Finite Group Databasebeta

← #227182

Subgroups — #227182

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#455C211yesmaximal, characteristic
2#17894C211yesmaximal, characteristic, complemented
3#17876C211yesmaximal, characteristic, complemented
4#17453C211yesmaximal, characteristic
5#17438C211yesmaximal, characteristic
6#9640C211yesmaximal, characteristic, complemented
7#9552C211yesmaximal, characteristic, complemented
8#9550C211yesmaximal, characteristic, complemented
9#9098C211yesmaximal, characteristic, complemented
10#9096C211yesmaximal, characteristic, complemented
11#8964C211yesmaximal, characteristic, complemented
12#8841C211yesmaximal, characteristic, complemented
13#8839C211yesmaximal, characteristic, complemented
14#8195C211yesmaximal, characteristic, complemented
15#8191C211yesmaximal, characteristic, complemented
16C4×H(5,𝔽2)C2211yescharacteristic
17C2×#455C411yescharacteristic
18#2989C2211yescharacteristic
19#2988C2211yescharacteristic
20#2983C2211yescharacteristic, complemented
21#2965C2211yescharacteristic
22C8×#41C2211yescharacteristic
23C4×#137C2211yescharacteristic
24C4×#137C2211yescharacteristic
25C4×#124C2211yescharacteristic
26#1885C411yescharacteristic
27#1882C411yescharacteristic
28#1882C411yescharacteristic
29#1860C411yescharacteristic
30#1860C411yescharacteristic
31#1856C411yescharacteristic
32#1842C2211yescharacteristic, complemented
33#1792C411yescharacteristic
34C4×#104C2211yescharacteristic
35C4×#104C2211yescharacteristic
36C4×#100C2211yescharacteristic
37#1744C2211yescharacteristic, complemented
38#1593—21no—
39#1592—21no—
40#1591—21no—
41#1590—21no—
42#1587—21no—
43#1585—21no—
44#1584—21no—
45#1582—21no—
46#1565C2211yescharacteristic, complemented
47#1565C2211yescharacteristic, complemented
48#1490—21no—
49#1489—21no—
50#1488—21no—
51#1486—21no—
52#1478—21no—
53#1478C2211yescharacteristic
54#1478C2211yescharacteristic
55#1477—21no—
56#1477C2211yescharacteristic
57#1477C2211yescharacteristic
58#1476—21no—
59#1476C2211yescharacteristic
60#1476C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic, complemented
62C2×Q8⋊C8C2211yescharacteristic, complemented
63C2×D4⋊C8—21no—
64C2×D4⋊C8C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66#1461C2211yescharacteristic, complemented
67#1451C2211yescharacteristic, complemented
68C8×M16C2211yescharacteristic, complemented
69#1283C2211yescharacteristic, complemented
70#1283C2211yescharacteristic, complemented
71#1278C2211yescharacteristic, complemented
72#1278C2211yescharacteristic, complemented
73#1276C2211yescharacteristic, complemented
74#1276C2211yescharacteristic, complemented
75C2×H(5,𝔽2)C2×C411yescharacteristic
76C2×#137C2×C411yescharacteristic
77C2×#137C2×C411yescharacteristic
78C2×#124C2×C411yescharacteristic
79#520—21no—
80#520—21no—
81#520C2×C411yescharacteristic
82#520C2×C411yescharacteristic
83#518—21no—
84#518—21no—
85#518—41no—
86#518C2×C411yescharacteristic
87C4×#41—21no—
88C4×#41—21no—
89C4×#41C2311yescharacteristic
90C4×#41C2311yescharacteristic
91C2×C4×Q8C2311yescharacteristic
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94C2×C4×D4—41no—
95C2×C4×D4C2311yescharacteristic
96#455—24no—
97#446—21no—
98#446—21no—
99#446—21no—
100#446—21no—
101#446—21no—
102#446—21no—
103C8×Q8—21no—
104C8×Q8—21no—
105#441—21no—
106#441—21no—
107#440—21no—
108#440—21no—
109C4×Q16—21no—
110C4×Q16—21no—
111C4×SD16—21no—
112C4×SD16—21no—
113#436—21no—
114#436—21no—
115#435—41no—
116#435—41no—
117C8×D4—21no—
118C8×D4—21no—
119C8×D4—41no—
120C8×D4—41no—
121#432—21no—
122#432—41no—
123#431—21no—
124#431—21no—
125#431—21no—
126#431C2311yescharacteristic
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#428C2×C411yescharacteristic
131#428C2×C411yescharacteristic
132#423—21no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8—41no—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×#104C2×C411yescharacteristic
138C2×#104C2×C411yescharacteristic
139C2×#104C2×C411yescharacteristic
140C2×#104C2×C411yescharacteristic
