Finite Group Databasebeta

← #227171

Subgroups — #227171

Order 512.

799 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#453C211yesmaximal, characteristic
2#17873C211yesmaximal, characteristic, complemented
3#17852C211yesmaximal, characteristic, complemented
4#17453C211yesmaximal, characteristic, complemented
5#17438C211yesmaximal, characteristic, complemented
6#9644C211yesmaximal, characteristic, complemented
7#9552C211yesmaximal, characteristic, complemented
8#9550C211yesmaximal, characteristic, complemented
9#9097C211yesmaximal, characteristic, complemented
10#9095C211yesmaximal, characteristic, complemented
11#8959C211yesmaximal, characteristic, complemented
12#8841C211yesmaximal, characteristic, complemented
13#8839C211yesmaximal, characteristic, complemented
14#8197C211yesmaximal, characteristic, complemented
15#8193C211yesmaximal, characteristic, complemented
16C4×H(5,𝔽2)C2211yescharacteristic
17C2×#453C411yescharacteristic
18#2989C2211yescharacteristic
19#2988C2211yescharacteristic
20#2972C2211yescharacteristic, complemented
21#2965C2211yescharacteristic
22C8×#41C2211yescharacteristic
23C4×AGL(1,ℤ/8)C2211yescharacteristic
24C4×AGL(1,ℤ/8)C2211yescharacteristic
25C4×#127C2211yescharacteristic
26#1887C411yescharacteristic
27#1883C411yescharacteristic
28#1883C411yescharacteristic
29#1859C411yescharacteristic
30#1859C411yescharacteristic
31#1856C411yescharacteristic
32#1842C2211yescharacteristic, complemented
33#1792C411yescharacteristic
34C4×#104C2211yescharacteristic
35C4×#104C2211yescharacteristic
36C4×#100C2211yescharacteristic
37#1744C2211yescharacteristic, complemented
38#1586—21no—
39#1585—21no—
40#1582—21no—
41#1581—21no—
42#1580—21no—
43SD16⋊C8—21no—
44C8×SD16—21no—
45C8×D8—21no—
46#1565C2211yescharacteristic, complemented
47#1565C2211yescharacteristic, complemented
48#1490—21no—
49#1489—21no—
50#1488—21no—
51#1486—21no—
52#1478—21no—
53#1478C2211yescharacteristic
54#1478C2211yescharacteristic
55#1477—21no—
56#1477C2211yescharacteristic
57#1477C2211yescharacteristic
58#1476—21no—
59#1476C2211yescharacteristic
60#1476C2211yescharacteristic
61C2×D4⋊C8—21no—
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic, complemented
65C2×D4⋊C8C2211yescharacteristic, complemented
66#1460C2211yescharacteristic, complemented
67#1451C2211yescharacteristic, complemented
68C8×M16C2211yescharacteristic, complemented
69#1283C2211yescharacteristic, complemented
70#1283C2211yescharacteristic, complemented
71#1278C2211yescharacteristic, complemented
72#1278C2211yescharacteristic, complemented
73#1276C2211yescharacteristic, complemented
74#1276C2211yescharacteristic, complemented
75C2×H(5,𝔽2)C2×C411yescharacteristic
76C2×AGL(1,ℤ/8)C2×C411yescharacteristic
77C2×AGL(1,ℤ/8)C2×C411yescharacteristic
78C2×#127C2×C411yescharacteristic
79#520—21no—
80#520—21no—
81#520C2×C411yescharacteristic
82#520C2×C411yescharacteristic
83#518—21no—
84#518—21no—
85#518—41no—
86#518C2×C411yescharacteristic
87C4×#41—21no—
88C4×#41—21no—
89C4×#41C2311yescharacteristic
90C4×#41C2311yescharacteristic
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×C4×D4—41no—
94C2×C4×D4—41no—
95C2×C4×D4C2311yescharacteristic
96C2×C4×D4C2311yescharacteristic
97#453—24no—
98#446—21no—
99#446—21no—
100C8×Q8—21no—
101C8×Q8—21no—
102#442—21no—
103#442—21no—
104#440—21no—
105#440—21no—
106C4×SD16—21no—
107C4×SD16—21no—
108C4×D8—21no—
109C4×D8—21no—
110#436—21no—
111#436—21no—
112#435—41no—
113#435—41no—
114C8×D4—21no—
115C8×D4—21no—
116C8×D4—21no—
117C8×D4—21no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4—41no—
121C8×D4—41no—
122#432—21no—
123#431—21no—
124#431—21no—
125#431—21no—
126#431C2311yescharacteristic
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#428C2×C411yescharacteristic
131#428C2×C411yescharacteristic
132#423—21no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8—41no—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×#104C2×C411yescharacteristic
138C2×#104C2×C411yescharacteristic
139C2×#104C2×C411yescharacteristic
140C2×#104C2×C411yescharacteristic
141#418C2×C411yescharacteristic
142#418C2×C411yescharacteristic
143#417C2×C411yescharacteristic
144#417C2×C411yescharacteristic
