Finite Group Databasebeta

← #6441491

Subgroups — #6441491

Order 512.

1972 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35558C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic, complemented
3#27089C214yesmaximal, complemented
4#27086C214yesmaximal, complemented
5#27072C214yesmaximal, complemented
6#27071C214yesmaximal, complemented
7#23735C211yesmaximal, characteristic, complemented
8#23540C211yesmaximal, characteristic
9#23506C211yesmaximal, characteristic, complemented
10C2×#2012C211yesmaximal, characteristic
11#21231C211yesmaximal, characteristic, complemented
12#21064C211yesmaximal, characteristic, complemented
13#21060C211yesmaximal, characteristic, complemented
14#21032C211yesmaximal, characteristic, complemented
15#21026C211yesmaximal, characteristic, complemented
16#20936C211yesmaximal, characteristic, complemented
17#20934C211yesmaximal, characteristic
18#20651C211yesmaximal, characteristic, complemented
19#20416C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×D8C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2513—22no—
38#2513C2214yescomplemented
39#2512—22no—
40#2512C2214yescomplemented
41#2480—24no—
42#2367C2214yescomplemented
43#2367C2214yescomplemented
44#2342—24no—
45#2326—22no—
46#2317C2211yescharacteristic
47#2310C2211yescharacteristic
48#2309C2211yescharacteristic
49#2309C2211yescharacteristic
50#2309C2211yescharacteristic
51C4×#124—22no—
52#2296C2211yescharacteristic
53C2×#390C2211yescharacteristic
54#2285C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2094C2212yescomplemented
58#2093C2212yescomplemented
59#2092C2212yescomplemented
60#2092C2212yescomplemented
61#2054C2214yes—
62#2054C2214yes—
63#2042C2214yescomplemented
64#2042C2214yescomplemented
65#2012C2214yes—
66#2012C2214yes—
67#1973C2212yescomplemented
68#1973C2212yescomplemented
69#1968C2212yescomplemented
70#1968C2212yescomplemented
71#1928C2214yescomplemented
72#1927C2214yescomplemented
73#1926C2214yescomplemented
74#1925C2214yescomplemented
75#1921C2212yescomplemented
76#1920C2212yescomplemented
77#1919C2212yescomplemented
78#1918C2212yescomplemented
79#1917C2214yescomplemented
80#1916C2214yescomplemented
81#1880C2214yescomplemented
82#1880C2214yescomplemented
83#1879C2214yes—
84#1879C2214yes—
85#1805C2214yescomplemented
86#1803C2214yescomplemented
87#1762C2214yescomplemented
88C4×#102C2214yescomplemented
89#1736C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—24no—
102C22×SD16—42no—
103C22×SD16C2312yes—
104C22×SD16C2312yes—
105C22×D8—24no—
106C22×D8—42no—
107C22×D8—42no—
108C22×D8C2314yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C2×#126—24no—
114C2×#124—24no—
115C2×#124—24no—
116C2×#124—21no—
117C2×#124—21no—
118C2×#124C2314yes—
119#519—24no—
120#519—42no—
121#518—24no—
122#518—42no—
123C2×C4×Q8—22no—
124C2×C4×Q8C2314yes—
125C2×C4×D4—22no—
126C2×C4×D4C2314yes—
127C2×#117—22no—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—22no—
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C42C2311yescharacteristic
139#428—42no—
140#427—24no—
141C2×#107C2311yescharacteristic
142C2×#107C2311yescharacteristic
143C2×#107C2312yes—
144C2×#106C2311yescharacteristic
145C2×#106C2311yescharacteristic
146C2×#106C2312yes—
147#424—42no—
148C2×C4⋊C8C2314yes—
149#419—24no—
150#419—42no—
151#418—24no—
152#418—24no—
153#418—42no—
154#418—42no—
155C2×#103—42no—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#102—24no—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169#405—42no—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#397—24no—
173#397—24no—
174#397—24no—
175#396—24no—
176#391—22no—
177#391C2312yes—
178#391C2312yes—
179#390—22no—
180#390—22no—
181#390—22no—
182#390—21no—
183#390—21no—
184#390C2312yes—
185#390C2312yes—
186#390C2314yes—
187#390C2312yes—
188#390C2312yes—
189#390C2314yes—
190#389—22no—
191#389C2312yes—
192#389C2312yes—
193#388—24no—
194#388—24no—
195#388—24no—
196#388—24no—
197#388—24no—
198#388—24no—
199#388—24no—
200#385—22no—
201#385—24no—
202#385—24no—
203#385—22no—
204#384D412yes—
205#384C2312yes—
206#382D412yes—
207#382C2312yes—
208#381—22no—
209#380D414yes—
210#379—22no—
211#379—22no—
212#379—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—24no—
223C4×C22⋊C4—24no—
224#376—21no—
225C43—21no—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2312yes—
237#336C2314yes—
238#336C2312yes—
239C25—21no—
240C25—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8#4114yes—
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4#4114yes—
263C22×D4C2412yes—
264C22×D4C2412yes—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—24no—
268C23×C4C2×D412yes—
269C23×C4C2411yescharacteristic
270C2×SD16—24no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—42no—
275C2×SD16—42no—
276C2×SD16—416no—
277C2×D8—28no—
278C2×D8—44no—
279C2×D8—44no—
280C2×D8—416no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8#4114yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2411yescharacteristic
297C22×C8C2411yescharacteristic
298#126—88no—
299#124—28no—
300#124—28no—
301#124—28no—
302#124—28no—
303C4×Q8—28no—
304C4×Q8—44no—
305C4×Q8—48no—