141#419C2×C411yescharacteristic
142#419C2×C411yescharacteristic
143#417C2×C411yescharacteristic
144#417C2×C411yescharacteristic
145C2×#100C2×C411yescharacteristic
146C2×#100C2×C411yescharacteristic
147C4×M16—21no—
148C4×M16C2311yescharacteristic
149C4×M16C2311yescharacteristic
150C2×C4×C8—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153#391C2×C411yescharacteristic
154#390C2×C411yescharacteristic
155#388—41no—
156#382C2×C411yescharacteristic
157C4×C22⋊C4—41no—
158C43C2311yescharacteristic
159#343C2×C411yescharacteristic
160#343C2×C411yescharacteristic
161#339C2×C411yescharacteristic
162#339C2×C411yescharacteristic
163#339C2×C411yescharacteristic
164#339C2×C411yescharacteristic
165#336—41no—
166#335—21no—
167#335—21no—
168#334—21no—
169#334—21no—
170Q8⋊C8—21no—
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173Q8⋊C8—21no—
174Q8⋊C8—21no—
175Q8⋊C8—21no—
176Q8⋊C8D411yescharacteristic
177Q8⋊C8D411yescharacteristic
178Q8⋊C8D411yescharacteristic
179Q8⋊C8D411yescharacteristic
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8—21no—
188D4⋊C8—21no—
189D4⋊C8—21no—
190D4⋊C8D411yescharacteristic
191D4⋊C8D411yescharacteristic
192D4⋊C8D411yescharacteristic
193D4⋊C8D411yescharacteristic
194#324—41no—
195#322—21no—
196#322—21no—
197#322—21no—
198C82—21no—
199H(5,𝔽2)—22no—
200H(5,𝔽2)—22no—
201H(5,𝔽2)—22no—
202H(5,𝔽2)—41no—
203C2×#41—21no—
204C2×#41—21no—
205C2×#41C22×C411yescharacteristic
206C2×#41C22×C411yescharacteristic
207C22×Q8C22×C411yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—41no—
211C22×D4C22×C411yescharacteristic
212C23×C4—21no—
213C23×C4—41no—
214#137—22no—
215#137—22no—
216#137—22no—
217#137—41no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×SD16—21no—
221C2×SD16—21no—
222C2×M16—41no—
223C2×M16—41no—
224C2×M16C22×C411yescharacteristic
225C2×M16C22×C411yescharacteristic
226C2×M16C22×C411yescharacteristic
227C2×M16C22×C411yescharacteristic
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—41no—
233C22×C8—41no—
234#124—22no—
235#124—22no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8C2×D411yescharacteristic
243C4×Q8C2×D411yescharacteristic
244C4×Q8#4111yescharacteristic
245C4×Q8#4111yescharacteristic
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4—41no—
263C4×D4—41no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—41no—
270C4×D4C2×D411yescharacteristic
271C4×D4C2×D411yescharacteristic
272C4×D4#4111yescharacteristic
273C4×D4#4111yescharacteristic
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—41no—
279#117—41no—
280#117C22×C411yescharacteristic
281#117C22×C411yescharacteristic
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—41no—
301C2×C42C22×C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C2411yescharacteristic
304#107—21no—
305#107—21no—
306#106—21no—
307#106—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—41no—
315C4⋊C8—41no—
316C4⋊C8—41no—
317C4⋊C8—41no—
318C4⋊C8—41no—
319C4⋊C8—41no—
320C4⋊C8—82no—
321C4⋊C8C2×D411yescharacteristic
322C4⋊C8C2×D411yescharacteristic
323C4⋊C8C2×D411yescharacteristic
324C4⋊C8C2×D411yescharacteristic
325#104—22no—
326#104—22no—
327#104—41no—
328#104M1614yescomplemented
329#103—21no—
330#103—21no—
331#103—21no—
332#103—21no—
333#103—21no—
334#103—21no—
335#102—21no—
336#102—21no—
337#100—22no—
338#100—22no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—21no—
342C22⋊C8—21no—
343C22⋊C8—41no—
344C22⋊C8—41no—
345C22⋊C8—41no—
346C22⋊C8—41no—
347#97—21no—
348#97—21no—
349#97—41no—
350#97—41no—
351#97—41no—
352#97—41no—
353C4×C8—21no—
354C4×C8—21no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—41no—
362C4×C8—41no—
363C4×C8C2×D411yescharacteristic
364C4×C8C2×D411yescharacteristic
365C4×C8C2×D411yescharacteristic
366C4×C8C2×D411yescharacteristic
367#95—41no—
368#95—41no—
369C24—21no—
370C24—41no—
371#41—21no—
372#41—21no—
373#41—24no—
374#41—21no—
375#41—21no—
376#41—21no—
377#41—21no—
378#41—42no—
379#41—42no—
380#41—42no—
381#41C2×M1612yes—
