Finite Group Databasebeta

← #6471412

Subgroups — #6471412

Order 512.

2414 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic, complemented
2C2×#2341C211yesmaximal, characteristic
3#32954C211yesmaximal, characteristic, complemented
4#32906C211yesmaximal, characteristic, complemented
5#32883C212yesmaximal
6#29950C211yesmaximal, characteristic, complemented
7#29894C212yesmaximal, complemented
8#29714C211yesmaximal, characteristic, complemented
9#27529C214yesmaximal, complemented
10#27233C214yesmaximal, complemented
11#25151C214yesmaximal, complemented
12#24904C214yesmaximal, complemented
13#22043C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic, complemented
15C4×#451C212yesmaximal, complemented
16C23×#41C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#475C2212yescomplemented
20C2×#462C2212yescomplemented
21C2×#452C2214yes—
22C2×#452C2214yes—
23C2×#451C2212yes—
24C2×#450C2212yes—
25C2×C4×Q16C2214yescomplemented
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2848C2212yes—
29#2646—21no—
30#2626—21no—
31#2589—22no—
32#2587—22no—
33#2497C2212yescomplemented
34#2492C2212yescomplemented
35#2429C2212yescomplemented
36#2423C2212yescomplemented
37#2341—22no—
38#2341—22no—
39#2341C2214yes—
40#2341C2214yes—
41#2329—21no—
42#2312—21no—
43C4×C22⋊Q8C2212yes—
44C2×#390—21no—
45C2×#390—21no—
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#380—21no—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4—21no—
51C2×C4×C22⋊C4C2211yescharacteristic
52#2093C2212yescomplemented
53#2077C2214yescomplemented
54#2075C2214yescomplemented
55#2071C2212yescomplemented
56#2067C2214yescomplemented
57#2055C2214yescomplemented
58#2036C2214yescomplemented
59#2035C2214yescomplemented
60#2023C2212yescomplemented
61#2013C2212yescomplemented
62#1968C2214yescomplemented
63#1934C2214yescomplemented
64#1926C2214yescomplemented
65#1923C2214yescomplemented
66#1923C2214yescomplemented
67#1880C2214yescomplemented
68#1878C2214yescomplemented
69#1878C2214yescomplemented
70#1872C2212yes—
71#1872C2212yes—
72#1865C2214yescomplemented
73#1863C2214yescomplemented
74#1853C2212yes—
75#1852C2212yes—
76#1803C2212yescomplemented
77#1803C2212yescomplemented
78#1797C2211yescharacteristic
79#1797C2211yescharacteristic
80#1788C2214yes—
81#1788C2214yes—
82#1787C2212yes—
83#1786C2212yes—
84#1779C2212yes—
85C4×#103C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C22×#41—24no—
88C22×#41—24no—
89C22×#41—22no—
90C22×#41—44no—
91C22×#41—44no—
92C22×#41C2314yes—
93C23×Q8C2311yescharacteristic
94C23×D4—21no—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4C2311yescharacteristic
98C22×Q16C2314yes—
99C22×SD16C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—44no—
110C2×C4×Q8—44no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2314yes—
114C2×C4×Q8C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#475—44no—
131#462—44no—
132#452—48no—
133#452—48no—
134#451—28no—
135#450—28no—
136C4×Q16—48no—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#400—22no—
156#399—22no—
157#398—22no—
158#398—22no—
159#396—22no—
160#396—22no—
161#393—22no—
162#391—21no—
163#391—22no—
164#391C2312yes—
165#391C2312yes—
166#390—24no—
167#390—24no—
168#390—24no—
169#390—24no—
170#390—22no—
171#390—21no—
172#390—21no—
173#390C2312yes—
174#390C2312yes—
175#390C2314yes—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389C2312yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#380—21no—
196#380—21no—
197#380—21no—
198#380—22no—
199#380—24no—
200#380—24no—
201#380C2312yes—
202#380C2314yes—
203#380C2312yes—
204#380C2312yes—
205#379—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—24no—
210C4×C22⋊C4—24no—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4C2312yes—
214C4×C22⋊C4C2312yes—
215C2×#95—21no—
216C2×#95D411yescharacteristic
217C2×#95D411yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#336C2312yes—
223#336C2312yes—
224#336C2314yes—
225C25—21no—
226C25C2×D411yescharacteristic
227C2×#41—216no—
228C2×#41—28no—
229C2×#41—28no—
230C2×#41—28no—
231C2×#41—28no—
232C2×#41—44no—
233C2×#41—44no—
234C2×#41—48no—
235C2×#41—48no—
236C2×#41—432no—
237C2×#41—432no—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8#4114yes—
244C22×Q8C2412yes—
245C22×Q8C2412yes—
246C22×D4—28no—
247C22×D4—24no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—42no—
251C22×D4—42no—
252C22×D4—44no—
253C22×D4—44no—
254C22×D4#4114yes—
255C22×D4C2412yes—
256C23×C4—21no—
257C23×C4—28no—
258C23×C4—24no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—42no—
264C23×C4—42no—
265C23×C4—44no—
266C23×C4—44no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D412yes—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275C2×Q16—28no—
276C2×Q16—28no—
277C2×Q16—28no—
278C2×SD16—28no—
279C2×SD16—44no—
280C2×SD16—44no—
281C22×C8C2414yes—
282C22×C8C2414yes—
283C4⋊Q8—42no—
284C4⋊Q8—42no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C4×Q8—28no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—44no—
294C4×Q8—42no—
295C4×Q8—42no—
296C4×Q8—44no—
297C4×Q8—48no—
298C4×Q8—44no—
299C4×Q8—44no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
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—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—82no—
345C2×C22⋊C4—82no—