145C2×#100C2×C411yescharacteristic
146C2×#100C2×C411yescharacteristic
147C2×C22⋊C8—41no—
148C4×M16—21no—
149C4×M16C2311yescharacteristic
150C4×M16C2311yescharacteristic
151C2×C4×C8—21no—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154#390—41no—
155#390C2×C411yescharacteristic
156#390C2×C411yescharacteristic
157#384C2×C411yescharacteristic
158C43C2311yescharacteristic
159#343C2×C411yescharacteristic
160#343C2×C411yescharacteristic
161#339C2×C411yescharacteristic
162#339C2×C411yescharacteristic
163#339C2×C411yescharacteristic
164#339C2×C411yescharacteristic
165#336—41no—
166#335—21no—
167#335—21no—
168#334—21no—
169#334—21no—
170Q8⋊C8—21no—
171Q8⋊C8—21no—
172Q8⋊C8D411yescharacteristic, complemented
173Q8⋊C8D411yescharacteristic, complemented
174Q8⋊C8D411yescharacteristic, complemented
175Q8⋊C8D411yescharacteristic, complemented
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8—21no—
188D4⋊C8—21no—
189D4⋊C8—21no—
190D4⋊C8D411yescharacteristic, complemented
191D4⋊C8D411yescharacteristic, complemented
192D4⋊C8D411yescharacteristic, complemented
193D4⋊C8D411yescharacteristic, complemented
194#323—41no—
195#322—21no—
196#322—21no—
197#322—21no—
198C82—21no—
199H(5,𝔽2)—22no—
200H(5,𝔽2)—22no—
201H(5,𝔽2)—22no—
202H(5,𝔽2)—41no—
203C2×#41—21no—
204C2×#41—21no—
205C2×#41C22×C411yescharacteristic
206C2×#41C22×C411yescharacteristic
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—41no—
210C22×D4—41no—
211C22×D4C22×C411yescharacteristic
212C22×D4C22×C411yescharacteristic
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—41no—
216AGL(1,ℤ/8)—22no—
217AGL(1,ℤ/8)—22no—
218AGL(1,ℤ/8)—41no—
219AGL(1,ℤ/8)M1614yescomplemented
220C2×SD16—21no—
221C2×SD16—21no—
222C2×D8—21no—
223C2×D8—21no—
224C2×M16—41no—
225C2×M16—41no—
226C2×M16C22×C411yescharacteristic
227C2×M16C22×C411yescharacteristic
228C2×M16C22×C411yescharacteristic
229C2×M16C22×C411yescharacteristic
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—41no—
235C22×C8—41no—
236C22×C8—41no—
237C22×C8—41no—
238#127—22no—
239#127—22no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8C2×D411yescharacteristic
243C4×Q8C2×D411yescharacteristic
244C4×Q8#4111yescharacteristic
245C4×Q8#4111yescharacteristic
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—41no—
270C4×D4—41no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4—82no—
275C4×D4—82no—
276C4×D4C2×D411yescharacteristic
277C4×D4C2×D411yescharacteristic
278C4×D4#4111yescharacteristic
279C4×D4#4111yescharacteristic
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—41no—
285#117—41no—
286#117C22×C411yescharacteristic
287#117C22×C411yescharacteristic
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—41no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—41no—
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C2411yescharacteristic
310#107—21no—
311#107—21no—
312#106—21no—
313#106—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—41no—
321C4⋊C8—41no—
322C4⋊C8—41no—
323C4⋊C8—41no—
324C4⋊C8—82no—
325C4⋊C8C2×D411yescharacteristic
326C4⋊C8C2×D411yescharacteristic
327C4⋊C8C2×D411yescharacteristic
328C4⋊C8C2×D411yescharacteristic
329#104—22no—
330#104—22no—
331#104—22no—
332#104—41no—
333#103—21no—
334#103—21no—
335#102—21no—
336#102—21no—
337#102—21no—
338#102—21no—
339#102—21no—
340#102—21no—
341#100—22no—
342#100—22no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347C22⋊C8—41no—
348C22⋊C8—41no—
349C22⋊C8—41no—
350C22⋊C8—41no—
351C22⋊C8—41no—
352C22⋊C8—41no—
353#97—21no—
354#97—21no—
355#97—41no—
356#97—41no—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8—41no—
366C4×C8—41no—
367C4×C8C2×D411yescharacteristic
368C4×C8C2×D411yescharacteristic
369C4×C8C2×D411yescharacteristic
370C4×C8C2×D411yescharacteristic
371C24—21no—
372C24—21no—
373C24—41no—
374#41—21no—
375#41—21no—
376#41—24no—
377#41—21no—
378#41—21no—
379#41—21no—
380#41—21no—
381#41—42no—
382#41—42no—
383#41—42no—
384#41C2×M1612yes—
385#41#13112yes—
386C2×Q8C4×D411yescharacteristic
387C2×Q8C4×D411yescharacteristic
388C2×D4—21no—
389C2×D4—24no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4C4×D411yescharacteristic
412C2×D4C4×D411yescharacteristic
413C2×D4C2×M1612yes—
