Finite Group Databasebeta

← #196631

Subgroups — #196631

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#423C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic, complemented
4C2×#1547C212yesmaximal, complemented
5#9141C212yesmaximal, complemented
6#9139C212yesmaximal, complemented
7#9133C212yesmaximal, complemented
8#9131C212yesmaximal, complemented
9#8045C212yesmaximal, complemented
10C23×M16—21no—
11C2×#532C2211yescharacteristic
12C2×#525—21no—
13C23×C42C2211yescharacteristic
14C2×#423—21no—
15C2×#423—21no—
16C2×#423—28no—
17C2×#423C2212yes—
18C2×#423C2212yes—
19C22×C4⋊C8—21no—
20C22×C4⋊C8C2212yes—
21C22×C4⋊C8C2212yes—
22C22×C22⋊C8—21no—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854—21no—
28#2854C412yes—
29#2850—21no—
30#2850—21no—
31#2847—21no—
32#2847C412yes—
33C4×#121—21no—
34C4×#121—21no—
35C4×#121C412yes—
36C4×#121C412yes—
37#2297—21no—
38#2285C2211yescharacteristic
39#1797—21no—
40#1797—21no—
41#1797C2212yes—
42#1797C2212yes—
43C2×#336—21no—
44#1547—28no—
45C2×#332C2214yescomplemented
46C2×#331C2214yescomplemented
47#1296C2214yescomplemented
48#1296C2214yescomplemented
49C22×#41—21no—
50C24×C4C2×C411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C22×M16—28no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—44no—
57C22×M16—44no—
58C23×C8—22no—
59C23×C8—22no—
60#532—24no—
61#532—24no—
62C2×C4⋊Q8C2311yescharacteristic
63C2×#127C2311yescharacteristic
64C2×#124—21no—
65#525—44no—
66#525—44no—
67C2×#123—22no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×#121—21no—
75C2×#121C2×C412yes—
76C2×#121C2×C412yes—
77C2×#121C2×C412yes—
78C2×#120—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—21no—
82C2×C4×D4C2×C412yes—
83C2×#117—21no—
84C22×C42—24no—
85C22×C42—24no—
86C22×C42C2×C411yescharacteristic
87C22×C42C2×C411yescharacteristic
88C22×C42C2×C411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#423—28no—
99#423—28no—
100#423—24no—
101#423—28no—
102#423D418yescomplemented
103C2×C4⋊C8—28no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—28no—
108C2×C4⋊C8—24no—
109C2×C4⋊C8—24no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8C2314yes—
112C2×C4⋊C8C2314yes—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—44no—
116#391—21no—
117#391—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#390—21no—
122#390C2×C412yes—
123#390C2×C412yes—
124#389—21no—
125#389—21no—
126#389C2×C412yes—
127#389C2×C412yes—
128#388—22no—
129#388—22no—
130#388—22no—
131#388—22no—
132#386—22no—
133#386—22no—
134#386—22no—
135#386—22no—
136#386—22no—
137#386—22no—
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#382—21no—
141#380—21no—
142#380—21no—
143#380—21no—
144#380C2×C412yes—
145#380C2×C412yes—
146#380C2×C412yes—
147C4×C4⋊C4—21no—
148C4×C4⋊C4C2×C412yes—
149C4×C22⋊C4—22no—
150C4×C22⋊C4—22no—
151#336—22no—
152#336—22no—
153#332—28no—
154#331—28no—
155C25C22⋊C411yescharacteristic
156C2×#41—42no—
157C2×#41—42no—
158C2×#41—42no—
159C2×#41—42no—
160C2×#41—42no—
161C2×#41—42no—
162C22×Q8—21no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4C22×C412yes—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4D814yes—
191C23×C4SD1614yes—
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C2×M16—28no—
202C2×M16—28no—
203C2×M16—432no—
204C2×M16—44no—
205C2×M16—44no—
206C2×M16—44no—
207C2×M16—44no—
208C22×C8—28no—
209C22×C8—24no—
210C22×C8—24no—
211C22×C8—44no—
212C22×C8—44no—
213C22×C8—44no—
214C22×C8—44no—
215C22×C8—44no—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218#127—22no—
219#127—22no—
220#124—42no—
221#124—42no—
222#123—48no—
223C22⋊Q8—42no—
224C22⋊Q8—42no—
225C22⋊Q8—42no—
226C22⋊Q8—42no—
227#121—28no—
228#121—44no—
229#121—44no—
230#121—44no—
231#121—42no—
232#121—42no—
233#121—42no—
234#121—42no—
235#121—42no—
236#121—42no—
237#121—44no—
238#120—48no—
239C4×D4—28no—
240C4×D4—48no—
241C4×D4—48no—
242C4×D4—44no—
243#117—42no—
244#117—42no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D414yes—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42#4114yes—
300C2×C42C2411yescharacteristic
301C4⋊C8—28no—
302C4⋊C8—28no—
303C4⋊C8—28no—
304C4⋊C8—24no—
305C4⋊C8—24no—
306C4⋊C8—24no—
307C4⋊C8—44no—
308C4⋊C8C2×D418yes—
309C22⋊C8—44no—
310C22⋊C8—44no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—41no—
323C24—41no—
324C24#10214yes—
