Finite Group Databasebeta

← #6433881

Subgroups — #6433881

Order 512.

1980 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34633C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic, complemented
3#30335C214yesmaximal, complemented
4#30334C214yesmaximal, complemented
5#30331C214yesmaximal, complemented
6#30330C214yesmaximal, complemented
7#23657C211yesmaximal, characteristic, complemented
8#23629C211yesmaximal, characteristic, complemented
9#23508C211yesmaximal, characteristic, complemented
10#23482C212yesmaximal, complemented
11#23429C211yesmaximal, characteristic, complemented
12#23392C212yesmaximal, complemented
13#21237C212yesmaximal, complemented
14#21093C212yesmaximal, complemented
15#20377C211yesmaximal, characteristic, complemented
16C22×#137C2212yes—
17C2×#532C2211yescharacteristic
18C2×#525C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C2×#428C2212yes—
22C2×#417C2212yes—
23C2×#417C2212yes—
24C2×#417C2212yes—
25C2×#419C2212yes—
26C2×#407C2212yes—
27C2×C4×M16C2212yes—
28#2625—22no—
29#2625—22no—
30#2625C2214yes—
31#2625C2214yes—
32#2621C2212yes—
33#2620C2212yes—
34#2620C2212yes—
35#2619C2212yes—
36#2618—21no—
37#2618—21no—
38#2617—21no—
39#2617—21no—
40C2×#400—21no—
41#2392—22no—
42#2392—22no—
43C2×#398—21no—
44C2×#397—21no—
45C2×#397—21no—
46#2297C2212yes—
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51#2285C2211yescharacteristic
52C2×#380C2211yescharacteristic
53#2085C2214yescomplemented
54#2084C2212yescomplemented
55#2083C2212yescomplemented
56#2074C2214yescomplemented
57#2072C2212yescomplemented
58#2072C2212yescomplemented
59#2043C2212yescomplemented
60#2042C2214yescomplemented
61#2041C2212yescomplemented
62#2036C2214yes—
63#2035C2214yes—
64#2034C2214yes—
65#2033C2214yes—
66#2016C2212yescomplemented
67#2015C2214yescomplemented
68#2014C2212yescomplemented
69#2001C2214yes—
70#2001C2214yes—
71#2000C2214yes—
72#2000C2214yes—
73#1973C2214yescomplemented
74#1971C2214yescomplemented
75#1969C2214yescomplemented
76#1968C2214yescomplemented
77#1936C2214yescomplemented
78#1935C2214yescomplemented
79#1934C2214yescomplemented
80#1933C2214yescomplemented
81#1750C2214yes—
82#1750C2214yes—
83#1736C2212yes—
84#1736C2212yes—
85#1728C2211yescharacteristic
86#1728C2211yescharacteristic
87C22×#41C2312yes—
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#137—28no—
91C2×#137—44no—
92C2×#137—44no—
93C22×Q16C2314yes—
94C22×SD16C2314yes—
95C22×M16C2312yes—
96C22×M16C2312yes—
97#532—44no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127C2311yescharacteristic
100C2×#124C2312yes—
101#525—44no—
102#525—44no—
103C2×#123—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—44no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2314yes—
109C2×#121—21no—
110C2×#121—21no—
111C2×#121C2312yes—
112C2×#121C2312yes—
113C2×#120—22no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×#117—44no—
119C2×#117D412yes—
120C2×#117D412yes—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4D411yescharacteristic
135C22×C22⋊C4D411yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#428—44no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141#419—44no—
142#417—28no—
143#417—44no—
144#417—44no—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153#407—28no—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C4×M16—44no—
157C2×#97C2314yes—
158C2×C4×C8C2314yes—
159#400—24no—
160#400—24no—
161#398—22no—
162#398—22no—
163#398—24no—
164#398—24no—
165#397—22no—
166#397—22no—
167#397—24no—
168#397—24no—
169#397—24no—
170#397—24no—
171#396—22no—
172#396—22no—
173#396—22no—
174#396—22no—
175#394—22no—
176#394—22no—
177#394—22no—
178#394—22no—
179#394—22no—
180#394—22no—
181#392—22no—
182#392—22no—
183#391—21no—
184#391—21no—
185#391—22no—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#391C2312yes—
190#391C2314yes—
191#390C2312yes—
192#390C2312yes—
193#389—21no—
194#389—21no—
195#389—22no—
196#389C2312yes—
197#389C2312yes—
198#389C2314yes—
199#386—22no—
200#386—22no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382C2312yes—
204#380—21no—
205#380—21no—
206#380—22no—
207#380C2314yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95—21no—
218C2×#95—21no—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#336C2312yes—
224#336C2312yes—
225#336C2312yes—
226#336C2312yes—
227C25C2×D411yescharacteristic
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—24no—
231C2×#41—24no—
232C2×#41—24no—
233C2×#41—24no—
234C22×Q8—21no—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2412yes—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264#137—432no—
265C2×Q16—28no—
266C2×Q16—44no—
267C2×Q16—44no—
268C2×SD16—28no—
269C2×SD16—44no—
270C2×SD16—44no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16—24no—
274C2×M16—24no—
275C2×M16—24no—
276C2×M16#4118yes—
277C22×C8C2414yes—
