Finite Group Databasebeta

← #222176

Subgroups — #222176

Order 512.

1432 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D8C211yesmaximal, characteristic
2#22401C211yesmaximal, characteristic, complemented
3D4×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×#1586C212yesmaximal, complemented
5#9665C212yesmaximal, complemented
6#8951C212yesmaximal, complemented
7#8947C212yesmaximal, complemented
8#8595C211yesmaximal, characteristic, complemented
9#8310C211yesmaximal, characteristic, complemented
10#8310C211yesmaximal, characteristic, complemented
11#8145C211yesmaximal, characteristic, complemented
12C23×D8C411yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×C4×D8—28no—
16C2×C4×D8C411yescharacteristic
17C2×C4×D8C411yescharacteristic
18C2×C4×D8C411yescharacteristic
19C2×C4×D8C411yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C2×#436C2212yescomplemented
25C2×C8×D4C2212yescomplemented
26C22×#107C411yescharacteristic
27C22×#102C411yescharacteristic
28C22×#102C411yescharacteristic
29C2×#407—21no—
30C22×C22⋊C8—21no—
31C22×C4×C8C2211yescharacteristic
32#1954C2212yescomplemented
33#1954C2212yescomplemented
34#1870—22no—
35#1869—22no—
36#1797C2211yescharacteristic
37#1797C2211yescharacteristic
38#1779—21no—
39#1779—21no—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42#1586—28no—
43C2×#334C2212yescomplemented
44C2×D4⋊C8C2212yescomplemented
45C2×D4⋊C8C2212yescomplemented
46C2×D4⋊C8C2212yescomplemented
47C2×D4⋊C8C2212yescomplemented
48C2×#322C2212yescomplemented
49#1277C2212yescomplemented
50#1273C2212yescomplemented
51C23×D4C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53C24×C4—21no—
54C24×C4—21no—
55C22×D8—24no—
56C22×D8—24no—
57C22×D8—24no—
58C22×D8C2×C411yescharacteristic
59C22×D8C2×C411yescharacteristic
60C22×D8C2×C411yescharacteristic
61C22×D8C2×C411yescharacteristic
62C22×M16—22no—
63C23×C8—22no—
64C23×C8C2×C411yescharacteristic
65C23×C8C2×C411yescharacteristic
66C2×C4×D4—24no—
67C2×C4×D4—24no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—42no—
73C2×C4×D4—42no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42C2311yescharacteristic
87C4×D8—24no—
88C4×D8—24no—
89C4×D8—24no—
90C4×D8—24no—
91C4×D8—24no—
92C4×D8—24no—
93C4×D8—24no—
94C4×D8D418yes—
95#436—24no—
96#436—42no—
97C8×D4—24no—
98C8×D4—42no—
99C2×#107—24no—
100C2×#107C2×C411yescharacteristic
101C2×#107C2×C411yescharacteristic
102C2×#107C2×C411yescharacteristic
103C2×#107C2×C411yescharacteristic
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116#407—44no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—22no—
121C2×C22⋊C8—22no—
122C2×C22⋊C8—22no—
123C2×C22⋊C8—22no—
124C2×C22⋊C8—22no—
125C2×C22⋊C8—22no—
126C2×C22⋊C8—44no—
127C2×C22⋊C8D411yescharacteristic
128C2×C22⋊C8D411yescharacteristic
129C2×C4×C8—24no—
130C2×C4×C8C2×C411yescharacteristic
131C2×C4×C8C2×C411yescharacteristic
132C2×C4×C8C2312yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2312yes—
135C2×C4×C8C2311yescharacteristic
136#336—21no—
137#336—21no—
138#336—21no—
139#336D412yes—
140#334—24no—
141D4⋊C8—24no—
142D4⋊C8—24no—
143D4⋊C8—42no—
144D4⋊C8—42no—
145#322—24no—
146C25—21no—
147C25—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—41no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4—41no—
161C22×D4M1614yescomplemented
162C22×D4C22×C411yescharacteristic
163C22×D4C22×C411yescharacteristic
164C22×D4C22×C411yescharacteristic
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—82no—
183C23×C4—82no—
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4#4111yescharacteristic
187C2×D8—232no—
188C2×D8—22no—
189C2×D8—22no—
190C2×D8—22no—
191C2×D8—22no—
192C2×D8—22no—
193C2×D8—22no—
194C2×D8—22no—
195C2×D8—22no—
196C2×D8—22no—
197C2×D8—22no—
198C2×D8C22⋊C414yes—
199C2×D8M1614yescomplemented
200C2×M16—44no—
201C2×M16—44no—
202C2×M16—82no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—41no—
212C22×C8—44no—
213C22×C8—44no—
214C22×C8—82no—
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—44no—
232C4×D4—44no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4C2×D414yes—
240C4×D4C2×D414yes—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42C22×C411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42C2411yescharacteristic
267#107—22no—
268#107—22no—
269#107—22no—
270#107C22⋊C414yes—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275#102—22no—
276#102—22no—
277#102—22no—
278#102—22no—
279#102—22no—
280#102—22no—
281#102C22⋊C414yes—
282#102C22⋊C414yes—
283C22⋊C8—24no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—44no—
