Finite Group Databasebeta

← #6640639

Subgroups — #6640639

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic, complemented
2C2×#2500C211yesmaximal, characteristic
3#32984C212yesmaximal, complemented
4#32283C214yesmaximal, complemented
5#32047C214yesmaximal, complemented
6#31931C214yesmaximal, complemented
7#31837C214yesmaximal, complemented
8#29598C212yesmaximal
9#29103C211yesmaximal, characteristic, complemented
10#28302C211yesmaximal, characteristic, complemented
11#28123C212yesmaximal, complemented
12#27961C211yesmaximal, characteristic, complemented
13#27910C211yesmaximal, characteristic, complemented
14#27653C212yesmaximal, complemented
15#21902C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#484C2212yes—
19C2×#484C2212yes—
20C2×#483C2212yes—
21C2×#450C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2848C2212yes—
25#2646—21no—
26#2626—21no—
27#2589—22no—
28#2587—22no—
29#2500—22no—
30#2500—22no—
31#2500C2214yes—
32#2500C2214yes—
33#2497C2212yescomplemented
34#2492C2212yescomplemented
35#2429C2212yescomplemented
36#2423C2212yescomplemented
37C2×#394—21no—
38C2×#394—21no—
39#2329—21no—
40#2312—21no—
41C4×C22⋊Q8C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#388—21no—
45C2×#388—21no—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2100C2214yescomplemented
50#2095C2214yescomplemented
51#2094C2212yescomplemented
52#2085C2214yescomplemented
53#2074C2214yescomplemented
54#2071C2212yescomplemented
55#2067C2214yescomplemented
56#2055C2214yescomplemented
57#2033C2214yes—
58#2033C2214yes—
59#2023C2212yescomplemented
60#2013C2212yescomplemented
61#2000C2214yes—
62#2000C2214yes—
63#1990C2212yescomplemented
64#1968C2214yescomplemented
65#1956C2212yescomplemented
66#1949C2214yescomplemented
67#1934C2214yescomplemented
68#1925C2214yescomplemented
69#1923C2214yescomplemented
70#1923C2214yescomplemented
71#1880C2214yescomplemented
72#1873C2214yescomplemented
73#1873C2214yescomplemented
74#1854C2214yescomplemented
75#1854C2214yescomplemented
76#1845C2212yescomplemented
77#1834C2212yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1805C2214yescomplemented
82#1797C2212yes—
83#1786C2212yes—
84#1779C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C22×SD16C2314yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8C2312yes—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×#121—22no—
98C2×#121—44no—
99C2×#121C2314yes—
100C2×C4×Q8C2312yes—
101C2×#117—22no—
102C2×#117—44no—
103C2×#117C2314yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4D411yescharacteristic
116C22×C22⋊C4D411yescharacteristic
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#484—44no—
121#484—44no—
122#483—44no—
123#450—44no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140#400—22no—
141#400—22no—
142#399—22no—
143#398—22no—
144#398—22no—
145#396—22no—
146#396—22no—
147#394—24no—
148#394—24no—
149#394—24no—
150#394—24no—
151#393—22no—
152#391—21no—
153#391—22no—
154#391C2312yes—
155#391C2312yes—
156#390—22no—
157#390—21no—
158#390—21no—
159#390C2314yes—
160#390C2312yes—
161#390C2312yes—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—24no—
173#388—24no—
174#388—24no—
175#388—24no—
176#387—22no—
177#387—22no—
178#387—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#382—22no—
190#382C2314yes—
191#380—21no—
192#380—21no—
193#380—21no—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#379—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4C2312yes—
204C4×C22⋊C4C2312yes—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95D411yescharacteristic
208C2×#95D411yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2312yes—
217#336C2312yes—
218C25—21no—
219C25C2×D411yescharacteristic
220C22×Q8—21no—
221C22×Q8C2412yes—
222C22×D4—24no—
223C22×D4—21no—
224C22×D4—24no—
225C22×D4—42no—
226C22×D4—42no—
227C22×D4—44no—
228C22×D4—82no—
229C22×D4—82no—
230C22×D4C2412yes—
