Finite Group Databasebeta

← #6441489

Subgroups — #6441489

Order 512.

1882 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35630C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic
3#27088C214yesmaximal, complemented
4#27087C214yesmaximal, complemented
5#27073C214yesmaximal, complemented
6#27070C214yesmaximal, complemented
7#23735C211yesmaximal, characteristic
8#23541C211yesmaximal, characteristic
9#23506C211yesmaximal, characteristic
10C2×#2015C211yesmaximal, characteristic
11#21232C211yesmaximal, characteristic, complemented
12#21064C211yesmaximal, characteristic, complemented
13#21061C211yesmaximal, characteristic, complemented
14#21032C211yesmaximal, characteristic, complemented
15#21026C211yesmaximal, characteristic, complemented
16#20935C211yesmaximal, characteristic, complemented
17#20933C211yesmaximal, characteristic
18#20651C211yesmaximal, characteristic, complemented
19#20416C211yesmaximal, characteristic
20C23×Q16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2514—22no—
38#2514C2214yescomplemented
39#2511—22no—
40#2511C2214yescomplemented
41#2493—24no—
42#2367C2214yes—
43#2367C2214yes—
44#2342—24no—
45#2326—22no—
46#2317C2211yescharacteristic
47#2310C2211yescharacteristic
48#2310C2211yescharacteristic
49#2310C2211yescharacteristic
50#2309C2211yescharacteristic
51C4×#124—22no—
52#2296C2211yescharacteristic
53C2×#391C2211yescharacteristic
54#2285C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2094C2212yes—
58#2093C2212yes—
59#2092C2212yes—
60#2092C2212yes—
61#2055C2214yes—
62#2055C2214yes—
63#2042C2214yes—
64#2042C2214yes—
65#2015C2214yes—
66#2015C2214yes—
67#1971C2212yescomplemented
68#1971C2212yescomplemented
69#1969C2212yescomplemented
70#1969C2212yescomplemented
71#1928C2214yescomplemented
72#1928C2214yescomplemented
73#1927C2214yescomplemented
74#1927C2214yescomplemented
75#1921C2212yescomplemented
76#1920C2212yescomplemented
77#1919C2212yescomplemented
78#1918C2212yescomplemented
79#1917C2214yescomplemented
80#1916C2214yescomplemented
81#1878C2214yescomplemented
82#1878C2214yescomplemented
83#1877C2214yes—
84#1877C2214yes—
85#1805C2214yescomplemented
86#1803C2214yescomplemented
87#1762C2214yes—
88C4×#102C2214yes—
89#1736C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×Q16—24no—
101C22×Q16—24no—
102C22×Q16—42no—
103C22×Q16C2312yes—
104C22×Q16C2312yes—
105C22×SD16—24no—
106C22×SD16—42no—
107C22×SD16—42no—
108C22×SD16C2314yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C2×#124—24no—
114C2×#124—24no—
115C2×#124—21no—
116C2×#124—21no—
117C2×#124C2314yes—
118#519—24no—
119#519—42no—
120#518—24no—
121#518—42no—
122C2×C4×Q8—22no—
123C2×C4×Q8C2314yes—
124C2×C4×D4—22no—
125C2×C4×D4C2314yes—
126C2×#117—22no—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137#428—42no—
138#427—24no—
139C2×#107C2311yescharacteristic
140C2×#107C2311yescharacteristic
141C2×#107C2312yes—
142C2×#106C2311yescharacteristic
143C2×#106C2311yescharacteristic
144C2×#106C2312yes—
145#424—42no—
146C2×C4⋊C8C2314yes—
147#419—24no—
148#419—24no—
149#419—42no—
150#419—42no—
151#418—24no—
152#418—42no—
153C2×#103—42no—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#102—24no—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167#405—42no—
168C2×#97C2312yes—
169C2×C4×C8C2312yes—
170#401—24no—
171#400—24no—
172#400—24no—
173#400—24no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391—21no—
178#391—21no—
179#391C2314yes—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#391C2314yes—
185#390—22no—
186#390C2312yes—
187#390C2312yes—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#388—24no—
192#388—24no—
193#388—24no—
194#388—24no—
195#387—24no—
196#387—24no—
197#387—24no—
198#385—22no—
199#385—24no—
200#385—24no—
201#385—22no—
202#384D412yes—
203#384C2312yes—
204#383—24no—
205#382D412yes—
206#382C2312yes—
207#381—22no—
208#381—22no—
209#381—22no—
210#380D414yes—
211#379—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2314yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—24no—
222C4×C22⋊C4—24no—
223#376—21no—
224C2×#95—22no—
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—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8#4114yes—
254C22×Q8C2412yes—
255C22×Q8C2412yes—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4#4114yes—
263C22×D4C2412yes—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—24no—
268C23×C4C2×D412yes—
269C23×C4C2411yescharacteristic
270C2×Q16—24no—
271C2×Q16—24no—
272C2×Q16—24no—
273C2×Q16—24no—
274C2×Q16—42no—
275C2×Q16—42no—
276C2×Q16—416no—
277C2×SD16—28no—
278C2×SD16—44no—
279C2×SD16—44no—
