Finite Group Databasebeta

← #228158

Subgroups — #228158

Order 512.

743 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23016C211yesmaximal, characteristic, complemented
2#22505C211yesmaximal, characteristic, complemented
3#18522C211yesmaximal, characteristic
4#17994C211yesmaximal, characteristic, complemented
5#17512C211yesmaximal, characteristic, complemented
6#9851C211yesmaximal, characteristic, complemented
7#9589C211yesmaximal, characteristic, complemented
8#9584C211yesmaximal, characteristic, complemented
9#9233C211yesmaximal, characteristic, complemented
10#9210C211yesmaximal, characteristic, complemented
11#9055C211yesmaximal, characteristic, complemented
12#8906C211yesmaximal, characteristic, complemented
13#8902C211yesmaximal, characteristic, complemented
14#8196C211yesmaximal, characteristic, complemented
15#8172C211yesmaximal, characteristic, complemented
16#3475C2211yescharacteristic
17#3167C2211yescharacteristic
18#3100C2211yescharacteristic
19#3035C2211yescharacteristic
20#2965C2211yescharacteristic
21C2×#444—21no—
22#2935C2211yescharacteristic
23D4×M16C2211yescharacteristic
24#2906C2211yescharacteristic
25C42×D4C2211yescharacteristic
26#1994C2211yescharacteristic
27#1986C2211yescharacteristic
28#1962C2211yescharacteristic
29#1950C2211yescharacteristic
30#1938—21no—
31#1896—21no—
32#1876—21no—
33#1871—21no—
34#1866—21no—
35#1826—21no—
36#1806C2211yescharacteristic
37#1767C2211yescharacteristic
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#1589—21no—
50#1581—21no—
51#1581—21no—
52C8×D8—21no—
53#1571C2211yescharacteristic
54#1567C2211yescharacteristic
55#1542C2211yescharacteristic
56#1502C2211yescharacteristic
57#1495—21no—
58#1493C2211yescharacteristic
59#1492C2211yescharacteristic
60#1489—21no—
61#1486C2211yescharacteristic
62#1483—21no—
63#1481C2211yescharacteristic
64#1477—21no—
65#1476C2211yescharacteristic
66C2×D4⋊C8C2211yescharacteristic
67C2×D4⋊C8C2211yescharacteristic
68C8×M16C2211yescharacteristic
69#1303C2211yescharacteristic
70#1301C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1278C2211yescharacteristic
74#1276C2211yescharacteristic
75C2×#135—21no—
76C2×#131—21no—
77C2×#131—21no—
78C22×M16—21no—
79#556C2×C411yescharacteristic
80#555C2×C411yescharacteristic
81#554C2×C411yescharacteristic
82#553—21no—
83#552—21no—
84#552—21no—
85#551—41no—
86#547C2×C411yescharacteristic
87#546—21no—
88#533C2311yescharacteristic
89#532C2311yescharacteristic
90#529—21no—
91#529—41no—
92#525—21no—
93C4×#41—21no—
94C4×#41C2311yescharacteristic
95C2×C4×D4—21no—
96C2×C4×D4C2311yescharacteristic
97C2×#117—21no—
98C22×C42—21no—
99#498C2×C411yescharacteristic, complemented
100#492C2×C411yescharacteristic, complemented
101#487C2×C411yescharacteristic, complemented
102#486C2×C411yescharacteristic, complemented
103#480—21no—
104#474C2×C411yescharacteristic
105#474C2×C411yescharacteristic
106#465—41no—
107#464—21no—
108#463—21no—
109#462C2×C411yescharacteristic, complemented
110#461C2×C411yescharacteristic, complemented
111#460C2×C411yescharacteristic, complemented
112#459C2×C411yescharacteristic
113#459C2×C411yescharacteristic
114#459C2×C411yescharacteristic, complemented
115#447—21no—
116#446—21no—
117#444—44no—
118C4×SD16—41no—
119C4×D8—21no—
120C4×D8—21no—
121#436—21no—
122#436—21no—
123#436—21no—
124#435—41no—
125#435—41no—
126#435—41no—
127C8×D4—21no—
128C8×D4—21no—
129#431—21no—
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132C2×#108—21no—
133#424—21no—
134#424—41no—
135#423—21no—
136#423C2311yescharacteristic
137#423C2311yescharacteristic
138C2×C4⋊C8—21no—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×#104—21no—
142C2×#104—21no—
143C2×#104—21no—
144C2×#104D411yescharacteristic
145C2×#104D411yescharacteristic
146#416—21no—
147#408—21no—
148#407—21no—
149C4×M16—21no—
150C4×M16C2311yescharacteristic
151C2×#97—21no—
152C2×#97—41no—
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2311yescharacteristic
157C4×C22⋊C4—21no—
158C4×C22⋊C4—41no—
159C43C2311yescharacteristic
160#341—21no—
161#339—21no—
162#339—21no—
163#339—21no—
164#339—21no—
165#339—21no—
166#339D411yescharacteristic
