Finite Group Databasebeta

← #228207

Subgroups — #228207

Order 512.

721 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23114C211yesmaximal, characteristic, complemented
2#22505C211yesmaximal, characteristic, complemented
3#18522C211yesmaximal, characteristic
4#18001C211yesmaximal, characteristic, complemented
5#17508C211yesmaximal, characteristic, complemented
6#9851C211yesmaximal, characteristic, complemented
7#9591C211yesmaximal, characteristic, complemented
8#9583C211yesmaximal, characteristic, complemented
9#9233C211yesmaximal, characteristic, complemented
10#9208C211yesmaximal, characteristic, complemented
11#9055C211yesmaximal, characteristic, complemented
12#8906C211yesmaximal, characteristic, complemented
13#8903C211yesmaximal, characteristic, complemented
14#8192C211yesmaximal, characteristic, complemented
15#8172C211yesmaximal, characteristic, complemented
16#3475C2211yescharacteristic
17#3167C2211yescharacteristic
18#3100C2211yescharacteristic
19#3035C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2935C2211yescharacteristic
22D4×M16C2211yescharacteristic
23#2925C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1994C2211yescharacteristic
26#1986C2211yescharacteristic
27#1975—21no—
28#1962C2211yescharacteristic
29#1950C2211yescharacteristic
30#1929—21no—
31#1896—21no—
32#1876—21no—
33#1871—21no—
34#1826—21no—
35#1806C2211yescharacteristic
36#1767C2211yescharacteristic
37#1765—21no—
38C4×#104C2211yescharacteristic
39C2×#339—21no—
40C4×#97C2211yescharacteristic
41#1654C411yescharacteristic, complemented
42#1652C411yescharacteristic, complemented
43#1646C411yescharacteristic, complemented
44#1644C411yescharacteristic, complemented
45#1628C411yescharacteristic, complemented
46#1625C411yescharacteristic, complemented
47#1624C411yescharacteristic, complemented
48#1622C411yescharacteristic, complemented
49#1594—21no—
50#1586—21no—
51#1586—21no—
52#1580—21no—
53#1570C2211yescharacteristic
54#1567C2211yescharacteristic
55#1542C2211yescharacteristic
56#1502C2211yescharacteristic
57#1494—21no—
58#1493C2211yescharacteristic
59#1492C2211yescharacteristic
60#1488—21no—
61#1486C2211yescharacteristic
62#1482—21no—
63#1481C2211yescharacteristic
64#1476—21no—
65#1476C2211yescharacteristic
66C2×D4⋊C8C2211yescharacteristic
67C2×D4⋊C8C2211yescharacteristic
68#1451C2211yescharacteristic
69#1303C2211yescharacteristic
70#1301C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1278C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#135—21no—
76C2×#131—21no—
77C22×M16—21no—
78#556C2×C411yescharacteristic
79#555C2×C411yescharacteristic
80#554C2×C411yescharacteristic
81#553—21no—
82#552—21no—
83#552—21no—
84#551—41no—
85#547C2×C411yescharacteristic
86#546—21no—
87#533C2311yescharacteristic
88#532C2311yescharacteristic
89#529—21no—
90#529—41no—
91#525—21no—
92C4×#41—21no—
93C4×#41C2311yescharacteristic
94C2×C4×D4—21no—
95C2×C4×D4C2311yescharacteristic
96C2×#117—21no—
97C22×C42—21no—
98#498C2×C411yescharacteristic, complemented
99#492C2×C411yescharacteristic, complemented
100#487C2×C411yescharacteristic, complemented
101#486C2×C411yescharacteristic, complemented
102#480—21no—
103#474C2×C411yescharacteristic
104#474C2×C411yescharacteristic
105#465—41no—
106#464—21no—
107#463—21no—
108#462C2×C411yescharacteristic, complemented
109#461C2×C411yescharacteristic, complemented
110#460C2×C411yescharacteristic, complemented
111#459C2×C411yescharacteristic
112#459C2×C411yescharacteristic
113#459C2×C411yescharacteristic, complemented
114#447—21no—
115C8×Q8—21no—
116C4×SD16—41no—
117C4×D8—21no—
118C4×D8—21no—
119#436—21no—
120#436—21no—
121#435—41no—
122#435—41no—
123C8×D4—21no—
124C8×D4—21no—
125C8×D4—21no—
126C8×D4—41no—
127#433—41no—
128#432—21no—
129#431—21no—
130#431—21no—
131#431C2311yescharacteristic
132#431C2311yescharacteristic
133C2×#108—21no—
134#423C2311yescharacteristic
135#423C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×#104—21no—
139C2×#104—21no—
140C2×#104—21no—
141C2×#104D411yescharacteristic
142C2×#104D411yescharacteristic
143#416—21no—
144#408—21no—
145#408—21no—
146#407—21no—
147#405—21no—
148C4×M16C2311yescharacteristic