287C22⋊C8—44no—
288C22⋊C8—44no—
289C22⋊C8—44no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8—82no—
293C22⋊C8—82no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—22no—
298C4×C8—22no—
299C4×C8—22no—
300C4×C8—22no—
301C4×C8C2×D414yes—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—24no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—81no—
317C24—81no—
318C24—81no—
319C24—81no—
320C24C4×D411yescharacteristic
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—24no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—44no—
352C2×D4—42no—
353C2×D4C2×C22⋊C412yes—
354C2×D4C2×C22⋊C412yes—
355C2×D4C2×M1612yes—
356C2×D4C2×M1612yes—
357C2×D4C2×M1612yes—
358C2×D4C2×M1612yes—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4C4×D411yescharacteristic
411C22×C4C4×D411yescharacteristic
412C22×C4C4×D411yescharacteristic
413C22×C4C2×M1612yes—
414C22×C4#13112yes—
415C22×C4AGL(1,ℤ/8)12yes—
416C22×C4AGL(1,ℤ/8)12yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C2×#4111yescharacteristic
420D8—28no—
421D8—28no—
422D8—28no—
423D8—28no—
424M16—88no—
425C2×C8—28no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—24no—
433C2×C8—21no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—44no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—88no—
452C2×C8C2×C22⋊C412yes—
453C2×C8C2×C22⋊C412yes—
454C2×C8C2×M1612yes—
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⋊C4C2×C22⋊C412yes—
464C4⋊C4C2×C22⋊C412yes—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C42—21no—
474C42—21no—
475C42—22no—
476C42—22no—
477C42C22×D412yes—
478C42H(5,𝔽2)12yes—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—48no—
490C23—48no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—48no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—84no—
515C23—84no—
516C23—84no—
517C23—84no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23#43512yes—
526C23#44212yes—
527C23#44312yes—
528C23C2×C4×D411yescharacteristic
529D4—22no—
530D4—22no—
531D4—22no—
532D4—22no—
533D4—22no—
534D4—22no—
535D4—22no—
536D4—22no—
537D4—22no—
538D4—48no—
539D4—48no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4#40714yescomplemented
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4#43512yes—
590C2×C4#44212yes—
591C2×C4C22×C22⋊C411yescharacteristic
592C2×C4#51811yescharacteristic
593C2×C4D4212yes—
594C2×C4#54612yes—
595C2×C4C22×M1611yescharacteristic
596C2×C4C22×M1611yescharacteristic
597C2×C4C2×#13111yescharacteristic
598C2×C4#56811yescharacteristic
599C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
600C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
601C2×C4#57612yes—
602C8—22no—
603C8—22no—
604C8—22no—
605C8—44no—
606C8—41no—
607C8—42no—
608C8—42no—
609C8—82no—
610C8—82no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—816no—
634C22—816no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22#158614yescomplemented
648C22D4×C22⋊C411yescharacteristic
649C22C2×#43511yescharacteristic
650C22D4×M1611yescharacteristic
651C22C2×#44211yescharacteristic
652C22#294811yescharacteristic
653C22C2×#44311yescharacteristic
654C22#295711yescharacteristic
655C4—22no—
656C4—22no—
657C4—22no—
658C4—42no—
659C4—42no—
660C4—41no—
661C4—81no—
662C4—81no—
663C4C2×#40711yescharacteristic
664C4#288011yescharacteristic
665C4#297311yescharacteristic
666C4#297411yescharacteristic
667C4#326412yes—
668C4#326412yes—
669C2—24no—
670C2—44no—
671C2—82no—
672C2—82no—
673C2—82no—
674C2C2×#158611yescharacteristic
675C2#1787711yescharacteristic
676C2#1804411yescharacteristic
677C2#1805411yescharacteristic
678C2#2239911yescharacteristic
679C2#2243611yescharacteristic
680C2#2245411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC23×D8C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22401C2semidirect
3. D4×C22⋊C8C2semidirect
4. C2×#1586C2semidirect
5. #9665C2semidirect
6. #8951C2semidirect
7. #8947C2semidirect
8. #8595C2semidirect
9. #8310C2semidirect
10. #8310C2semidirect
11. #8145C2semidirect
24. C2×#436C22semidirect
25. C2×C8×D4C22semidirect
32. #1954C22semidirect
33. #1954C22semidirect
43. C2×#334C22semidirect
44. C2×D4⋊C8C22semidirect
45. C2×D4⋊C8C22semidirect
46. C2×D4⋊C8C22semidirect
47. C2×D4⋊C8C22semidirect
48. C2×#322C22semidirect
49. #1277C22semidirect
50. #1273C22semidirect
161. C22×D4M16semidirect
199. C2×D8M16semidirect
546. D4#407semidirect
647. C22#1586semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D82
#224012
D4×C22⋊C82
C2×#15862
#96652
#89512
#89472
#85952
#83102
#81452