306C4×D4—28no—
307C4×D4—44no—
308C4×D4—48no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—48no—
321#117#4114yes—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—216no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—48no—
357C2×C22⋊C4—84no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—24no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2412yes—
373C2×C42C2412yes—
374C2×C42C2412yes—
375#107—22no—
376#107—22no—
377#107—42no—
378#106—22no—
379#106—22no—
380#106—42no—
381C4⋊C8—44no—
382#103—28no—
383#103—44no—
384#103—42no—
385#103—42no—
386#102—28no—
387#102—24no—
388#102—24no—
389#102—28no—
390#102—42no—
391#102—42no—
392#102—42no—
393#102—42no—
394#97—42no—
395C4×C8—42no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95C2×D414yes—
405#95C2×D414yes—
406C24—22no—
407C24—24no—
408C24—22no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—82no—
415C24—82no—
416C24—82no—
417C24—82no—
418C24C22×D411yescharacteristic
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—28no—
422C2×Q8—28no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8C2×#4114yes—
432C2×D4—24no—
433C2×D4—28no—
434C2×D4—28no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—42no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4C2×#4114yes—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—48no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4—82no—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D414yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4C2511yescharacteristic
489SD16—48no—
490SD16—48no—
491D8—416no—
492M16—44no—
493M16—44no—
494C2×C8—28no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—44no—
509C2×C8C2×#4112yes—
510C2×C8C2×#4112yes—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—48no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—82no—
532C4⋊C4—84no—
533C4⋊C4C2×#4114yes—
534C22⋊C4—216no—
535C22⋊C4—216no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C42—22no—
552C42—22no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—48no—
561C42—42no—
562C42—42no—
563C42—42no—
564C42—82no—
565C42—82no—
566C42—84no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—48no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—88no—
581C23—88no—
582C23—84no—
583C23—164no—
584C23—164no—
585C23—162no—
586C23—162no—
587C23—162no—
588C23—162no—
589C23#52512yes—
590C23#54612yes—
591C23#55012yes—
592C23C23×D411yescharacteristic
593Q8—24no—
594Q8—24no—
595Q8—42no—
596Q8—42no—
597Q8—48no—
598D4—24no—
599D4—24no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—48no—
607D4—84no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—48no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—84no—
641C2×C4—168no—
642C2×C4—162no—
643C2×C4#52512yes—
644C2×C4#52914yes—
645C2×C4#53814yes—
646C2×C4#54612yes—
647C2×C4#55012yes—
648C2×C4#55114yes—
649C2×C4#55314yes—
650C2×C4C22×#4111yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C8—22no—
657C8—22no—
658C8—42no—
659C8—42no—
660C8—42no—
661C22—24no—
662C22—24no—
663C22—22no—
664C22—22no—
665C22—44no—
666C22—48no—
667C22—84no—
668C22—82no—
669C22—82no—
670C22—84no—
671C22—82no—
672C22—84no—
673C22—82no—
674C22—82no—
675C22—168no—
676C22—168no—
677C22#246714yes—
678C22C2×#52511yescharacteristic
679C22C2×#54611yescharacteristic
680C22C2×#55011yescharacteristic
681C22#358412yes—
682C22#358512yes—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—42no—
688C4—42no—
689C4—82no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4—81no—
694C4—82no—
695C4#246714yes—
696C4C2×#52911yescharacteristic
697C4C2×#53811yescharacteristic
698C4C2×#55111yescharacteristic
699C4C2×#55311yescharacteristic
700C2—24no—
701C2—84no—
702C2—162no—
703C2C2×#246711yescharacteristic
704C2#6058012yes—
705C2#6120812yes—
706C2#6141211yescharacteristic
707C2#6141711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35558C2semidirect
2. #34211C2semidirect
3. #27089C2semidirect
4. #27086C2semidirect
5. #27072C2semidirect
6. #27071C2semidirect
7. #23735C2semidirect
9. #23506C2semidirect
11. #21231C2semidirect
12. #21064C2semidirect
13. #21060C2semidirect
14. #21032C2semidirect
15. #21026C2semidirect
16. #20936C2semidirect
18. #20651C2semidirect
19. #20416C2semidirect
38. #2513C22semidirect
40. #2512C22semidirect
42. #2367C22semidirect
43. #2367C22semidirect
57. #2094C22semidirect
58. #2093C22semidirect
59. #2092C22semidirect
60. #2092C22semidirect
63. #2042C22semidirect
64. #2042C22semidirect
67. #1973C22semidirect
68. #1973C22semidirect
69. #1968C22semidirect
70. #1968C22semidirect
71. #1928C22semidirect
72. #1927C22semidirect
73. #1926C22semidirect
74. #1925C22semidirect
75. #1921C22semidirect
76. #1920C22semidirect
77. #1919C22semidirect
78. #1918C22semidirect
79. #1917C22semidirect
80. #1916C22semidirect
81. #1880C22semidirect
82. #1880C22semidirect
85. #1805C22semidirect
86. #1803C22semidirect
87. #1762C22semidirect
88. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#355582
#342112
#270892
#270862
#270722
#270712
#237352
#235402
#235062
C2×#20122
#212312
#210642
#210602
#210322
#210262
#209362
#209342
#206512
#204162