Finite Group Databasebeta

← #217816

Subgroups — #217816

Order 512.

896 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21598C211yesmaximal, characteristic, complemented
2C4×#450C211yesmaximal, characteristic
3#21190C211yesmaximal, 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#8942C211yesmaximal, characteristic, complemented
13#8833C211yesmaximal, characteristic, complemented
14#8310C211yesmaximal, characteristic
15#8146C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#450C411yescharacteristic
18C2×#446C2211yescharacteristic, complemented
19C2×C4×SD16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#407—21no—
23C2×#407—21no—
24C2×C4×M16C2211yescharacteristic
25C4×C22⋊Q8C2211yescharacteristic
26#1984C2211yescharacteristic, complemented
27#1955—21no—
28#1955—21no—
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic
31#1954C2211yescharacteristic
32#1954C2211yescharacteristic
33#1939C2211yescharacteristic, complemented
34#1917—21no—
35#1877C411yescharacteristic
36#1877C411yescharacteristic
37#1872C411yescharacteristic
38#1864C411yescharacteristic
39#1864C411yescharacteristic
40#1853C411yescharacteristic
41#1851—21no—
42#1797C2211yescharacteristic
43#1786C411yescharacteristic
44#1770—21no—
45C8×C4⋊C4C2211yescharacteristic, complemented
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic, complemented
52C2×#334C2211yescharacteristic, complemented
53C2×Q8⋊C8C2211yescharacteristic, complemented
54C2×Q8⋊C8C2211yescharacteristic, complemented
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic, complemented
60C2×C82C2211yescharacteristic, complemented
61#1279C2211yescharacteristic, complemented
62#1279C2211yescharacteristic, complemented
63#1277C2211yescharacteristic, complemented
64#1277C2211yescharacteristic, complemented
65#1273C2211yescharacteristic, complemented
66#1273C2211yescharacteristic, complemented
67C23×D4C2×C411yescharacteristic
68C24×C4—21no—
69C22×SD16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×M16—21no—
72C22×M16D411yescharacteristic
73C22×M16D411yescharacteristic
74C2×C22⋊Q8C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—41no—
79C2×C4×D4—41no—
80C2×C4×D4—41no—
81C2×C4×D4—41no—
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4C2×C411yescharacteristic
84C2×C4×D4C2311yescharacteristic
85C2×C4×D4C2311yescharacteristic
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C22⋊C4—21no—
88C22×C42C2311yescharacteristic
89#450—24no—
90#446—41no—
91#446—41no—
92C4×SD16—41no—
93C4×SD16—41no—
94C4×SD16—41no—
95C4×SD16—41no—
96C2×#106C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98#423—41no—
99#423—41no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115#407—44no—
116#407—44no—
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⋊C8—21no—
127C2×C22⋊C8C2×C411yescharacteristic
128C2×C22⋊C8C2×C411yescharacteristic
129C4×M16—41no—
130C4×M16—41no—
131C2×#97—21no—
132C2×#97D411yescharacteristic
133C2×#97D411yescharacteristic
134C2×C4×C8—21no—
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141#391C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#387—21no—
144#385—21no—
145#380C2×C411yescharacteristic
146C4×C4⋊C4—21no—
147C4×C4⋊C4C2311yescharacteristic
148C4×C22⋊C4—21no—
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336C2×C411yescharacteristic
160#336C2×C411yescharacteristic
161#334—41no—
162#334—41no—
163Q8⋊C8—41no—
164Q8⋊C8—41no—
165D4⋊C8—41no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168D4⋊C8—41no—
169#322—41no—
170C82—41no—
171C25—21no—
172C22×Q8C22×C411yescharacteristic
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—21no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4C22×C411yescharacteristic
181C22×D4C22×C411yescharacteristic
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4#4111yescharacteristic
196C2×SD16—22no—
197C2×SD16—22no—
198C2×SD16—22no—
199C2×SD16—41no—
200C2×SD16—41no—
201C2×SD16—41no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—41no—
205C2×M16—41no—
206C2×M16—41no—
207C2×M16—42no—
208C2×M16—42no—
209C2×M16—42no—
210C2×M16SD1614yescomplemented
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C2×D411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22⋊Q8—22no—
226C22⋊Q8—41no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4—41no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—42no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42C22×C411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42C2411yescharacteristic
276#106—41no—
277#106—41no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—42no—
