Finite Group Databasebeta

← #6434285

Subgroups — #6434285

Order 512.

1763 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35313C211yesmaximal, characteristic, complemented
2#35264C211yesmaximal, characteristic
3#30353C214yesmaximal, complemented
4#30350C214yesmaximal, complemented
5#30347C214yesmaximal, complemented
6#30343C214yesmaximal, complemented
7#23614C211yesmaximal, characteristic, complemented
8#23611C211yesmaximal, characteristic, complemented
9#23555C211yesmaximal, characteristic, complemented
10#23545C211yesmaximal, characteristic, complemented
11#23541C211yesmaximal, characteristic, complemented
12#23509C211yesmaximal, characteristic
13#23457C211yesmaximal, characteristic, complemented
14#23423C211yesmaximal, characteristic
15#21237C211yesmaximal, characteristic, complemented
16C2×#1973C211yesmaximal, characteristic, complemented
17#21095C211yesmaximal, characteristic, complemented
18#21086C211yesmaximal, characteristic, complemented
19#20427C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C2×#535C2211yescharacteristic
24C2×#428C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#416C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×C4×M16C2211yescharacteristic
37C22×#97C2211yescharacteristic
38#2863C2214yescomplemented
39#2861C2212yes—
40#2858C2212yes—
41#2645C2214yescomplemented
42#2641C2212yes—
43#2640C2212yes—
44#2514—22no—
45#2511—22no—
46#2493—24no—
47#2491—22no—
48#2490—22no—
49#2472—24no—
50#2317C2211yescharacteristic
51#2316C2211yescharacteristic
52#2311C2211yescharacteristic
53#2310C2211yescharacteristic
54#2310C2211yescharacteristic
55#2309C2211yescharacteristic
56#2297C2211yescharacteristic
57#2296C2211yescharacteristic
58C2×#391C2211yescharacteristic
59C2×#390C2211yescharacteristic
60C4×#117C2211yescharacteristic
61#2068C2214yescomplemented
62#2067C2214yes—
63#2067C2214yescomplemented
64#2065C2214yes—
65#2059C2214yescomplemented
66#2057C2214yescomplemented
67#2056C2214yescomplemented
68#2055C2214yes—
69#2055C2214yes—
70#2055C2214yescomplemented
71#2042C2214yes—
72#2041C2214yes—
73#2026C2214yescomplemented
74#2024C2214yescomplemented
75#2015C2214yes—
76#2014C2214yes—
77#1973C2214yes—
78#1973C2214yes—
79#1973C2212yescomplemented
80#1971C2212yescomplemented
81#1969C2212yescomplemented
82#1968C2212yescomplemented
83#1936C2214yes—
84#1936C2212yescomplemented
85#1936C2212yescomplemented
86#1935C2212yescomplemented
87#1935C2212yescomplemented
88#1934C2214yes—
89#1768C2214yescomplemented
90#1767C2214yescomplemented
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C2×#137—24no—
101C2×#137—42no—
102C2×#137—42no—
103C22×Q16C2312yes—
104C22×SD16—24no—
105C22×SD16—42no—
106C22×SD16—42no—
107C22×SD16C2314yes—
108C22×SD16C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#544—42no—
114#544—42no—
115#535—42no—
116#535—42no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2312yes—
119C2×#127C2312yes—
120C2×#125—21no—
121C2×#125C2312yes—
122C2×#124—24no—
123C2×#124C2312yes—
124C2×C22⋊Q8—22no—
125C2×#121—21no—
126C2×#121—21no—
127C2×#120—22no—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137#428—42no—
138C2×#107—42no—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#106C2312yes—
142#424—24no—
143#424—24no—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2314yes—
146#419—42no—
147#418—24no—
148#418—42no—
149#417—42no—
150#416—24no—
151#416—42no—
