Finite Group Databasebeta

← #217824

Subgroups — #217824

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23570C211yesmaximal, characteristic
2#21598C211yesmaximal, characteristic, complemented
3#21189C211yesmaximal, characteristic
4#9661C211yesmaximal, characteristic, complemented
5#9655C211yesmaximal, characteristic, complemented
6#9558C211yesmaximal, characteristic, complemented
7#9554C211yesmaximal, characteristic, complemented
8#9542C211yesmaximal, characteristic, complemented
9#9542C211yesmaximal, characteristic, complemented
10#8952C211yesmaximal, characteristic, complemented
11#8952C211yesmaximal, characteristic, complemented
12#8937C211yesmaximal, characteristic, complemented
13#8830C211yesmaximal, characteristic, complemented
14#8313C211yesmaximal, characteristic
15#8146C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#408—21no—
22C2×#408—21no—
23C2×C4×M16C2211yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25#2058C411yescharacteristic
26#2041C411yescharacteristic
27#2034C411yescharacteristic
28#2034C411yescharacteristic
29#2024C411yescharacteristic
30#2014C411yescharacteristic
31#2000C411yescharacteristic
32#2000C411yescharacteristic
33#1984C2211yescharacteristic
34#1955—21no—
35#1955—21no—
36#1954C2211yescharacteristic
37#1954C2211yescharacteristic
38#1954C2211yescharacteristic
39#1954C2211yescharacteristic
40#1939C2211yescharacteristic
41#1917—21no—
42#1851—21no—
43#1797C2211yescharacteristic
44#1770—21no—
45C8×C4⋊C4C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×#41—21no—
68C22×SD16C2×C411yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×M16D411yescharacteristic
71C22×M16D411yescharacteristic
72C23×C8—21no—
73#532—41no—
74#532—41no—
75#532—41no—
76#532—41no—
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×#127C2×C411yescharacteristic
79C2×#124—21no—
80C2×C22⋊Q8C2×C411yescharacteristic
81C2×#121—21no—
82C2×#121C2×C411yescharacteristic
83C2×#121C2×C411yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4C2311yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#446—41no—
90#446—41no—
91C4×SD16—41no—
92C4×SD16—41no—
93C4×SD16—41no—
94C4×SD16—41no—
95C2×#106C2×C411yescharacteristic
96C2×#106C2×C411yescharacteristic
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—41no—
101C2×C4⋊C8—41no—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114#408—44no—
115#408—44no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8C2×C411yescharacteristic
127C2×C22⋊C8C2×C411yescharacteristic
128C4×M16—41no—
129C4×M16—41no—
130C2×#97—21no—
131C2×#97D411yescharacteristic
132C2×#97D411yescharacteristic
133C2×C4×C8—21no—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#391C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#387—21no—
143#385—21no—
144#380C2×C411yescharacteristic
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2311yescharacteristic
147C4×C22⋊C4—21no—
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336C2×C411yescharacteristic
159#336C2×C411yescharacteristic
160#334—41no—
161#334—41no—
162Q8⋊C8—41no—
163Q8⋊C8—41no—
164D4⋊C8—41no—
165D4⋊C8—41no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168#322—41no—
169C82—41no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—42no—
173C2×#41—42no—
174C2×#41—42no—
175C2×#41—42no—
176C22×Q8—21no—
177C22×Q8C22×C411yescharacteristic
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C22×C411yescharacteristic
183C22×D4C22×C411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4#4111yescharacteristic
189C2×SD16—22no—
190C2×SD16—41no—
191C2×SD16—41no—
192C2×SD16—41no—
193C2×SD16—41no—
194C2×SD16—41no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—41no—
198C2×M16—41no—
199C2×M16—41no—
200C2×M16SD1614yescomplemented
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—42no—
208C22×C8—42no—
209C22×C8—42no—
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C4⋊Q8—41no—
219C4⋊Q8—41no—
220#127—41no—
221#127—41no—
222#124—42no—
223#124—42no—
224C22⋊Q8—22no—
225C22⋊Q8—41no—
226#121—41no—
227#121—41no—
228#121—41no—
229#121—41no—
230#121—42no—
231#121—42no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—42no—
