Finite Group Databasebeta

← #6463840

Subgroups — #6463840

Order 512.

2064 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
1C2×#3083C212yesmaximal, complemented
2#34272C211yesmaximal, characteristic, complemented
3#34207C211yesmaximal, characteristic
4#28215C214yesmaximal, complemented
5#28211C214yesmaximal, complemented
6#28209C212yesmaximal, complemented
7#28207C212yesmaximal, complemented
8#28206C212yesmaximal, complemented
9#28201C212yesmaximal, complemented
10#23658C211yesmaximal, characteristic, complemented
11C2×#2083C211yesmaximal, characteristic, complemented
12#23508C211yesmaximal, characteristic
13#23504C211yesmaximal, characteristic
14#21067C212yesmaximal, complemented
15#21062C212yesmaximal, complemented
16#20754C211yesmaximal, characteristic, complemented
17#20652C212yesmaximal, complemented
18C2×#544C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×#520C2212yes—
21#3083—216no—
22C2×#479C2214yes—
23C2×#477C2214yes—
24C2×#478C2214yes—
25C2×#474C2214yes—
26C2×#428C2212yes—
27C22×#107C2212yes—
28C2×#424C2212yes—
29C2×#417C2212yes—
30C2×#417C2212yes—
31C2×#416C2212yes—
32C2×#416C2212yes—
33#2370—21no—
34#2370—21no—
35#2369—21no—
36#2368—22no—
37#2365—21no—
38#2361—22no—
39#2361—22no—
40#2360—22no—
41#2354—21no—
42#2354—21no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45#2331C2212yes—
46#2330C2212yes—
47#2326C2214yes—
48#2317C2211yescharacteristic
49#2311C2212yes—
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2212yes—
52C4×#127C2212yes—
53C4×#125C2212yes—
54#2296C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2274C2211yescharacteristic
58#2274C2211yescharacteristic
59C4×#117C2211yescharacteristic
60#2084C2212yes—
61#2084C2212yes—
62#2083C2212yes—
63#2083C2212yes—
64#2083C2212yes—
65#2083C2212yes—
66#2083C2214yes—
67#2043C2211yescharacteristic
68#2043C2212yes—
69#2043C2211yescharacteristic
70#2042C2212yes—
71#2042C2214yes—
72#2042C2212yes—
73#2041C2211yescharacteristic
74#2041C2212yes—
75#2041C2211yescharacteristic
76#1928C2214yes—
77#1928C2214yes—
78#1928C2214yes—
79#1927C2214yes—
80#1926C2214yes—
81#1925C2214yes—
82#1925C2214yes—
83#1925C2214yes—
84#1832C2214yes—
85#1831C2212yes—
86#1831C2212yes—
87#1805C2214yes—
88#1804C2214yes—
89#1803C2214yes—
90#1802C2214yes—
91#1736C2212yes—
92C2×#340C2212yes—
93#1729C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C22×#41C2312yes—
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—24no—
100#544D418yescomplemented
101#532—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#125C2312yes—
108C2×#124C2312yes—
109C2×C22⋊Q8—22no—
110C2×#121—21no—
111C2×#121—21no—
112#520—28no—
113#520—28no—
114C2×C4×Q8—22no—
115C2×C4×Q8C2314yes—
116C2×C4×D4—24no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4C2314yes—
122C2×#117—22no—
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#479—216no—
136#478—216no—
137#477—216no—
138#474—216no—
139#428—28no—
140C2×#107—28no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145#424—28no—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148#417—28no—
149#417—28no—
150#416—28no—
151#416—28no—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164#391—22no—
165#391—22no—
166#391—22no—
167#391C2314yes—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389C2311yescharacteristic
181#389C2312yes—
182#389C2311yescharacteristic
183#389C2311yescharacteristic
184#389C2312yes—
185#389C2311yescharacteristic
186#388—24no—
187#388—24no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#386—24no—
195#385—21no—
196#385—21no—
197#385—22no—
198#385—24no—
199#385—24no—
200#384—21no—
201#384—21no—
202#384C2312yes—
203#384C2312yes—
204#384C2312yes—
205#382C2312yes—
206#380—22no—
207#380—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4D412yes—
231#376C2312yes—
232C2×#95—21no—
233C43C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#336C2311yescharacteristic
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2312yes—
242#336C2312yes—
243#336C2314yes—
244#336C2314yes—
245C2×#41—24no—
246C2×#41—24no—
247C2×#41—24no—
248C2×#41—24no—
249C2×#41—24no—
250C2×#41#4118yes—
251C22×Q8—22no—
252C22×Q8C2412yes—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4C2412yes—
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—24no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8—24no—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2414yes—
274C4⋊Q8—24no—
275C4⋊Q8—28no—
276C4⋊Q8—24no—