149C2×#97—21no—
150C2×#97—41no—
151C2×C4×C8—21no—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C4×C4⋊C4—21no—
155C4×C4⋊C4C2311yescharacteristic
156C4×C22⋊C4—21no—
157C4×C22⋊C4—41no—
158C43C2311yescharacteristic
159#341—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#339—21no—
164#339—21no—
165#339D411yescharacteristic
166#339D411yescharacteristic
167#336—21no—
168#335—41no—
169#334—21no—
170#334—21no—
171#333C2×C411yescharacteristic, complemented
172#333C2×C411yescharacteristic, complemented
173#332C2×C411yescharacteristic, complemented
174#332C2×C411yescharacteristic, complemented
175#327—41no—
176Q8⋊C8—21no—
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179Q8⋊C8—21no—
180Q8⋊C8C2×C411yescharacteristic, complemented
181Q8⋊C8C2×C411yescharacteristic, complemented
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8C2×C411yescharacteristic
187D4⋊C8C2×C411yescharacteristic
188D4⋊C8C2×C411yescharacteristic
189D4⋊C8C2×C411yescharacteristic
190D4⋊C8C2×C411yescharacteristic, complemented
191D4⋊C8C2×C411yescharacteristic, complemented
192D4⋊C8D411yescharacteristic
193D4⋊C8D411yescharacteristic
194D4⋊C8D411yescharacteristic
195D4⋊C8D411yescharacteristic
196#322—21no—
197#322—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×#41C2×D411yescharacteristic
201C22×D4#4111yescharacteristic
202C23×C4—21no—
203C23×C4—41no—
204#135—81no—
205#135—81no—
206C2×Q16—21no—
207C2×SD16—21no—
208C2×SD16—21no—
209C2×SD16—41no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—21no—
213#131—42no—
214#131—81no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—41no—
218C2×M16—41no—
219C2×M16—42no—
220C2×M16—81no—
221C2×M16C2×D411yescharacteristic
222C2×M16C2×D411yescharacteristic
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—41no—
228C22×C8—41no—
229C22×C8—41no—
230C4⋊Q8C22×C411yescharacteristic
231C4⋊Q8C22×C411yescharacteristic
232C4⋊Q8C22×C411yescharacteristic
233#127C22×C411yescharacteristic
234#126—41no—
235#126—41no—
236#126—41no—
237#125—21no—
238#125—21no—
239#125—41no—
240#124—21no—
241#124—21no—
242#124—41no—
243#123—41no—
244#123—41no—
245#123—41no—
246#123—41no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—41no—
253#121—21no—
254#121—21no—
255#121—21no—
256#121—21no—
257#121—21no—
258#121—41no—
259#120—41no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8C22×C411yescharacteristic
264C4×Q8#4111yescharacteristic
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—41no—
273C4×D4—42no—
274C4×D4—41no—
275C4×D4—41no—
276C4×D4—42no—
277C4×D4C22×C411yescharacteristic
278C4×D4C22×C411yescharacteristic
279C4×D4C22×C411yescharacteristic
280C4×D4C2×D411yescharacteristic
281C4×D4C2×D411yescharacteristic
282C4×D4#4111yescharacteristic
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—41no—
289#117—41no—
290#117—41no—
291#117C2×D411yescharacteristic
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4#4111yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—41no—
304C2×C42—42no—
305C2×C42—41no—
306C2×C42—81no—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2411yescharacteristic
310#108—81no—
311#107—21no—
312#107—21no—
313#106—41no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—41no—
322C4⋊C8C22×C411yescharacteristic
323C4⋊C8C22×C411yescharacteristic
324C4⋊C8C22×C411yescharacteristic
325C4⋊C8C22×C411yescharacteristic
326C4⋊C8C2×D411yescharacteristic
327C4⋊C8C2×D411yescharacteristic
328#104—41no—
329#104—41no—
330#104—42no—
331#104—81no—
332#104—81no—
333#103—21no—
334#103—21no—
335#103—41no—
336#102—21no—
337#102—21no—
338#102—21no—
339#102—21no—
340#102—21no—
341#102—21no—
342#102—41no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—41no—
346C22⋊C8—41no—
347C22⋊C8—41no—
348C22⋊C8—41no—
349C22⋊C8—41no—
350C22⋊C8—41no—
351C22⋊C8—41no—
352#97—21no—
353#97—41no—
354#97—44no—
355#97—81no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—41no—
362C4×C8—41no—
363C4×C8C22×C411yescharacteristic
364C4×C8C22×C411yescharacteristic
365C4×C8C2×D411yescharacteristic
366C4×C8C2×D411yescharacteristic
367#95—21no—
368#95—41no—
369C24—21no—
370#41—21no—
371#41—21no—
372#41—21no—
373#41—21no—