152C2×#103—42no—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#102—42no—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2312yes—
166C4×M16—42no—
167C2×#97—42no—
168C2×#97C2312yes—
169C2×#97C2312yes—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#401—24no—
173#400—24no—
174#400—24no—
175#400—24no—
176#397—24no—
177#394—24no—
178#394—24no—
179#393—24no—
180#391—21no—
181#391—22no—
182#391—21no—
183#391—22no—
184#391C2314yes—
185#391C2314yes—
186#391C2312yes—
187#391C2311yescharacteristic
188#391C2312yes—
189#391C2311yescharacteristic
190#391C2312yes—
191#390—22no—
192#390—24no—
193#390—22no—
194#390C2312yes—
195#390C2312yes—
196#390C2312yes—
197#390C2311yescharacteristic
198#390C2312yes—
199#390C2312yes—
200#390C2311yescharacteristic
201#389—22no—
202#389C2314yes—
203#389C2314yes—
204#389C2312yes—
205#389C2312yes—
206#388—24no—
207#388—24no—
208#387—24no—
209#387—24no—
210#387—24no—
211#386—22no—
212#386—22no—
213#385—22no—
214#385—22no—
215#384—21no—
216#383—24no—
217#382—21no—
218#381—22no—
219#381—22no—
220#380—22no—
221#380—22no—
222#379—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4D412yes—
225C4×C4⋊C4D412yes—
226C4×C22⋊C4—22no—
227C4×C22⋊C4D414yes—
228#376C2312yes—
229C2×#95—22no—
230C43C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#340C2314yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8C2411yescharacteristic
255C22×Q8C2412yes—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4C2412yes—
267C22×D4C2411yescharacteristic
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4C2×D412yes—
273C23×C4C2411yescharacteristic
274#137—416no—
275C2×Q16—24no—
276C2×Q16—42no—
277C2×Q16—42no—
278C2×SD16—28no—
279C2×SD16—24no—
280C2×SD16—44no—
281C2×SD16—44no—
282C2×SD16—42no—
283C2×SD16—42no—
284C2×SD16—416no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16#4114yes—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8#4114yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C4⋊Q8—42no—
302C4⋊Q8—42no—
303C4⋊Q8—44no—
304#127—42no—
305#127—42no—
306#125—42no—
307#125—42no—
308#125—44no—
309#124—42no—
310#124—42no—
311#124—88no—
312C22⋊Q8—48no—
313#121—44no—
314#121—44no—
315#120—48no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—84no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42C2×D412yes—
369C2×C42C2×D414yes—
370C2×C42C2×D414yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—44no—
376#107—42no—
377#106—42no—
378C4⋊C8—28no—
379C4⋊C8—28no—
380#103—24no—
381#103—44no—
382#103—42no—
383#103—42no—
384#103—42no—
385#103—42no—
386#102—24no—
387#102—28no—
388#102—44no—
389#102—42no—
390#102—42no—
391#102—44no—
392#97—42no—
393#97—42no—
394#97—42no—
395C4×C8—42no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95C2×D414yes—
410#95C2×D414yes—
411C24—22no—
412C24—21no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—82no—
417C24—82no—
418C24C22×D411yescharacteristic
419#41—44no—
420#41—44no—
421#41—44no—
422#41—44no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—84no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—84no—
469C2×D4—84no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4C22×D412yes—
492C22×C4C22×D414yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C42−1+412yes—
503C22×C4C2511yescharacteristic
504Q16—48no—
505SD16—416no—
506SD16—48no—
507M16—28no—
508M16—44no—
509C2×C8—28no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—44no—
521C2×C8C2×#4114yes—
