Finite Group Databasebeta

← #196174

Subgroups — #196174

Order 512.

1449 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60276C211yesmaximal, characteristic, complemented
2C2×C4×AGL(1,ℤ/8)C211yesmaximal, characteristic
3#17437C211yesmaximal, characteristic, complemented
4C2×#1478C211yesmaximal, characteristic, complemented
5C2×#1460C211yesmaximal, characteristic, complemented
6#9095C218yesmaximal, complemented
7#7932C211yesmaximal, characteristic, complemented
8#7882C211yesmaximal, characteristic, complemented
9C22×AGL(1,ℤ/8)C411yescharacteristic
10C2×C4×#41C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#2988—22no—
13#2988—22no—
14#2982—21no—
15#2972—22no—
16#2972—21no—
17C8×#41—22no—
18C8×#41C2214yescomplemented
19C4×AGL(1,ℤ/8)—22no—
20C4×AGL(1,ℤ/8)—22no—
21C4×AGL(1,ℤ/8)C414yes—
22C4×AGL(1,ℤ/8)C2214yes—
23C2×#442—21no—
24C2×#440—21no—
25C2×C4×SD16—21no—
26C2×C4×D8—21no—
27C2×#431—22no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#428C411yescharacteristic
33C2×#417C411yescharacteristic
34C2×#418C411yescharacteristic
35C2×C4×M16C2211yescharacteristic
36#1479—21no—
37#1478—22no—
38#1478—22no—
39#1478—21no—
40#1477C2214yescomplemented
41#1476—22no—
42C2×D4⋊C8C2214yescomplemented
43#1460—22no—
44#1460C2214yescomplemented
45C2×#323—22no—
46#1283C2214yescomplemented
47#1278C2214yescomplemented
48C22×#41C2×C411yescharacteristic
49C23×D4C2×C411yescharacteristic
50C24×C4—21no—
51C2×AGL(1,ℤ/8)—22no—
52C2×AGL(1,ℤ/8)—24no—
53C2×AGL(1,ℤ/8)C818yescomplemented
54C2×AGL(1,ℤ/8)C2×C414yes—
55C22×SD16—21no—
56C22×D8—21no—
57C22×M16C2×C411yescharacteristic
58C22×M16C2×C411yescharacteristic
59C4×#41—21no—
60C4×#41—21no—
61C4×#41—21no—
62C4×#41—21no—
63C4×#41C2×C412yes—
64C4×#41D412yes—
65C4×#41D412yes—
66C4×#41C2312yes—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4C2×C412yes—
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×C4×D4C2312yes—
85C2×#117—21no—
86C2×#117C2×C411yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C22⋊C4—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91C8×Q8—24no—
92C8×Q8—24no—
93#442—24no—
94#442—24no—
95#440—24no—
96#440—24no—
97C4×SD16—24no—
98C4×SD16—24no—
99C4×D8—24no—
100C4×D8—24no—
101C8×D4—28no—
102C8×D4—24no—
103C8×D4—24no—
104C8×D4—28no—
105#431—24no—
106#431—24no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—24no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431—21no—
119#431D412yes—
120#431D412yes—
121#431D412yes—
122#431C2312yes—
123#431C2312yes—
124#428—22no—
125#428C2×C414yes—
126C2×#107—21no—
127C2×#106—21no—
128C2×C4⋊C8—24no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8C2312yes—
133#418—22no—
134#418C2×C414yes—
135#417—22no—
136#417C2×C414yes—
137C2×#103—21no—
138C2×#102—21no—
139C2×#102—21no—
140C2×#102—21no—
141C2×C22⋊C8—24no—
142C2×C22⋊C8—21no—
143C2×C22⋊C8—21no—
144C2×C22⋊C8—21no—
145C2×C22⋊C8D412yes—
146C4×M16—21no—
147C4×M16—21no—
148C4×M16C2×C412yes—
149C4×M16C2312yes—
150C2×#97—21no—
151C2×C4×C8—21no—
152C2×C4×C8—21no—
153C2×C4×C8—24no—
154C2×C4×C8—21no—
155C2×C4×C8—21no—
156C2×C4×C8C2312yes—
157Q8⋊C8—24no—
158D4⋊C8—24no—
159D4⋊C8—28no—
160#323—28no—
161C25—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41C22⋊C412yes—
167C2×#41C2×C814yes—
168C2×#41M1614yes—
169C2×#41C22×C412yes—
170C22×Q8C22⋊C411yescharacteristic
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4C22⋊C411yescharacteristic
181C22×D4C2×C814yescomplemented
182C22×D4C22×C412yes—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—41no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4—41no—