287C4⋊C8—42no—
288C4⋊C8—42no—
289#103—41no—
290#103—41no—
291#102—22no—
292#102—22no—
293#102—41no—
294#102—41no—
295#102—41no—
296#102M1614yescomplemented
297C22⋊C8—22no—
298C22⋊C8—41no—
299C22⋊C8—42no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309#97—41no—
310#97—41no—
311#97—42no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—42no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330C24—21no—
331C24—21no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—81no—
338C24—81no—
339C24C4×D411yescharacteristic
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4C2×M1612yes—
364C2×D4#13112yes—
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—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4C4×D411yescharacteristic
411C22×C4C4×D411yescharacteristic
412C22×C4C4×D411yescharacteristic
413C22×C4C2×SD1612yes—
414C22×C4#13512yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4111yescharacteristic
418SD16—44no—
419SD16—44no—
420M16—44no—
421M16—44no—
422M16—84no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—22no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8C2×M1612yes—
462C2×C8C2×SD1612yes—
463C2×C8C2×SD1612yes—
464C2×C8AGL(1,ℤ/8)12yes—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—81no—
479C4⋊C4C2×M1612yes—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—81no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42H(5,𝔽2)11yescharacteristic
499C42H(5,𝔽2)11yescharacteristic
500C42H(5,𝔽2)11yescharacteristic
501C42H(5,𝔽2)11yescharacteristic
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—48no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23C4×SD1612yes—
530C23C2×C4×D411yescharacteristic
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535D4—22no—
536D4—22no—
537D4—22no—
538D4—22no—
539D4—48no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
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—81no—
581C2×C4—81no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4C4×SD1612yes—
594C2×C4#51811yescharacteristic
595C2×C4#51811yescharacteristic
596C2×C4#53411yescharacteristic
597C2×C4#53511yescharacteristic
598C2×C4#53811yescharacteristic
599C2×C4#53811yescharacteristic
600C2×C4C22×M1611yescharacteristic
601C2×C4C2×#13111yescharacteristic
602C2×C4#56811yescharacteristic
603C2×C4#56811yescharacteristic
604C2×C4C22×SD1611yescharacteristic
605C2×C4C2×#13511yescharacteristic
606C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
607C2×C4#57511yescharacteristic
608C8—22no—
609C8—22no—
610C8—22no—
611C8—22no—
612C8—42no—
613C8—41no—
614C8—41no—
615C8—41no—
616C8—41no—
617C8—81no—
618C8—81no—
619C8—81no—
620C8—81no—
621C8—81no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—816no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22#232111yescharacteristic
645C22#293311yescharacteristic
646C22D4×M1611yescharacteristic
647C22C2×C4×SD1611yescharacteristic
648C22C4×AGL(1,ℤ/8)11yescharacteristic
649C22#295611yescharacteristic
650C22#295711yescharacteristic
651C4—22no—
652C4—21no—
653C4—21no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4#297411yescharacteristic
663C4#297511yescharacteristic
664C4#297611yescharacteristic
665C4#297811yescharacteristic
666C4#318711yescharacteristic
667C4#318811yescharacteristic
668C4#320811yescharacteristic
669C4#321111yescharacteristic
670C2—42no—
671C2—82no—
672C2—82no—
673C2#1799511yescharacteristic
674C2#1800811yescharacteristic
675C2M16×SD1611yescharacteristic
676C2#1804611yescharacteristic
677C2#2203811yescharacteristic
678C2#2204311yescharacteristic
679C2#2206211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC23×D4C2×C4
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
1. #21598C2semidirect
4. #9661C2semidirect
5. #9655C2semidirect
6. #9558C2semidirect
7. #9554C2semidirect
8. #9542C2semidirect
9. #9542C2semidirect
10. #8952C2semidirect
11. #8952C2semidirect
12. #8942C2semidirect
13. #8833C2semidirect
15. #8146C2semidirect
18. C2×#446C22semidirect
26. #1984C22semidirect
33. #1939C22semidirect
45. C8×C4⋊C4C22semidirect
51. C2×#334C22semidirect
52. C2×#334C22semidirect
53. C2×Q8⋊C8C22semidirect
54. C2×Q8⋊C8C22semidirect
59. C2×#322C22semidirect
60. C2×C82C22semidirect
61. #1279C22semidirect
62. #1279C22semidirect
63. #1277C22semidirect
64. #1277C22semidirect
65. #1273C22semidirect
66. #1273C22semidirect
210. C2×M16SD16semidirect
296. #102M16semidirect

Maximal subgroups

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

subgroupindex
#215982
C4×#4502
#211902
#96612
#96552
#95582
#95542
#95422
#89522
#89422
#88332
#83102
#81462