414C2×D4#13112yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—161no—
439C22×C4—161no—
440C22×C4C23×C411yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443SD16—24no—
444SD16—42no—
445D8—24no—
446D8—42no—
447M16—21no—
448M16—21no—
449M16—21no—
450M16—21no—
451M16—21no—
452M16—21no—
453M16—21no—
454M16—82no—
455M16—161no—
456M16C2×M1612yes—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—81no—
474C2×C8—81no—
475C2×C8—82no—
476C2×C8—81no—
477C2×C8—81no—
478C2×C8—161no—
479C2×C8—161no—
480C2×C8—161no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—82no—
492C4⋊C4C4×D411yescharacteristic
493C4⋊C4C4×D411yescharacteristic
494C4⋊C4C4×D411yescharacteristic
495C4⋊C4C4×D411yescharacteristic
496C22⋊C4—21no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—41no—
525C42—41no—
526C42—81no—
527C42C22×D411yescharacteristic
528C42C22×D411yescharacteristic
529C42C2×#4111yescharacteristic
530C42C2×#4111yescharacteristic
531C23—21no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—44no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—81no—
541C23—81no—
542C23—81no—
543C23—81no—
544C23—161no—
545C23—161no—
546C23#51811yescharacteristic
547Q8—21no—
548Q8—21no—
549Q8—21no—
550Q8#43512yes—
551D4—22no—
552D4—22no—
553D4—22no—
554D4—21no—
555D4—22no—
556D4—21no—
557D4—22no—
558D4—21no—
559D4—22no—
560D4—44no—
561D4—44no—
562D4—41no—
563D4—41no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—88no—
569D4#43512yes—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—161no—
605C2×C4—161no—
606C2×C4—161no—
607C2×C4—161no—
608C2×C4—161no—
609C2×C4—161no—
610C2×C4C2×C4×D411yescharacteristic
611C2×C4C2×C4×D411yescharacteristic
612C2×C4D4211yescharacteristic
613C2×C4#54611yescharacteristic
614C2×C4#54611yescharacteristic
615C2×C4#55111yescharacteristic
616C2×C4C22×M1611yescharacteristic
617C2×C4C2×#13111yescharacteristic
618C2×C4C2×#13111yescharacteristic
619C2×C4#56811yescharacteristic
620C8—22no—
621C8—41no—
622C8—41no—
623C8—41no—
624C8—41no—
625C8—41no—
626C8—41no—
627C8—41no—
628C8—41no—
629C8—41no—
630C8—41no—
631C8—41no—
632C8—81no—
633C8—81no—
634C8—81no—
635C8—81no—
636C22—21no—
637C22—21no—
638C22—22no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—42no—
643C22—42no—
644C22—41no—
645C22—41no—
646C22—82no—
647C22—82no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—161no—
653C22—161no—
654C22—164no—
655C22#235811yescharacteristic
656C22#297411yescharacteristic
657C22#297611yescharacteristic
658C4—22no—
659C4—21no—
660C4—21no—
661C4—21no—
662C4—42no—
663C4—42no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—81no—
670C4—81no—
671C4C2×#43511yescharacteristic
672C4#292911yescharacteristic
673C4#293311yescharacteristic
674C4D4×M1611yescharacteristic
675C4#329111yescharacteristic
676C4#329211yescharacteristic
677C2—21no—
678C2—21no—
679C2—42no—
680C2—81no—
681C2—82no—
682C2—82no—
683C2#2284611yescharacteristic
684C2#2289711yescharacteristic
685C2#2289811yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radicalC2×#453C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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
2. #17873C2semidirect
3. #17852C2semidirect
4. #17453C2semidirect
5. #17438C2semidirect
6. #9644C2semidirect
7. #9552C2semidirect
8. #9550C2semidirect
9. #9097C2semidirect
10. #9095C2semidirect
11. #8959C2semidirect
12. #8841C2semidirect
13. #8839C2semidirect
14. #8197C2semidirect
15. #8193C2semidirect
20. #2972C22semidirect
32. #1842C22semidirect
37. #1744C22semidirect
46. #1565C22semidirect
47. #1565C22semidirect
64. C2×D4⋊C8C22semidirect
65. C2×D4⋊C8C22semidirect
66. #1460C22semidirect
67. #1451C22semidirect
68. C8×M16C22semidirect
69. #1283C22semidirect
70. #1283C22semidirect
71. #1278C22semidirect
72. #1278C22semidirect
73. #1276C22semidirect
74. #1276C22semidirect
172. Q8⋊C8D4semidirect
173. Q8⋊C8D4semidirect
174. Q8⋊C8D4semidirect
175. Q8⋊C8D4semidirect
190. D4⋊C8D4semidirect
191. D4⋊C8D4semidirect
192. D4⋊C8D4semidirect
193. D4⋊C8D4semidirect
219. AGL(1,ℤ/8)M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4532
#178732
#178522
#174532
#174382
#96442
#95522
#95502
#90972
#90952
#89592
#88412
#88392
#81972
#81932