382#41#13112yes—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8C4×D411yescharacteristic
390C2×Q8C4×D411yescharacteristic
391C2×D4—21no—
392C2×D4—24no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—42no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4C4×D411yescharacteristic
407C2×D4C4×D411yescharacteristic
408C2×D4#13112yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—161no—
430C22×C4—161no—
431C22×C4C23×C411yescharacteristic
432C22×C4H(5,𝔽2)11yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434Q16—42no—
435Q16—42no—
436SD16—42no—
437SD16—42no—
438M16—21no—
439M16—21no—
440M16—21no—
441M16—21no—
442M16—21no—
443M16—21no—
444M16—21no—
445M16—82no—
446M16—161no—
447M16C2×M1612yes—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—81no—
465C2×C8—81no—
466C2×C8—82no—
467C2×C8—81no—
468C2×C8—81no—
469C2×C8—161no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4C4×D411yescharacteristic
483C4⋊C4C4×D411yescharacteristic
484C4⋊C4C4×D411yescharacteristic
485C4⋊C4C4×D411yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—44no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—82no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—81no—
516C42C2×M1612yes—
517C42C22×D411yescharacteristic
518C42C22×D411yescharacteristic
519C42C2×#4111yescharacteristic
520C42C2×#4111yescharacteristic
521C23—21no—
522C23—21no—
523C23—22no—
524C23—44no—
525C23—41no—
526C23—41no—
527C23—81no—
528C23—81no—
529C23—161no—
530C23—161no—
531C23#51811yescharacteristic
532Q8—21no—
533Q8—21no—
534Q8—21no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8#43612yes—
540D4—22no—
541D4—22no—
542D4—21no—
543D4—22no—
544D4—21no—
545D4—22no—
546D4—21no—
547D4—44no—
548D4—44no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4#43612yes—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—161no—
590C2×C4—161no—
591C2×C4—161no—
592C2×C4—161no—
593C2×C4—161no—
594C2×C4C2×C4×D411yescharacteristic
595C2×C4C2×C4×D411yescharacteristic
596C2×C4D4211yescharacteristic
597C2×C4#54611yescharacteristic
598C2×C4#54611yescharacteristic
599C2×C4#55111yescharacteristic
600C2×C4C22×M1611yescharacteristic
601C2×C4C2×#13111yescharacteristic
602C2×C4C2×#13111yescharacteristic
603C2×C4#56811yescharacteristic
604C8—22no—
605C8—41no—
606C8—41no—
607C8—41no—
608C8—41no—
609C8—41no—
610C8—41no—
611C8—41no—
612C8—41no—
613C8—41no—
614C8—41no—
615C8—41no—
616C8—81no—
617C8—81no—
618C8—81no—
619C8—81no—
620C22—21no—
621C22—21no—
622C22—22no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—42no—
627C22—41no—
628C22—41no—
629C22—82no—
630C22—81no—
631C22—81no—
632C22—161no—
633C22—164no—
634C22#235811yescharacteristic
635C22#297411yescharacteristic
636C22#297611yescharacteristic
637C4—22no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4C2×#43611yescharacteristic
652C4#292911yescharacteristic
653C4#293311yescharacteristic
654C4D4×M1611yescharacteristic
655C4#328811yescharacteristic
656C4#328911yescharacteristic
657C2—21no—
658C2—21no—
659C2—42no—
660C2—81no—
661C2—82no—
662C2#2284811yescharacteristic
663C2#2289011yescharacteristic
664C2#2289811yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radicalC2×H(5,𝔽2)C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17894C2semidirect
3. #17876C2semidirect
6. #9640C2semidirect
7. #9552C2semidirect
8. #9550C2semidirect
9. #9098C2semidirect
10. #9096C2semidirect
11. #8964C2semidirect
12. #8841C2semidirect
13. #8839C2semidirect
14. #8195C2semidirect
15. #8191C2semidirect
20. #2983C22semidirect
32. #1842C22semidirect
37. #1744C22semidirect
46. #1565C22semidirect
47. #1565C22semidirect
61. C2×Q8⋊C8C22semidirect
62. C2×Q8⋊C8C22semidirect
66. #1461C22semidirect
67. #1451C22semidirect
68. C8×M16C22semidirect
69. #1283C22semidirect
70. #1283C22semidirect
71. #1278C22semidirect
72. #1278C22semidirect
73. #1276C22semidirect
74. #1276C22semidirect
328. #104M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4552
#178942
#178762
#174532
#174382
#96402
#95522
#95502
#90982
#90962
#89642
#88412
#88392
#81952
#81912