346C2×C22⋊C4—82no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—42no—
357C2×C42—42no—
358C2×C42—44no—
359C2×C42—44no—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2412yes—
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—44no—
367#106—44no—
368C4⋊C8—42no—
369C4⋊C8—42no—
370#103—28no—
371#103—28no—
372#103—28no—
373#103—44no—
374#103—44no—
375#102—28no—
376#102—44no—
377#102—44no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380C22⋊C8—44no—
381C22⋊C8—44no—
382C4×C8—44no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—22no—
399C24—24no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—41no—
404C24—41no—
405C24—41no—
406C24—41no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24C22×D412yes—
414C24C22×D411yescharacteristic
415#41—416no—
416#41—416no—
417#41—464no—
418#41—416no—
419#41—416no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—216no—
423C2×Q8—216no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—42no—
433C2×Q8—81no—
434C2×Q8—81no—
435C2×Q8C2×#4114yes—
436C2×Q8H(5,𝔽2)14yes—
437C2×Q8H(5,𝔽2)14yes—
438C2×D4—216no—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—48no—
444C2×D4—48no—
445C2×D4—48no—
446C2×D4—432no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—48no—
450C2×D4C2×#4114yes—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—216no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—432no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4#13514yes—
496C22×C4#13714yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D412yes—
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C22×C4C2511yescharacteristic
508Q16—416no—
509SD16—88no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—81no—
532C4⋊C4—82no—
533C4⋊C4—81no—
534C4⋊C4—81no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—81no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—48no—
550C22⋊C4—84no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C22⋊C4—84no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—42no—
562C42—41no—
563C42—41no—
564C42—48no—
565C42—44no—
566C42—44no—
567C42—82no—
568C42—82no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—24no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—48no—
582C23—44no—
583C23—44no—
584C23—48no—
585C23—42no—
586C23—42no—
587C23—44no—
588C23—48no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—82no—
594C23—82no—
595C23—82no—
596C23—82no—
597C23—84no—
598C23—84no—
599C23—84no—
600C23—84no—
601C23#52512yes—
602C23C2×#13512yes—
603C23C2×#13712yes—
604C23C23×D411yescharacteristic
605Q8—216no—
606Q8—24no—
607Q8—24no—
608Q8—24no—
609Q8—44no—
610Q8—44no—
611Q8—162no—
612Q8#53818yes—
613D4—28no—
614D4—432no—
615D4—48no—
616D4—44no—
617C2×C4—216no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—432no—
646C2×C4—48no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—82no—
669C2×C4—82no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4#52512yes—
673C2×C4C2×#13512yes—
674C2×C4C2×#13512yes—
675C2×C4C2×#13714yes—
676C2×C4C22×#4111yescharacteristic
677C2×C4C22×#4111yescharacteristic
678C2×C4C2×H(5,𝔽2)12yes—
679C2×C4C2×H(5,𝔽2)12yes—
680C8—44no—
681C8—44no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—42no—
689C22—44no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—44no—
696C22—44no—
697C22—42no—
698C22—88no—
699C22—816no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22#310714yes—
705C22C2×#52511yescharacteristic
706C22#348411yescharacteristic
707C22#348411yescharacteristic
708C22C22×#13511yescharacteristic
709C22C22×#13711yescharacteristic
710C22#358511yescharacteristic
711C22#358611yescharacteristic
712C4—22no—
713C4—22no—
714C4—24no—
715C4—28no—
716C4—22no—
717C4—42no—
718C4—41no—
719C4—41no—
720C4—42no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4C2×#53812yes—
728C4#349012yes—
729C2—24no—
730C2—82no—
731C2—84no—
732C2#3498811yescharacteristic
733C2C2×#310711yescharacteristic
734C2#6056511yescharacteristic
735C2#6156411yescharacteristic
736C2#6156611yescharacteristic
737C2#6157311yescharacteristic
738C2#6158811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35014C2semidirect
3. #32954C2semidirect
4. #32906C2semidirect
6. #29950C2semidirect
7. #29894C2semidirect
8. #29714C2semidirect
9. #27529C2semidirect
10. #27233C2semidirect
11. #25151C2semidirect
12. #24904C2semidirect
13. #22043C2semidirect
14. #21902C2semidirect
15. C4×#451C2semidirect
19. C2×#475C22semidirect
20. C2×#462C22semidirect
25. C2×C4×Q16C22semidirect
33. #2497C22semidirect
34. #2492C22semidirect
35. #2429C22semidirect
36. #2423C22semidirect
52. #2093C22semidirect
53. #2077C22semidirect
54. #2075C22semidirect
55. #2071C22semidirect
56. #2067C22semidirect
57. #2055C22semidirect
58. #2036C22semidirect
59. #2035C22semidirect
60. #2023C22semidirect
61. #2013C22semidirect
62. #1968C22semidirect
63. #1934C22semidirect
64. #1926C22semidirect
65. #1923C22semidirect
66. #1923C22semidirect
67. #1880C22semidirect
68. #1878C22semidirect
69. #1878C22semidirect
72. #1865C22semidirect
73. #1863C22semidirect
76. #1803C22semidirect
77. #1803C22semidirect
85. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#23412
#329542
#329062
#328832
#299502
#298942
#297142
#275292
#272332
#251512
#249042
#220432
#219022
C4×#4512