167#339D411yescharacteristic
168#336—21no—
169#335—21no—
170#335—21no—
171#335—41no—
172#333C2×C411yescharacteristic, complemented
173#333C2×C411yescharacteristic, complemented
174#332C2×C411yescharacteristic, complemented
175#332C2×C411yescharacteristic, complemented
176#327—41no—
177Q8⋊C8—21no—
178Q8⋊C8—21no—
179Q8⋊C8—21no—
180Q8⋊C8—21no—
181Q8⋊C8C2×C411yescharacteristic, complemented
182Q8⋊C8C2×C411yescharacteristic, complemented
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8C2×C411yescharacteristic
188D4⋊C8C2×C411yescharacteristic
189D4⋊C8C2×C411yescharacteristic
190D4⋊C8C2×C411yescharacteristic
191D4⋊C8C2×C411yescharacteristic, complemented
192D4⋊C8C2×C411yescharacteristic, complemented
193D4⋊C8D411yescharacteristic
194D4⋊C8D411yescharacteristic
195D4⋊C8D411yescharacteristic
196D4⋊C8D411yescharacteristic
197#322—21no—
198C82—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41C2×D411yescharacteristic
202C22×D4#4111yescharacteristic
203C23×C4—21no—
204C23×C4—41no—
205#135—42no—
206#135—42no—
207C2×Q16—21no—
208C2×SD16—21no—
209C2×SD16—21no—
210C2×SD16—41no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214#131—42no—
215#131—42no—
216#131—81no—
217#131—81no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—41no—
221C2×M16—41no—
222C2×M16—41no—
223C2×M16—41no—
224C2×M16—42no—
225C2×M16—81no—
226C2×M16C2×D411yescharacteristic
227C2×M16C2×D411yescharacteristic
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—41no—
233C22×C8—41no—
234C22×C8—41no—
235C4⋊Q8C22×C411yescharacteristic
236C4⋊Q8C22×C411yescharacteristic
237C4⋊Q8C22×C411yescharacteristic
238#127C22×C411yescharacteristic
239#126—41no—
240#126—41no—
241#126—41no—
242#125—21no—
243#125—21no—
244#125—41no—
245#124—21no—
246#124—21no—
247#124—41no—
248#123—41no—
249#123—41no—
250#123—41no—
251#123—41no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—41no—
258#121—21no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—41no—
264#120—41no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8C22×C411yescharacteristic
269C4×Q8#4111yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—41no—
278C4×D4—42no—
279C4×D4—41no—
280C4×D4—41no—
281C4×D4—42no—
282C4×D4C22×C411yescharacteristic
283C4×D4C22×C411yescharacteristic
284C4×D4C22×C411yescharacteristic
285C4×D4C2×D411yescharacteristic
286C4×D4C2×D411yescharacteristic
287C4×D4#4111yescharacteristic
288#117—21no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117—21no—
293#117—41no—
294#117—41no—
295#117—41no—
296#117C2×D411yescharacteristic
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4#4111yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—41no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—41no—
309C2×C42—42no—
310C2×C42—41no—
311C2×C42—81no—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2411yescharacteristic
315#108—42no—
316#107—21no—
317#107—21no—
318#106—41no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—41no—
327C4⋊C8—41no—
328C4⋊C8—41no—
329C4⋊C8C22×C411yescharacteristic
330C4⋊C8C22×C411yescharacteristic
331C4⋊C8C22×C411yescharacteristic
332C4⋊C8C22×C411yescharacteristic
333C4⋊C8C2×D411yescharacteristic
334C4⋊C8C2×D411yescharacteristic
335#104—22no—
336#104—41no—
337#104—42no—
338#104—81no—
339#104—81no—
340#103—21no—
341#103—21no—
342#103—41no—
343#102—21no—
344#102—21no—
345#102—21no—
346#102—21no—
347#102—21no—
348#102—21no—
349#102—41no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8—41no—
353C22⋊C8—41no—
354C22⋊C8—41no—
355C22⋊C8—41no—
356C22⋊C8—41no—
357#97—21no—
358#97—41no—
359#97—42no—
360#97—44no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8—21no—
366C4×C8C22×C411yescharacteristic
367C4×C8C22×C411yescharacteristic
368C4×C8C2×D411yescharacteristic
369C4×C8C2×D411yescharacteristic
370#95—21no—
371#95—41no—
372C24—21no—
373#41—21no—
374#41—21no—
375#41—21no—
376#41—21no—
377#41—42no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8C4×D411yescharacteristic
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4C4×D411yescharacteristic
398C2×D4C4×D411yescharacteristic
399C2×D4C4×D411yescharacteristic