374#41—42no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—81no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8C4×D411yescharacteristic
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4C4×D411yescharacteristic
395C2×D4C4×D411yescharacteristic
396C2×D4C4×D411yescharacteristic
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—44no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4C22×D411yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421Q16—81no—
422SD16—82no—
423SD16—81no—
424SD16—81no—
425D8—42no—
426D8—81no—
427D8—81no—
428M16—21no—
429M16—21no—
430M16—21no—
431M16—21no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—82no—
437M16—82no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—82no—
460C2×C8—82no—
461C2×C8—82no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4C4×D411yescharacteristic
480C4⋊C4C4×D411yescharacteristic
481C4⋊C4C4×D411yescharacteristic
482C4⋊C4C4×D411yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—44no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—41no—
508C42—41no—
509C42—42no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—82no—
516C42C23×C411yescharacteristic
517C42C22×D411yescharacteristic
518C42C2×#4111yescharacteristic
519C42H(5,𝔽2)11yescharacteristic
520C23—21no—
521C23—21no—
522C23—21no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—81no—
528C23#54611yescharacteristic
529Q8—21no—
530Q8—21no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—81no—
536D4—21no—
537D4—21no—
538D4—21no—
539D4—21no—
540D4—21no—
541D4—42no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—81no—
549D4—82no—
550D4—81no—
551D4—81no—
552D4—161no—
553D4#44412yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
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—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—81no—
587C2×C4—82no—
588C2×C4—88no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—161no—
594C2×C4C2×C4×D411yescharacteristic
595C2×C4C2×C4×D411yescharacteristic
596C2×C4#51811yescharacteristic
597C2×C4D4211yescharacteristic
598C2×C4#54611yescharacteristic
599C2×C4#55111yescharacteristic
600C2×C4#57511yescharacteristic
601C2×C4#57511yescharacteristic
602C2×C4#57611yescharacteristic
603C2×C4#57711yescharacteristic
604C8—22no—
605C8—22no—
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—
619C22—21no—
620C22—21no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—82no—
631C22—82no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22#235811yescharacteristic
636C22#330711yescharacteristic
637C22#331911yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—81no—
648C4—82no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4#294311yescharacteristic
653C4#294611yescharacteristic
654C4C2×#44411yescharacteristic
655C4#295611yescharacteristic
656C4#326711yescharacteristic
657C4#326811yescharacteristic
658C2—21no—
659C2—21no—
660C2—42no—
661C2—81no—
662C2—81no—
663C2—161no—
664C2#2286011yescharacteristic
665C2#2288611yescharacteristic
666C2#2289911yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radical#18522C2
LCSG ▷ C42 ▷ C22 ▷ 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
1. #23114C2semidirect
2. #22505C2semidirect
4. #18001C2semidirect
5. #17508C2semidirect
6. #9851C2semidirect
7. #9591C2semidirect
8. #9583C2semidirect
9. #9233C2semidirect
10. #9208C2semidirect
11. #9055C2semidirect
12. #8906C2semidirect
13. #8903C2semidirect
14. #8192C2semidirect
15. #8172C2semidirect
41. #1654C4semidirect
42. #1652C4semidirect
43. #1646C4semidirect
44. #1644C4semidirect
45. #1628C4semidirect
46. #1625C4semidirect
47. #1624C4semidirect
48. #1622C4semidirect
98. #498C2×C4semidirect
99. #492C2×C4semidirect
100. #487C2×C4semidirect
101. #486C2×C4semidirect
108. #462C2×C4semidirect
109. #461C2×C4semidirect
110. #460C2×C4semidirect
113. #459C2×C4semidirect
171. #333C2×C4semidirect
172. #333C2×C4semidirect
173. #332C2×C4semidirect
174. #332C2×C4semidirect
180. Q8⋊C8C2×C4semidirect
181. Q8⋊C8C2×C4semidirect
190. D4⋊C8C2×C4semidirect
191. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#231142
#225052
#185222
#180012
#175082
#98512
#95912
#95832
#92332
#92082
#90552
#89062
#89032
#81922
#81722