Finite Group Databasebeta

← #192763

Subgroups — #192763

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34196C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21963C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8896C211yesmaximal, characteristic, complemented
7#8895C211yesmaximal, characteristic, complemented
8#8892C211yesmaximal, characteristic, complemented
9#8891C211yesmaximal, characteristic, complemented
10#8888C211yesmaximal, characteristic, complemented
11#8887C211yesmaximal, characteristic, complemented
12#8884C211yesmaximal, characteristic, complemented
13#8883C211yesmaximal, characteristic, complemented
14#8041C211yesmaximal, characteristic
15#7899C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#419—21no—
21#2355—21no—
22#2355—21no—
23#2355C2211yescharacteristic
24#2355C2211yescharacteristic
25Q8×C4⋊C4C2211yescharacteristic
26Q8×C4⋊C4C2211yescharacteristic
27#2344—22no—
28#2344—21no—
29Q8×C22⋊C4—21no—
30#2320—22no—
31#2320—21no—
32#2285—21no—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35#2283—21no—
36#2283C2211yescharacteristic
37#2276—21no—
38#1949—21no—
39#1949—21no—
40#1949—22no—
41#1948C2211yescharacteristic
42#1948C2211yescharacteristic
43#1947C2211yescharacteristic
44#1947C2211yescharacteristic
45#1854—22no—
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1809—21no—
51#1809—21no—
52#1805—21no—
53#1805—21no—
54#1803C2211yescharacteristic
55#1803C2211yescharacteristic
56#1803C2211yescharacteristic
57#1803C2211yescharacteristic
58#1736—21no—
59#1487C418yescomplemented
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×Q8⋊C8C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72C23×Q8D411yescharacteristic
73C22×M16—21no—
74C22×M16—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8C2×C414yes—
82C2×C4×Q8D411yescharacteristic
83C2×C4×Q8D411yescharacteristic
84C2×C4×Q8Q814yes—
85C2×C4×Q8C2311yescharacteristic
86C2×C4×Q8C2311yescharacteristic
87C2×#117—22no—
88C2×#117—21no—
89C2×#117D412yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4D411yescharacteristic
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94#431—22no—
95#431C2×C414yes—
96C2×#107—21no—
97C2×#107—21no—
98C2×#107—22no—
99C2×#106—21no—
100C2×#106—21no—
101C2×#106—22no—
102#423—22no—
103#423—22no—
104#423—22no—
105#423C2×C414yes—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113#419—44no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#103—21no—
118C2×C22⋊C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—22no—
122#391—22no—
123#391—21no—
124#391—21no—
125#389—21no—
126#389—21no—
127#389—22no—
128#389—22no—
129#387—22no—
130#387—22no—
131#387—22no—
132#387—22no—
133#387—22no—
134#387—22no—
135#384—21no—
136#384—21no—
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#383—22no—
142#383—21no—
143#382—22no—
144#382—21no—
145#382—21no—
146#382—21no—
147#382D412yes—
148#382D411yescharacteristic
149#382D411yescharacteristic
150#381—22no—
151#381—21no—
152#380—22no—
153#380—21no—
154#380—21no—
155#380—21no—
156#380—21no—
157#380D411yescharacteristic
158#380D411yescharacteristic
159C4×C4⋊C4—22no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4C2311yescharacteristic
173C4×C4⋊C4C2311yescharacteristic
174C4×C22⋊C4—22no—
175C4×C22⋊C4—21no—
176#376—22no—
177#376—21no—
178#340—21no—
179#340—21no—
180#340—21no—
181#340—21no—
182#340—21no—
183#340—21no—
184#340—21no—
185#340—21no—
186#340—22no—
187#340—22no—
188Q8⋊C8C2×C414yes—
189Q8⋊C8C2×C414yes—
190Q8⋊C8C2×C414yes—
191Q8⋊C8C2×C414yes—
192C22×Q8—21no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—41no—
197C22×Q8—41no—
198C22×Q8C4⋊C414yes—
199C22×Q8C2×D411yescharacteristic
200C22×Q8C2×D411yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C2×M16—24no—
207C2×M16—42no—
208C2×M16—42no—
209C2×M16—42no—
210C2×M16—81no—
211C2×M16—81no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—42no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8—42no—
232C4×Q8C4⋊C414yes—
233C4×Q8C4⋊C414yes—
234C4×Q8C22×C412yes—
235C4×Q8C22×C412yes—
236C4×Q8C2×Q812yes—
237C4×Q8C2×Q812yes—
238#117—42no—
239#117—42no—
240#117—84no—
241#117—82no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4C4⋊C414yes—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D411yescharacteristic
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—41no—