277C4⋊Q8C2×D418yes—
278#127—24no—
279#125—28no—
280#125—24no—
281#125C2×D418yes—
282#124—24no—
283#124—24no—
284C22⋊Q8—216no—
285#121—28no—
286#121—28no—
287C4×Q8—28no—
288C4×Q8—28no—
289C4×Q8—84no—
290C4×D4—28no—
291C4×D4—28no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—416no—
295C4×D4—84no—
296C4×D4—84no—
297#117—22no—
298#117—24no—
299#117—24no—
300#117—48no—
301#117C2×D414yes—
302#117C2×D414yes—
303#117#4114yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4—82no—
332C2×C4⋊C4—82no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—21no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372#107—28no—
373#107—28no—
374#107—28no—
375#106—28no—
376C4⋊C8—28no—
377C4⋊C8—28no—
378#103—24no—
379#103—24no—
380#103—28no—
381#103—24no—
382#103—24no—
383#103—28no—
384#102—24no—
385#102—24no—
386#102—28no—
387#102—24no—
388#102—24no—
389#102—28no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—21no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—22no—
409C24C22×D411yescharacteristic
410#41—216no—
411#41—216no—
412#41—48no—
413#41—48no—
414C2×Q8—28no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—82no—
421C2×Q8—82no—
422C2×Q8C2×#4114yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—28no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C2×#4114yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—48no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—82no—
464C22×C4—81no—
465C22×C4—82no—
466C22×C4—81no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4114yes—
479C22×C42−1+414yes—
480C22×C4C2511yescharacteristic
481M16—48no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—48no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—82no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4#12114yes—
524C4⋊C4#12114yes—
525C4⋊C4C22×D412yes—
526C4⋊C4C22×D414yes—
527C4⋊C4C22×D412yes—
528C22⋊C4—28no—
529C22⋊C4—28no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—42no—
533C22⋊C4—48no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4#12118yes—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—24no—
545C42—42no—
546C42—42no—
547C42—48no—
548C42—42no—
549C42—82no—
550C42—82no—
551C42—82no—
552C42—82no—
553C42C22×D414yes—
554C42C2×#4114yes—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—24no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23C2×#12112yes—
567C23#52512yes—
568C23#53612yes—
569C23C23×D411yescharacteristic
570Q8—24no—
571Q8—24no—
572Q8—42no—
573Q8—42no—
574Q8—88no—
575D4—24no—
576D4—24no—
577D4—48no—
578D4—48no—
579D4—416no—
580D4—42no—
581D4—42no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—416no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—84no—
630C2×C4—168no—
631C2×C4—164no—
632C2×C4—162no—
633C2×C4C2×#12114yes—
634C2×C4C2×#12114yes—
635C2×C4C2×#12112yes—
636C2×C4#52512yes—
637C2×C4#53612yes—
638C2×C4#54818yes—
639C2×C4#55218yes—
640C2×C4#57612yes—
641C2×C4#57812yes—
642C2×C4C23×D411yescharacteristic
643C2×C4C22×#4112yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×2−1+412yes—
646C8—42no—
647C8—42no—
648C8—44no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#253614yes—
663C22C22×#12111yescharacteristic
664C22C2×#52511yescharacteristic
665C22C2×#53611yescharacteristic
666C22C2×#57611yescharacteristic
667C22C2×#57811yescharacteristic
668C22#358511yescharacteristic
669C22#358611yescharacteristic
670C4—24no—
671C4—24no—
672C4—22no—
673C4—22no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—41no—
679C4—41no—
680C4—42no—
681C4—42no—
682C4—84no—
683C4—81no—
684C4—82no—
685C4—81no—
686C4#253614yes—
687C4C2×#54812yes—
688C4C2×#55212yes—
689C2—24no—
690C2—84no—
691C2C2×#253611yescharacteristic
692C2#6042811yescharacteristic
693C2#6043211yescharacteristic
694C2#6058411yescharacteristic
695C2#6058911yescharacteristic
696C2#6088011yescharacteristic
697C2#6088611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3083C2semidirect
2. #34272C2semidirect
4. #28215C2semidirect
5. #28211C2semidirect
6. #28209C2semidirect
7. #28207C2semidirect
8. #28206C2semidirect
9. #28201C2semidirect
10. #23658C2semidirect
11. C2×#2083C2semidirect
14. #21067C2semidirect
15. #21062C2semidirect
16. #20754C2semidirect
17. #20652C2semidirect
100. #544D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30832
#342722
#342072
#282152
#282112
#282092
#282072
#282062
#282012
#236582
C2×#20832
#235082
#235042
#210672
#210622
#207542
#206522