522C2×C8C2×#4114yes—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—84no—
545C4⋊C4—84no—
546C4⋊C4—82no—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—48no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—84no—
557C22⋊C4—82no—
558C22⋊C4—84no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C42—24no—
562C42—22no—
563C42—22no—
564C42—21no—
565C42—21no—
566C42—22no—
567C42—22no—
568C42—21no—
569C42—21no—
570C42—44no—
571C42—44no—
572C42—42no—
573C42—42no—
574C42—42no—
575C23—22no—
576C23—22no—
577C23—42no—
578C23—44no—
579C23—42no—
580C23—44no—
581C23—41no—
582C23—42no—
583C23—42no—
584C23—41no—
585C23—41no—
586C23—42no—
587C23—88no—
588C23—84no—
589C23—164no—
590C23—162no—
591C23—162no—
592C23#53212yes—
593C23#53512yes—
594C23#53712yes—
595C23C23×D411yescharacteristic
596Q8—42no—
597Q8—42no—
598Q8—41no—
599Q8—41no—
600Q8—42no—
601Q8—42no—
602Q8—41no—
603Q8—41no—
604Q8—84no—
605Q8—164no—
606D4—42no—
607D4—41no—
608D4—42no—
609D4—41no—
610D4—42no—
611D4—42no—
612D4—41no—
613D4—41no—
614D4—84no—
615D4—84no—
616D4—164no—
617D4—164no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—162no—
655C2×C4—162no—
656C2×C4#53212yes—
657C2×C4#53512yes—
658C2×C4#53712yes—
659C2×C4#53814yes—
660C2×C4#53814yes—
661C2×C4#54214yes—
662C2×C4#54214yes—
663C2×C4C22×#4111yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4C2×2−1+411yescharacteristic
669C8—24no—
670C8—24no—
671C8—42no—
672C8—42no—
673C22—24no—
674C22—24no—
675C22—22no—
676C22—22no—
677C22—82no—
678C22—84no—
679C22—82no—
680C22—84no—
681C22—81no—
682C22—82no—
683C22—81no—
684C22—82no—
685C22—81no—
686C22—82no—
687C22—168no—
688C22#266214yes—
689C22C2×#53211yescharacteristic
690C22C2×#53511yescharacteristic
691C22C2×#53711yescharacteristic
692C22#358512yes—
693C22#358612yes—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—82no—
700C4—82no—
701C4—81no—
702C4—82no—
703C4—81no—
704C4—82no—
705C4—82no—
706C4—81no—
707C4#266214yes—
708C4C2×#53811yescharacteristic
709C4C2×#53811yescharacteristic
710C4C2×#54211yescharacteristic
711C4C2×#54211yescharacteristic
712C2—24no—
713C2—161no—
714C2—162no—
715C2C2×#266211yescharacteristic
716C2#6069712yes—
717C2#6082611yescharacteristic
718C2#6082711yescharacteristic
719C2#6093812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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. #35313C2semidirect
3. #30353C2semidirect
4. #30350C2semidirect
5. #30347C2semidirect
6. #30343C2semidirect
7. #23614C2semidirect
8. #23611C2semidirect
9. #23555C2semidirect
10. #23545C2semidirect
11. #23541C2semidirect
13. #23457C2semidirect
15. #21237C2semidirect
16. C2×#1973C2semidirect
17. #21095C2semidirect
18. #21086C2semidirect
19. #20427C2semidirect
38. #2863C22semidirect
41. #2645C22semidirect
61. #2068C22semidirect
63. #2067C22semidirect
65. #2059C22semidirect
66. #2057C22semidirect
67. #2056C22semidirect
70. #2055C22semidirect
73. #2026C22semidirect
74. #2024C22semidirect
79. #1973C22semidirect
80. #1971C22semidirect
81. #1969C22semidirect
82. #1968C22semidirect
84. #1936C22semidirect
85. #1936C22semidirect
86. #1935C22semidirect
87. #1935C22semidirect
89. #1768C22semidirect
90. #1767C22semidirect

Maximal subgroups

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

subgroupindex
#353132
#352642
#303532
#303502
#303472
#303432
#236142
#236112
#235552
#235452
#235412
#235092
#234572
#234232
#212372
C2×#19732
#210952
#210862
#204272