277C2×C42—24no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—24no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—41no—
291C2×C42C22×C412yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×Q812yes—
296C2×C42#4112yes—
297C2×C42#4112yes—
298C2×C42C2411yescharacteristic
299#107—81no—
300#107—81no—
301#107—82no—
302#106—81no—
303#106—81no—
304#106—82no—
305C4⋊C8—24no—
306C4⋊C8—24no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—42no—
314C4⋊C8C22×C412yes—
315C4⋊C8C22×C412yes—
316#103—42no—
317#103—42no—
318#103—42no—
319#103—42no—
320C22⋊C8—24no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8C22×C412yes—
324C4×C8C22×C412yes—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—21no—
334#95—22no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—22no—
341#95—21no—
342#95—21no—
343#95—41no—
344#95—41no—
345C24#12011yescharacteristic
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—24no—
351C2×Q8—28no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8C2×C4⋊C412yes—
362C2×Q8C2×C4⋊C412yes—
363C2×Q8AGL(1,ℤ/8)12yes—
364C2×Q8#13712yes—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4C2×C4⋊C412yes—
394C22×C4C4×D412yes—
395C22×C4C4×D412yes—
396C22×C4#12011yescharacteristic
397C22×C4#12011yescharacteristic
398C22×C4#12011yescharacteristic
399C22×C4C22⋊Q812yes—
400C22×C4C22⋊Q812yes—
401C22×C4#12312yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405M16—84no—
406M16—84no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4C2×C4⋊C412yes—
467C4⋊C4C2×C4⋊C412yes—
468C22⋊C4—42no—
469C22⋊C4—44no—
470C22⋊C4—42no—
471C22⋊C4—82no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—24no—
477C42—24no—
478C42—44no—
479C42—44no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—82no—
485C42—82no—
486C42—82no—
487C42—82no—
488C42C2×C4⋊C412yes—
489C42C4×D412yes—
490C42C4×D412yes—
491C42C22⋊Q812yes—
492C42C22⋊Q812yes—
493C42#12312yes—
494C42C23×C411yescharacteristic
495C42C22×Q811yescharacteristic
496C42C2×#4111yescharacteristic
497C42C2×#4111yescharacteristic
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23#38512yes—
505C23C2×#12011yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8—22no—
510Q8—22no—
511Q8—48no—
512Q8—41no—
513Q8—41no—
514Q8#42814yes—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4#38512yes—
572C2×C4#38512yes—
573C2×C4#38512yes—
574C2×C4C22×C4⋊C411yescharacteristic
575C2×C4C2×C4×D411yescharacteristic
576C2×C4C2×C4×D411yescharacteristic
577C2×C4C2×C22⋊Q811yescharacteristic
578C2×C4C2×C22⋊Q811yescharacteristic
579C2×C4C2×#12311yescharacteristic
580C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
581C2×C4C2×#13711yescharacteristic
582C2×C4#57511yescharacteristic
583C2×C4#57511yescharacteristic
584C2×C4#57711yescharacteristic
585C2×C4#57711yescharacteristic
586C2×C4#57811yescharacteristic
587C2×C4#57811yescharacteristic
588C8—42no—
589C8—42no—
590C8—42no—
591C8—42no—
592C8—81no—
593C8—81no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22C2×#38511yescharacteristic
602C22#300611yescharacteristic
603C22#300711yescharacteristic
604C22#301311yescharacteristic
605C22#301311yescharacteristic
606C22#301911yescharacteristic
607C22#302011yescharacteristic
608C4—22no—
609C4—24no—
610C4—21no—
611C4—21no—
612C4—22no—
613C4—22no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—82no—
619C4—81no—
620C4—82no—
621C4—81no—
622C4—81no—
623C4—161no—
624C4—161no—
625C4C2×#42811yescharacteristic
626C4#291111yescharacteristic
627C4#294911yescharacteristic
628C4#295011yescharacteristic
629C4#308211yescharacteristic
630C4#308211yescharacteristic
631C4#309411yescharacteristic
632C4#309511yescharacteristic
633C2—42no—
634C2#1881211yescharacteristic
635C2#1881311yescharacteristic
636C2#1882411yescharacteristic
637C2#1882511yescharacteristic
638C2#2084611yescharacteristic
639C2#2086411yescharacteristic
640C2#2088011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8896C2semidirect
7. #8895C2semidirect
8. #8892C2semidirect
9. #8891C2semidirect
10. #8888C2semidirect
11. #8887C2semidirect
12. #8884C2semidirect
13. #8883C2semidirect
59. #1487C4semidirect

Maximal subgroups

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

subgroupindex
#341962
#219842
#219632
#206472
C2×#14872
#88962
#88952
#88922
#88912
#88882
#88872
#88842
#88832
#80412
#78992