231C23×C4—24no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C2×SD16—44no—
246C2×SD16—44no—
247C2×SD16—44no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—44no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257#121—48no—
258#121—48no—
259C4×Q8—44no—
260#117—48no—
261#117—48no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—82no—
314C2×C22⋊C4—82no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42—44no—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—44no—
332#107—44no—
333#106—44no—
334C4⋊C8—44no—
335#103—44no—
336#103—44no—
337#103—44no—
338#102—44no—
339#102—44no—
340#102—44no—
341C22⋊C8—24no—
342C22⋊C8—24no—
343C22⋊C8—44no—
344C22⋊C8—44no—
345C4×C8—42no—
346C4×C8—42no—
347#95—24no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—41no—
368C24—41no—
369C24—41no—
370C24—41no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24C22×D412yes—
379C24C22×D411yescharacteristic
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—42no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—48no—
394C2×D4—42no—
395C2×D4—88no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—816no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—416no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4#13514yes—
438C22×C4#13714yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4H(5,𝔽2)14yes—
448C22×C4C2511yescharacteristic
449SD16—88no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C42—22no—
502C42—22no—
503C42—24no—
504C42—41no—
505C42—41no—
506C42—42no—
507C42—48no—
508C42—82no—
509C42—82no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23#53512yes—
543C23C2×#13512yes—
544C23C2×#13712yes—
545C23C23×D411yescharacteristic
546Q8—44no—
547Q8—44no—
548Q8—162no—
549D4—44no—
550D4—44no—
551D4—84no—
552D4—816no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4#53512yes—
605C2×C4C2×#13512yes—
606C2×C4C2×#13512yes—
607C2×C4C2×#13712yes—
608C2×C4C2×#13712yes—
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4#58512yes—
611C2×C4#58512yes—
612C8—44no—
613C8—44no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—88no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—168no—
636C22#321014yes—
637C22C2×#53511yescharacteristic
638C22#348511yescharacteristic
639C22#348511yescharacteristic
640C22C22×#13511yescharacteristic
641C22C22×#13711yescharacteristic
642C22#358511yescharacteristic
643C22#358511yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4#353314yes—
660C2—24no—
661C2—82no—
662C2—162no—
663C2#3695611yescharacteristic
664C2C2×#321011yescharacteristic
665C2#6080611yescharacteristic
666C2#6169911yescharacteristic
667C2#6170011yescharacteristic
668C2#6170811yescharacteristic
669C2#6172111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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. #32984C2semidirect
4. #32283C2semidirect
5. #32047C2semidirect
6. #31931C2semidirect
7. #31837C2semidirect
9. #29103C2semidirect
10. #28302C2semidirect
11. #28123C2semidirect
12. #27961C2semidirect
13. #27910C2semidirect
14. #27653C2semidirect
15. #21902C2semidirect
33. #2497C22semidirect
34. #2492C22semidirect
35. #2429C22semidirect
36. #2423C22semidirect
49. #2100C22semidirect
50. #2095C22semidirect
51. #2094C22semidirect
52. #2085C22semidirect
53. #2074C22semidirect
54. #2071C22semidirect
55. #2067C22semidirect
56. #2055C22semidirect
59. #2023C22semidirect
60. #2013C22semidirect
63. #1990C22semidirect
64. #1968C22semidirect
65. #1956C22semidirect
66. #1949C22semidirect
67. #1934C22semidirect
68. #1925C22semidirect
69. #1923C22semidirect
70. #1923C22semidirect
71. #1880C22semidirect
72. #1873C22semidirect
73. #1873C22semidirect
74. #1854C22semidirect
75. #1854C22semidirect
76. #1845C22semidirect
77. #1834C22semidirect
81. #1805C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#25002
#329842
#322832
#320472
#319312
#318372
#295982
#291032
#283022
#281232
#279612
#279102
#276532
#219022