280C2×SD16—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#124—28no—
299#124—28no—
300#124—28no—
301#124—28no—
302C4×Q8—28no—
303C4×Q8—44no—
304C4×Q8—48no—
305C4×D4—28no—
306C4×D4—44no—
307C4×D4—48no—
308#117—22no—
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—48no—
320#117#4114yes—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D414yes—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—216no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—48no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2412yes—
365C2×C42C2412yes—
366C2×C42C2412yes—
367#107—22no—
368#107—22no—
369#107—42no—
370#106—22no—
371#106—22no—
372#106—42no—
373C4⋊C8—44no—
374#103—28no—
375#103—28no—
376#103—44no—
377#103—42no—
378#103—42no—
379#103—42no—
380#103—42no—
381#102—28no—
382#102—24no—
383#102—24no—
384#102—42no—
385#102—42no—
386#97—42no—
387C4×C8—42no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95C2×D414yes—
403#95C2×D414yes—
404C24—22no—
405C24—24no—
406C24—42no—
407C24—42no—
408C24—82no—
409C24—82no—
410C24C22×D411yescharacteristic
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—28no—
414C2×Q8—28no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—42no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8C2×#4114yes—
437C2×D4—24no—
438C2×D4—28no—
439C2×D4—28no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4C2×#4114yes—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—48no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—82no—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D414yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4C2511yescharacteristic
485Q16—48no—
486Q16—48no—
487SD16—416no—
488M16—44no—
489M16—44no—
490C2×C8—28no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—44no—
505C2×C8C2×#4112yes—
506C2×C8C2×#4112yes—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4C2×#4114yes—
530C22⋊C4—216no—
531C22⋊C4—216no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—48no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C42—22no—
542C42—22no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—48no—
551C42—42no—
552C42—42no—
553C42—42no—
554C42—84no—
555C42—82no—
556C42—82no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—48no—
565C23—88no—
566C23—164no—
567C23—162no—
568C23—162no—
569C23#52512yes—
570C23#54612yes—
571C23#55012yes—
572C23C23×D411yescharacteristic
573Q8—24no—
574Q8—24no—
575Q8—42no—
576Q8—42no—
577Q8—48no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—84no—
583D4—24no—
584D4—24no—
585D4—42no—
586D4—42no—
587D4—48no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—48no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—168no—
626C2×C4—162no—
627C2×C4#52512yes—
628C2×C4#52914yes—
629C2×C4#53814yes—
630C2×C4#54612yes—
631C2×C4#55012yes—
632C2×C4#55114yes—
633C2×C4#55314yes—
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C8—22no—
641C8—22no—
642C8—42no—
643C8—42no—
644C8—42no—
645C22—24no—
646C22—24no—
647C22—22no—
648C22—22no—
649C22—44no—
650C22—48no—
651C22—84no—
652C22—82no—
653C22—82no—
654C22—168no—
655C22#246714yes—
656C22C2×#52511yescharacteristic
657C22C2×#54611yescharacteristic
658C22C2×#55011yescharacteristic
659C22#358412yes—
660C22#358612yes—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—82no—
668C4—82no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4—81no—
673C4—81no—
674C4#246714yes—
675C4C2×#52911yescharacteristic
676C4C2×#53811yescharacteristic
677C4C2×#55111yescharacteristic
678C4C2×#55311yescharacteristic
679C2—24no—
680C2—84no—
681C2C2×#246711yescharacteristic
682C2#6058112yes—
683C2#6120712yes—
684C2#6141311yescharacteristic
685C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #35630C2semidirect
3. #27088C2semidirect
4. #27087C2semidirect
5. #27073C2semidirect
6. #27070C2semidirect
11. #21232C2semidirect
12. #21064C2semidirect
13. #21061C2semidirect
14. #21032C2semidirect
15. #21026C2semidirect
16. #20935C2semidirect
18. #20651C2semidirect
38. #2514C22semidirect
40. #2511C22semidirect
67. #1971C22semidirect
68. #1971C22semidirect
69. #1969C22semidirect
70. #1969C22semidirect
71. #1928C22semidirect
72. #1928C22semidirect
73. #1927C22semidirect
74. #1927C22semidirect
75. #1921C22semidirect
76. #1920C22semidirect
77. #1919C22semidirect
78. #1918C22semidirect
79. #1917C22semidirect
80. #1916C22semidirect
81. #1878C22semidirect
82. #1878C22semidirect
85. #1805C22semidirect
86. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#356302
#342112
#270882
#270872
#270732
#270702
#237352
#235412
#235062
C2×#20152
#212322
#210642
#210612
#210322
#210262
#209352
#209332
#206512
#204162