325C24C2×C22⋊C411yescharacteristic
326C24C2×C22⋊C411yescharacteristic
327C24C2×C22⋊C411yescharacteristic
328#41—84no—
329#41—84no—
330#41—84no—
331#41—84no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×D4—24no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—41no—
351C2×D4—42no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
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—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—416no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4#10214yes—
423C22×C4#10214yes—
424C22×C4#10214yes—
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C4×D414yes—
435C22×C4#12012yes—
436C22×C4#12114yes—
437C22×C4#12312yes—
438C22×C4C2×D812yes—
439C22×C4C2×D812yes—
440C22×C4C2×D812yes—
441C22×C4C2×SD1612yes—
442C22×C4C2×SD1612yes—
443C22×C4C2×SD1612yes—
444C22×C4AGL(1,ℤ/8)12yes—
445C22×C4AGL(1,ℤ/8)12yes—
446C22×C4#13712yes—
447C22×C4#13712yes—
448C22×C4C23×C411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451M16—416no—
452M16—416no—
453C2×C8—28no—
454C2×C8—416no—
455C2×C8—44no—
456C2×C8—44no—
457C2×C8—44no—
458C2×C8—44no—
459C2×C8—44no—
460C2×C8—88no—
461C4⋊C4—24no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C42—22no—
486C42—22no—
487C42—21no—
488C42—21no—
489C42—22no—
490C42—22no—
491C42—21no—
492C42—21no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—44no—
500C42C4×D414yes—
501C42#12012yes—
502C42#12114yes—
503C42#12312yes—
504C42C22×D412yes—
505C42C2×#4112yes—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—28no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—28no—
521C23—21no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23#38612yes—
535C23#41312yes—
536C23C2×#10212yes—
537C23C2×#10212yes—
538C23C2×#10212yes—
539C23#41712yes—
540C23#41712yes—
541C23C22×C22⋊C411yescharacteristic
542Q8—81no—
543Q8—81no—
544Q8—81no—
545Q8—81no—
546D4—84no—
547D4—88no—
548D4—88no—
549D4—88no—
550D4—81no—
551D4—81no—
552D4—81no—
553D4—81no—
554D4—82no—
555D4—82no—
556C2×C4—28no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—28no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—88no—
612C2×C4#38612yes—
613C2×C4#38612yes—
614C2×C4#38612yes—
615C2×C4#41312yes—
616C2×C4C2×#10212yes—
617C2×C4C2×#10212yes—
618C2×C4C2×#10212yes—
619C2×C4#41712yes—
620C2×C4#41712yes—
621C2×C4#44714yes—
622C2×C4#44814yes—
623C2×C4#48014yes—
624C2×C4#48114yes—
625C2×C4C2×C4×D412yes—
626C2×C4C2×#12011yescharacteristic
627C2×C4C2×#12112yes—
628C2×C4C2×#12311yescharacteristic
629C2×C4C22×D811yescharacteristic
630C2×C4C22×SD1611yescharacteristic
631C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
632C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
633C2×C4C2×#13711yescharacteristic
634C2×C4C2×#13711yescharacteristic
635C2×C4#57512yes—
636C8—48no—
637C8—84no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—416no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22#154714yescomplemented
666C22#187414yes—
667C22C2×#38611yescharacteristic
668C22C2×#41311yescharacteristic
669C22C2×#41311yescharacteristic
670C22C22×#10211yescharacteristic
671C22C2×#41711yescharacteristic
672C22C2×#41711yescharacteristic
673C22#289611yescharacteristic
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—22no—
681C4—41no—
682C4—41no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4#187414yes—
687C4C2×#44711yescharacteristic
688C4C2×#44811yescharacteristic
689C4#305512yes—
690C4#305612yes—
691C4C2×#48011yescharacteristic
692C4C2×#48111yescharacteristic
693C2—24no—
694C2—24no—
695C2—42no—
696C2—162no—
697C2C2×#154711yescharacteristic
698C2#1770411yescharacteristic
699C2#1775211yescharacteristic
700C2#1775311yescharacteristic
701C2C2×#187411yescharacteristic
702C2#2097111yescharacteristic
703C2#2098911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#423C2semidirect
3. C2×#1797C2semidirect
4. C2×#1547C2semidirect
5. #9141C2semidirect
6. #9139C2semidirect
7. #9133C2semidirect
8. #9131C2semidirect
9. #8045C2semidirect
45. C2×#332C22semidirect
46. C2×#331C22semidirect
47. #1296C22semidirect
48. #1296C22semidirect
102. #423D4semidirect
665. C22#1547semidirect

Maximal subgroups

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

subgroupindex
C22×#4232
#366242
C2×#17972
C2×#15472
#91412
#91392
#91332
#91312
#80452