193C23×C4—84no—
194C23×C4C22×C411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4#4111yescharacteristic
197AGL(1,ℤ/8)—216no—
198AGL(1,ℤ/8)—28no—
199AGL(1,ℤ/8)C2×C8116yescomplemented
200C2×SD16—28no—
201C2×SD16—24no—
202C2×D8—28no—
203C2×D8—24no—
204C2×M16—21no—
205C2×M16—22no—
206C2×M16—21no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16C2×C814yescomplemented
210C2×M16C22×C412yes—
211C2×M16C22×C412yes—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—28no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—41no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8C22⋊C412yes—
224C4×Q8C22⋊C412yes—
225C4×Q8C2×D412yes—
226C4×Q8C2×D412yes—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—41no—
248C4×D4—41no—
249C4×D4—41no—
250C4×D4—41no—
251C4×D4—41no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—41no—
255C4×D4C22⋊C412yes—
256C4×D4C22⋊C412yes—
257C4×D4C2×D412yes—
258C4×D4C2×D412yes—
259#117—21no—
260#117—21no—
261#117—22no—
262#117—22no—
263#117—22no—
264#117—22no—
265#117C22⋊C412yes—
266#117C22×C412yes—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C22⋊C411yescharacteristic
275C2×C4⋊C4C22⋊C411yescharacteristic
276C2×C4⋊C4C22×C412yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42#4111yescharacteristic
302C2×C42#4111yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42C2411yescharacteristic
305#107—24no—
306#106—24no—
307C4⋊C8—28no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8—22no—
315C4⋊C8—41no—
316#103—24no—
317#102—24no—
318#102—24no—
319#102—24no—
320C22⋊C8—28no—
321C22⋊C8—22no—
322C22⋊C8—22no—
323C22⋊C8—22no—
324C22⋊C8—22no—
325C22⋊C8—41no—
326C22⋊C8—41no—
327C22⋊C8—41no—
328C22⋊C8C2×D414yes—
329#97—22no—
330#97—22no—
331C4×C8—28no—
332C4×C8—22no—
333C4×C8—22no—
334C4×C8—22no—
335C4×C8—22no—
336C4×C8—22no—
337C4×C8—22no—
338C4×C8—41no—
339C4×C8—41no—
340C4×C8—41no—
341C4×C8C2×D414yes—
342C24—21no—
343C24—22no—
344C24—24no—
345C24—41no—
346C24—41no—
347C24—42no—
348C24—82no—
349C24—82no—
350C24C4×D411yescharacteristic
351#41—22no—
352#41—22no—
353#41—24no—
354#41—24no—
355#41—24no—
356#41—24no—
357#41—24no—
358#41C22⋊C818yes—
359#41C22×C814yes—
360#41C2×M1614yes—
361C2×Q8—21no—
362C2×Q8—22no—
363C2×Q8C22⋊C814yes—
364C2×Q8C2×C22⋊C412yes—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4C22⋊C814yes—
389C2×D4C2×C22⋊C412yes—
390C2×D4C22×C814yes—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—82no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—81no—
415C22×C4—82no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—82no—
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C4×D411yescharacteristic
423C22×C4C4×D411yescharacteristic
424C22×C4C4×D411yescharacteristic
425C22×C4C4×D411yescharacteristic
426C22×C4C4×D411yescharacteristic
427C22×C4#12011yescharacteristic
428C22×C4#12011yescharacteristic
429C22×C4#12111yescharacteristic
430C22×C4#12111yescharacteristic
431C22×C4#12111yescharacteristic
432C22×C4#12111yescharacteristic
433C22×C4#12311yescharacteristic
434C22×C4#12311yescharacteristic
435C22×C4C22×C812yes—
436C22×C4C2×M1612yes—
437C22×C4#13112yes—
438C22×C4#13112yes—
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4111yescharacteristic
442SD16—216no—
443D8—216no—
444M16—22no—
445M16—24no—
446M16—22no—
447M16—22no—
448M16—42no—
449M16C22×C814yes—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—82no—
459C2×C8—82no—
460C2×C8—88no—
461C2×C8—82no—
462C2×C8—82no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4C2×C22⋊C412yes—