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—44no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4C22×D411yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4H(5,𝔽2)11yescharacteristic
424Q16—42no—
425SD16—42no—
426SD16—42no—
427SD16—82no—
428D8—42no—
429D8—42no—
430D8—81no—
431M16—21no—
432M16—21no—
433M16—21no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—41no—
438M16—82no—
439M16—82no—
440M16—82no—
441M16—82no—
442M16AGL(1,ℤ/8)12yes—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—82no—
465C2×C8—82no—
466C2×C8—82no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4C4×D411yescharacteristic
485C4⋊C4C4×D411yescharacteristic
486C4⋊C4C4×D411yescharacteristic
487C4⋊C4C4×D411yescharacteristic
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—44no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—41no—
513C42—41no—
514C42—42no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—82no—
521C42C23×C411yescharacteristic
522C42C22×D411yescharacteristic
523C42C2×#4111yescharacteristic
524C42H(5,𝔽2)11yescharacteristic
525C23—21no—
526C23—21no—
527C23—21no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—81no—
533C23#54611yescharacteristic
534Q8—21no—
535Q8—21no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—81no—
541D4—21no—
542D4—21no—
543D4—21no—
544D4—21no—
545D4—21no—
546D4—42no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—81no—
554D4—82no—
555D4—81no—
556D4—81no—
557D4—161no—
558D4#44412yescomplemented
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
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—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—88no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—161no—
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×C4×D411yescharacteristic
601C2×C4#51811yescharacteristic
602C2×C4D4211yescharacteristic
603C2×C4#54611yescharacteristic
604C2×C4#55111yescharacteristic
605C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
606C2×C4#57511yescharacteristic
607C2×C4#57611yescharacteristic
608C2×C4#57811yescharacteristic
609C8—22no—
610C8—22no—
611C8—22no—
612C8—41no—
613C8—41no—
614C8—41no—
615C8—41no—
616C8—41no—
617C8—41no—
618C8—41no—
619C8—41no—
620C8—41no—
621C8—81no—
622C8—81no—
623C8—81no—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—82no—
636C22—82no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22#235811yescharacteristic
641C22#332211yescharacteristic
642C22#333011yescharacteristic
643C4—21no—
644C4—21no—
645C4—21no—
646C4—42no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—81no—
653C4—82no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4C4×AGL(1,ℤ/8)11yescharacteristic
658C4#295111yescharacteristic
659C4C2×#44411yescharacteristic
660C4#295611yescharacteristic
661C4#326411yescharacteristic
662C4#326911yescharacteristic
663C2—21no—
664C2—21no—
665C2—42no—
666C2—81no—
667C2—81no—
668C2—161no—
669C2#2287811yescharacteristic
670C2#2288611yescharacteristic
671C2#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. #23016C2semidirect
2. #22505C2semidirect
4. #17994C2semidirect
5. #17512C2semidirect
6. #9851C2semidirect
7. #9589C2semidirect
8. #9584C2semidirect
9. #9233C2semidirect
10. #9210C2semidirect
11. #9055C2semidirect
12. #8906C2semidirect
13. #8902C2semidirect
14. #8196C2semidirect
15. #8172C2semidirect
41. #1654C4semidirect
42. #1652C4semidirect
43. #1646C4semidirect
44. #1644C4semidirect
45. #1628C4semidirect
46. #1625C4semidirect
47. #1624C4semidirect
48. #1622C4semidirect
99. #498C2×C4semidirect
100. #492C2×C4semidirect
101. #487C2×C4semidirect
102. #486C2×C4semidirect
109. #462C2×C4semidirect
110. #461C2×C4semidirect
111. #460C2×C4semidirect
114. #459C2×C4semidirect
172. #333C2×C4semidirect
173. #333C2×C4semidirect
174. #332C2×C4semidirect
175. #332C2×C4semidirect
181. Q8⋊C8C2×C4semidirect
182. Q8⋊C8C2×C4semidirect
191. D4⋊C8C2×C4semidirect
192. D4⋊C8C2×C4semidirect
558. D4#444semidirect

Maximal subgroups

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

subgroupindex
#230162
#225052
#185222
#179942
#175122
#98512
#95892
#95842
#92332
#92102
#90552
#89062
#89022
#81962
#81722