278C22×C8C2414yes—
279C4⋊Q8—42no—
280#127—42no—
281#124—42no—
282#124—42no—
283#123—48no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C22⋊Q8—42no—
287C22⋊Q8—42no—
288C22⋊Q8—42no—
289C22⋊Q8—42no—
290C22⋊Q8—48no—
291#121—42no—
292#121—42no—
293#121—42no—
294#121—42no—
295#121—44no—
296#121—44no—
297#120—48no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—48no—
301C4×D4—48no—
302#117—24no—
303#117—24no—
304#117—44no—
305#117—44no—
306#117—42no—
307#117—42no—
308#117—42no—
309#117—42no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—82no—
333C2×C4⋊C4—82no—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—44no—
359C2×C22⋊C4—82no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2412yes—
373#107—44no—
374#106—44no—
375#103—28no—
376#103—44no—
377#103—44no—
378#103—44no—
379#103—44no—
380#102—28no—
381#102—44no—
382#102—44no—
383C22⋊C8—28no—
384C22⋊C8—28no—
385#97—44no—
386C4×C8—44no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24C22×D412yes—
412C24C22×D411yescharacteristic
413#41—48no—
414#41—48no—
415#41—48no—
416#41—48no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—48no—
431C2×Q8—48no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—81no—
446C2×D4—81no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4—82no—
452C2×D4—82no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D412yes—
508C22×C4C22×D411yescharacteristic
509C22×C4C2×#4114yes—
510C22×C4H(5,𝔽2)14yes—
511C22×C4H(5,𝔽2)12yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C4C2511yescharacteristic
514Q16—416no—
515SD16—416no—
516M16—216no—
517M16—48no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8C2×#4118yes—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—48no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—81no—
545C4⋊C4—81no—
546C4⋊C4—81no—
547C4⋊C4—81no—
548C4⋊C4—84no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—48no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—84no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—44no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—82no—
598C23—82no—
599C23—82no—
600C23—82no—
601C23#53212yes—
602C23#53412yes—
603C23#53512yes—
604C23C23×D411yescharacteristic
605Q8—44no—
606Q8—42no—
607Q8—42no—
608Q8—44no—
609Q8—42no—
610Q8—42no—
611Q8—84no—
612D4—42no—
613D4—42no—
614D4—42no—
615D4—42no—
616D4—84no—
617D4—84no—
618D4—88no—
619D4—88no—
620D4—88no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—44no—
643C2×C4—48no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
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—42no—
669C2×C4—82no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4—82no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—88no—
678C2×C4#53212yes—
679C2×C4#53412yes—
680C2×C4#53512yes—
681C2×C4#53818yes—
682C2×C4#54018yes—
683C2×C4C22×#4112yes—
684C2×C4C2×H(5,𝔽2)11yescharacteristic
685C2×C4C2×H(5,𝔽2)12yes—
686C2×C4C2×H(5,𝔽2)11yescharacteristic
687C8—28no—
688C8—44no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—88no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22#266114yes—
711C22C2×#53211yescharacteristic
712C22C2×#53411yescharacteristic
713C22C2×#53511yescharacteristic
714C22#358511yescharacteristic
715C22#358611yescharacteristic
716C22#358611yescharacteristic
717C22#358611yescharacteristic
718C4—42no—
719C4—42no—
720C4—41no—
721C4—41no—
722C4—42no—
723C4—41no—
724C4—41no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4#266114yes—
733C4C2×#53812yes—
734C4C2×#54012yes—
735C2—24no—
736C2—82no—
737C2—162no—
738C2C2×#266111yescharacteristic
739C2#6070311yescharacteristic
740C2#6070511yescharacteristic
741C2#6075411yescharacteristic
742C2#6075511yescharacteristic
743C2#6080911yescharacteristic
744C2#6081011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34633C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2625C2semidirect
3. #30335C2semidirect
4. #30334C2semidirect
5. #30331C2semidirect
6. #30330C2semidirect
7. #23657C2semidirect
8. #23629C2semidirect
9. #23508C2semidirect
10. #23482C2semidirect
11. #23429C2semidirect
12. #23392C2semidirect
13. #21237C2semidirect
14. #21093C2semidirect
15. #20377C2semidirect
53. #2085C22semidirect
54. #2084C22semidirect
55. #2083C22semidirect
56. #2074C22semidirect
57. #2072C22semidirect
58. #2072C22semidirect
59. #2043C22semidirect
60. #2042C22semidirect
61. #2041C22semidirect
66. #2016C22semidirect
67. #2015C22semidirect
68. #2014C22semidirect
73. #1973C22semidirect
74. #1971C22semidirect
75. #1969C22semidirect
76. #1968C22semidirect
77. #1936C22semidirect
78. #1935C22semidirect
79. #1934C22semidirect
80. #1933C22semidirect

Maximal subgroups

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

subgroupindex
#346332
C2×#26252
#303352
#303342
#303312
#303302
#236572
#236292
#235082
#234822
#234292
#233922
#212372
#210932
#203772