2412yes—
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2412yes—
260C2×SD16—24no—
261C2×SD16—28no—
262C2×SD16—24no—
263C2×SD16—42no—
264C2×SD16—44no—
265C2×SD16—44no—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—44no—
270C4⋊Q8—42no—
271C4⋊Q8—44no—
272C4⋊Q8—82no—
273#127—22no—
274#127#4114yes—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277#121—28no—
278#121—24no—
279#121—24no—
280#120—216no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—48no—
291C4×D4—44no—
292C4×D4—48no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—82no—
296C4×D4#4114yes—
297#117—44no—
298#117—44no—
299#117—84no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
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—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—48no—
344C2×C22⋊C4—48no—
345C2×C22⋊C4—84no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—44no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—28no—
365#107—44no—
366#107—44no—
367#106—24no—
368C4⋊C8—24no—
369C4⋊C8—24no—
370#103—24no—
371#103—24no—
372#103—42no—
373#103—44no—
374#102—24no—
375#102—28no—
376#102—28no—
377#102—44no—
378#102—44no—
379C22⋊C8—28no—
380C22⋊C8—44no—
381#97—44no—
382C4×C8—24no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—22no—
392#95—21no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—28no—
398C24—21no—
399C24—22no—
400C24—44no—
401C24—44no—
402C24—82no—
403C24—82no—
404C24—82no—
405C24—82no—
406C24C22×D412yes—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—44no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—432no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—81no—
434C2×D4—81no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—88no—
438C2×D4—88no—
439C2×D4#13714yes—
440C2×D4C2×#4112yes—
441C2×D4C2×#4114yes—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C2511yescharacteristic
481SD16—48no—
482SD16—88no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8H(5,𝔽2)14yes—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—82no—
515C4⋊C4—81no—
516C4⋊C4—84no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4C2×#4112yes—
520C22⋊C4—28no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—44no—
533C22⋊C4—81no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—24no—
544C42—22no—
545C42—42no—
546C42—42no—
547C42—41no—
548C42—44no—
549C42—44no—
550C42—42no—
551C42—84no—
552C42AGL(1,ℤ/8)14yes—
553C42C2×#4112yes—
554C42C2×#4112yes—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—416no—
564C23—41no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—41no—
573C23—42no—
574C23—41no—
575C23—42no—
576C23—88no—
577C23—88no—
578C23—164no—
579C23—164no—
580C23—164no—
581C23—164no—
582C23C2×#13714yes—
583C23C23×D411yescharacteristic
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—164no—
589D4—28no—
590D4—24no—
591D4—28no—
592D4—416no—
593D4—48no—
594D4—44no—
595D4—42no—
596D4—84no—
597D4—88no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—48no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—84no—
644C2×C4—164no—
645C2×C4—164no—
646C2×C4—164no—
647C2×C4#52512yes—
648C2×C4#52512yes—
649C2×C4C2×AGL(1,ℤ/8)12yes—
650C2×C4C2×AGL(1,ℤ/8)12yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4C2×#13712yes—
653C2×C4#57612yes—
654C2×C4C22×#4111yescharacteristic
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×H(5,𝔽2)12yes—
657C2×C4#58512yes—
658C8—24no—
659C8—42no—
660C8—44no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—44no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—82no—
675C22—81no—
676C22—82no—
677C22—81no—
678C22—82no—
679C22—81no—
680C22—82no—
681C22—81no—
682C22—1616no—
683C22C2×#52511yescharacteristic
684C22#348511yescharacteristic
685C22#349811yescharacteristic
686C22C22×AGL(1,ℤ/8)11yescharacteristic
687C22C22×#13711yescharacteristic
688C22C2×#57611yescharacteristic
689C22#358611yescharacteristic
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—41no—
699C4—81no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—81no—
704C4—84no—
705C4#311014yes—
706C4#349412yes—
707C4#351412yes—
708C2—44no—
709C2—84no—
710C2—82no—
711C2#3602211yescharacteristic
712C2C2×#311011yescharacteristic
713C2#6059611yescharacteristic
714C2#6172211yescharacteristic
715C2#6172611yescharacteristic
716C2#6242311yescharacteristic
717C2#6244811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
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. #36194C2semidirect
3. #30113C2semidirect
4. #29980C2semidirect
5. #28508C2semidirect
6. #28311C2semidirect
7. #28160C2semidirect
8. #27582C2semidirect
9. #27245C2semidirect
10. #26803C2semidirect
11. #25016C2semidirect
13. #23761C2semidirect
14. #23677C2semidirect
15. #23101C2semidirect
16. #22839C2semidirect
17. #22811C2semidirect
20. C2×#484C22semidirect
21. C2×#474C22semidirect
25. #2868C22semidirect
26. #2857C22semidirect
31. #2492C22semidirect
38. #2355C22semidirect
46. C4×C4⋊Q8C22semidirect
51. #2285C22semidirect
53. #2099C22semidirect
55. #2093C22semidirect
56. #2093C22semidirect
57. #2084C22semidirect
58. #2076C22semidirect
60. #2071C22semidirect
61. #2066C22semidirect
62. #2057C22semidirect
64. #2043C22semidirect
65. #2036C22semidirect
69. #2013C22semidirect
72. #1973C22semidirect
75. #1947C22semidirect
76. #1928C22semidirect
78. #1925C22semidirect
79. #1924C22semidirect
80. #1877C22semidirect
81. #1877C22semidirect
82. #1852C22semidirect
83. #1845C22semidirect
84. #1831C22semidirect
85. #1817C22semidirect
86. #1803C22semidirect
89. #1776C22semidirect
90. #1770C22semidirect
91. C4×C4⋊C8C22semidirect
92. #1762C22semidirect
93. C4×#103C22semidirect
94. #1749C22semidirect

Maximal subgroups

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

subgroupindex
#361942
#343102
#301132
#299802
#285082
#283112
#281602
#275822
#272452
#268032
#250162
#242402
#237612
#236772
#231012
#228392
#228112