474C4⋊C4C2×C22⋊C412yes—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42C2×C22⋊C411yescharacteristic
492C42C2×C22⋊C411yescharacteristic
493C42C4×D411yescharacteristic
494C42C4×D411yescharacteristic
495C42#12011yescharacteristic
496C42#12011yescharacteristic
497C42#12111yescharacteristic
498C42#12111yescharacteristic
499C42#12111yescharacteristic
500C42#12111yescharacteristic
501C42#12311yescharacteristic
502C42#12311yescharacteristic
503C42C22×D411yescharacteristic
504C42C22×D411yescharacteristic
505C42C22×D411yescharacteristic
506C42C2×#4111yescharacteristic
507C23—22no—
508C23—24no—
509C23—24no—
510C23—41no—
511C23—41no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—81no—
517C23—81no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—88no—
525C23—88no—
526C23#38611yescharacteristic
527C23#38611yescharacteristic
528C23C8×D412yes—
529C23#43512yes—
530C23C2×C4×D411yescharacteristic
531Q8—22no—
532Q8—24no—
533Q8C2×C22⋊C814yes—
534D4—28no—
535D4—24no—
536D4—24no—
537D4—24no—
538D4—22no—
539D4—24no—
540D4—24no—
541D4—42no—
542D4—44no—
543D4—44no—
544D4—42no—
545D4C2×C22⋊C814yes—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
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—41no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—84no—
578C2×C4—82no—
579C2×C4#38611yescharacteristic
580C2×C4#38611yescharacteristic
581C2×C4#38611yescharacteristic
582C2×C4#38611yescharacteristic
583C2×C4#38611yescharacteristic
584C2×C4#38611yescharacteristic
585C2×C4C2×C22⋊C812yes—
586C2×C4C2×C22⋊C812yes—
587C2×C4#40812yes—
588C2×C4#40812yes—
589C2×C4C8×D412yes—
590C2×C4C8×D412yes—
591C2×C4C8×D412yes—
592C2×C4#43512yes—
593C2×C4#43512yes—
594C2×C4#43512yes—
595C2×C4C22×C22⋊C411yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×#12011yescharacteristic
598C2×C4C2×#12111yescharacteristic
599C2×C4C2×#12111yescharacteristic
600C2×C4C2×#12311yescharacteristic
601C2×C4C23×C811yescharacteristic
602C2×C4C22×M1611yescharacteristic
603C2×C4C2×#13111yescharacteristic
604C2×C4C2×#13111yescharacteristic
605C8—24no—
606C8—44no—
607C8—44no—
608C8—42no—
609C8—82no—
610C8—82no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—84no—
624C22—84no—
625C22—88no—
626C22—88no—
627C22—84no—
628C22—82no—
629C22—82no—
630C22#186912yes—
631C22#186912yes—
632C22C2×#38611yescharacteristic
633C22C2×C8×D411yescharacteristic
634C22C2×#43511yescharacteristic
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—24no—
643C4—42no—
644C4—42no—
645C4—82no—
646C4#186912yes—
647C4#186912yes—
648C4C22×C22⋊C811yescharacteristic
649C4C2×#40811yescharacteristic
650C4C2×C8×D411yescharacteristic
651C4C2×#43511yescharacteristic
652C4#306812yes—
653C2—22no—
654C2—22no—
655C2—21no—
656C2—21no—
657C2—44no—
658C2—84no—
659C2C2×#186911yescharacteristic
660C2#2100011yescharacteristic
661C2#2100511yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC22×AGL(1,ℤ/8)C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60276C2semidirect
3. #17437C2semidirect
4. C2×#1478C2semidirect
5. C2×#1460C2semidirect
6. #9095C2semidirect
7. #7932C2semidirect
8. #7882C2semidirect
18. C8×#41C22semidirect
40. #1477C22semidirect
42. C2×D4⋊C8C22semidirect
44. #1460C22semidirect
46. #1283C22semidirect
47. #1278C22semidirect
53. C2×AGL(1,ℤ/8)C8semidirect
181. C22×D4C2×C8semidirect
199. AGL(1,ℤ/8)C2×C8semidirect
209. C2×M16C2×C8semidirect

Maximal subgroups

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

subgroupindex
#602762
C2×C4×AGL(1,ℤ/8)2
#174372
C2×#14782
C2×#14602
#90952
#79322
#78822