245C4×D4—42no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42C22×C411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42#4111yescharacteristic
270C2×C42C2411yescharacteristic
271#106—41no—
272#106—41no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283C4⋊C8—42no—
284#103—41no—
285#103—41no—
286#102—22no—
287#102—22no—
288#102—22no—
289#102—41no—
290#102—41no—
291#102—41no—
292C22⋊C8—22no—
293C22⋊C8—41no—
294C22⋊C8—42no—
295C22⋊C8—42no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298C22⋊C8—42no—
299C22⋊C8—42no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304#97—41no—
305#97—41no—
306#97—42no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—42no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325C24—21no—
326C24—21no—
327C24C4×D411yescharacteristic
328#41—84no—
329#41—84no—
330#41—84no—
331#41—84no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—81no—
364C2×D4—81no—
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—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4C4×D411yescharacteristic
398C22×C4C4×D411yescharacteristic
399C22×C4C4×D411yescharacteristic
400C22×C4C2×SD1612yes—
401C22×C4#13512yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405SD16—44no—
406SD16—44no—
407M16—44no—
408M16—44no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—84no—
448C2×C8#13112yes—
449C2×C8C2×SD1612yes—
450C2×C8C2×SD1612yes—
451C2×C8AGL(1,ℤ/8)12yes—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—81no—
466C4⋊C4#13112yes—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—81no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42H(5,𝔽2)11yescharacteristic
486C42H(5,𝔽2)11yescharacteristic
487C42H(5,𝔽2)11yescharacteristic
488C42H(5,𝔽2)11yescharacteristic
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C4×SD1612yes—
510C23C2×C4×D411yescharacteristic
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—84no—
516D4—41no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—84no—
525D4—84no—
526D4—84no—
527D4—84no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4C4×SD1612yes—
575C2×C4#51811yescharacteristic
576C2×C4#51811yescharacteristic
577C2×C4#53411yescharacteristic
578C2×C4#53511yescharacteristic
579C2×C4#53811yescharacteristic
580C2×C4#53811yescharacteristic
581C2×C4C2×#13111yescharacteristic
582C2×C4C2×#13111yescharacteristic
583C2×C4#56811yescharacteristic
584C2×C4#56811yescharacteristic
585C2×C4C22×SD1611yescharacteristic
586C2×C4C2×#13511yescharacteristic
587C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
588C2×C4#57511yescharacteristic
589C8—22no—
590C8—22no—
591C8—22no—
592C8—22no—
593C8—42no—
594C8—41no—
595C8—41no—
596C8—41no—
597C8—41no—
598C8—81no—
599C8—81no—
600C8—81no—
601C8—81no—
602C8—81no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22#232111yescharacteristic
625C22#293311yescharacteristic
626C22#293511yescharacteristic
627C22C2×C4×SD1611yescharacteristic
628C22C4×AGL(1,ℤ/8)11yescharacteristic
629C22#295611yescharacteristic
630C22#295711yescharacteristic
631C4—22no—
632C4—21no—
633C4—21no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4#297611yescharacteristic
643C4#297811yescharacteristic
644C4#297911yescharacteristic
645C4#298111yescharacteristic
646C4#318711yescharacteristic
647C4#318811yescharacteristic
648C4#320811yescharacteristic
649C4#321111yescharacteristic
650C2—42no—
651C2—161no—
652C2—161no—
653C2#1799711yescharacteristic
654C2#1800711yescharacteristic
655C2#1806711yescharacteristic
656C2#1807811yescharacteristic
657C2#2203911yescharacteristic
658C2#2204311yescharacteristic
659C2#2206211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21598C2semidirect
4. #9661C2semidirect
5. #9655C2semidirect
6. #9558C2semidirect
7. #9554C2semidirect
8. #9542C2semidirect
9. #9542C2semidirect
10. #8952C2semidirect
11. #8952C2semidirect
12. #8937C2semidirect
13. #8830C2semidirect
15. #8146C2semidirect
200. C2×M16SD16semidirect

Maximal subgroups

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

subgroupindex
#235702
#215982
#211892
#96612
#96552
#95582
#95542
#95422
#89522
#89372
#88302
#83132
#81462