Finite Group Databasebeta

← #6462204

Subgroups — #6462204

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#29573C212yesmaximal, complemented
4#29555C212yesmaximal, complemented
5#29514C212yesmaximal, complemented
6#29494C212yesmaximal, complemented
7#29166C212yesmaximal, complemented
8#28580C212yesmaximal
9#27862C212yesmaximal, complemented
10#25503C212yesmaximal, complemented
11#23839C211yesmaximal, characteristic, complemented
12#23823C211yesmaximal, characteristic, complemented
13#23822C211yesmaximal, characteristic, complemented
14#23803C211yesmaximal, characteristic, complemented
15#21540C214yesmaximal, complemented
16#21124C214yesmaximal, complemented
17#20364C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529—21no—
20C2×#468C2214yes—
21C2×#440C2214yes—
22C2×C4×M16C2212yes—
23#2861—21no—
24#2860—21no—
25#2860C2212yes—
26#2859C2211yescharacteristic
27#2858C2211yescharacteristic
28#2857C2211yescharacteristic
29#2856C2211yescharacteristic
30#2854—21no—
31#2854C2212yes—
32#2850—21no—
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×#127C2211yescharacteristic
39C4×#125—21no—
40C4×#125—21no—
41C4×#124—21no—
42C4×C22⋊Q8—21no—
43C4×C22⋊Q8—21no—
44C4×C22⋊Q8C2212yes—
45C4×C22⋊Q8C2212yes—
46C4×#121—21no—
47C4×#121C2212yes—
48#2296—21no—
49#2295—21no—
50#2295C2212yes—
51#2293—21no—
52#2293C2212yes—
53#2285C2211yescharacteristic
54C42×Q8—21no—
55C42×Q8C2212yes—
56C4×#117—21no—
57C2×C43C2211yescharacteristic
58#2100C2212yes—
59#2099C2212yes—
60#2095C2212yes—
61#2093C2212yes—
62#2084C2212yes—
63#2083C2212yes—
64#2072C2212yes—
65#2071C2212yes—
66#2058C2212yes—
67#2057C2212yes—
68#2043C2212yes—
69#2041C2212yes—
70#2024C2212yes—
71#2023C2212yes—
72#2014C2212yes—
73#2013C2212yes—
74#1968C2214yes—
75#1942C2214yes—
76#1934C2214yes—
77#1933C2214yes—
78#1926C2214yes—
79#1925C2214yes—
80#1921C2214yes—
81#1918C2214yes—
82#1895C2214yes—
83#1894C2214yes—
84#1880C2214yes—
85#1880C2214yes—
86#1865C2214yes—
87#1863C2214yes—
88#1809C2214yes—
89#1776C2214yes—
90#1763C2214yes—
91#1762C2214yes—
92#1745C2211yescharacteristic
93#1745C2211yescharacteristic
94#1745C2212yes—
95#1744C2212yes—
96#1744C2212yes—
97#1728C2212yes—
98#1728C2212yes—
99C22×SD16C2314yes—
100C22×M16C2314yes—
101#533—28no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106#529—48no—
107C2×#126—22no—
108C2×#125—21no—
109C2×#125—21no—
110C2×#124—21no—
111C2×#124—21no—
112C2×#123—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2312yes—
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×D4—22no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×#117—21no—
131C2×#117—21no—
132C2×#117—22no—
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42D411yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#468—216no—
141#440—216no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C4×M16—28no—
152C2×#97C2312yes—
153C2×#97C2312yes—
154C2×C4×C8C2314yes—
155#391—21no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391C2312yes—
160#391C2312yes—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#386—22no—
200#386—22no—
201#385—22no—
202#385—22no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#382—21no—
208#382—21no—
209#380—21no—
210#380—21no—
211#380C2312yes—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235#376—22no—
236C43—22no—
237C43D411yescharacteristic
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43C2311yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#336C2312yes—
252#336C2312yes—
253#336C2312yes—
254#336C2312yes—
255#336C2314yes—
256#336C2314yes—
257C22×Q8—21no—
258C22×Q8—21no—
259C22×Q8C2412yes—
260C22×D4—22no—
261C22×D4—21no—
262C22×D4—21no—
263C22×D4C2412yes—
264C23×C4—22no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2411yescharacteristic
272C2×SD16—28no—
273C2×SD16—28no—
274C2×SD16—44no—
275C2×M16—28no—
276C2×M16—28no—
277C2×M16—44no—
278C22×C8C2414yes—
279C22×C8C2414yes—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—42no—
283#127—42no—
284#126—48no—
285#125—28no—
286#125—44no—
287#124—44no—
288#124—44no—
289#123—48no—
290C22⋊Q8—28no—
291C22⋊Q8—24no—
292C22⋊Q8—44no—
293C22⋊Q8#4118yes—
294#121—28no—
295#121—44no—
296#121—44no—
297C4×Q8—28no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—28no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×D4—28no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—48no—
309#117—28no—
310#117—48no—
311#117—44no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—42no—
365C2×C42—42no—
366C2×C42—42no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395#107—28no—
396#107—28no—
397#106—44no—
398#103—28no—
399#103—28no—
400#103—44no—
401#102—28no—
402#102—28no—
403#102—44no—
404#97—24no—
405#97#4118yes—
406C4×C8—28no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423C24—22no—
424C24C22×D411yescharacteristic
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8C2×#4114yes—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—42no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4114yes—
492C22×C4C2511yescharacteristic
493SD16—416no—
494M16—416no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8C2×#4118yes—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—28no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4C2×#4114yes—
531C22⋊C4—28no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C42—28no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—44no—
571C42—44no—
572C42—44no—
573C42—44no—
574C42—44no—
575C42—44no—
576C42AGL(1,ℤ/8)14yes—
577C42#13714yes—
578C42C2×#4114yes—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23#52512yes—
593C23C23×D411yescharacteristic
594Q8—28no—
595Q8—48no—
596Q8—44no—
597Q8—84no—
598D4—44no—
599D4—44no—
600D4—84no—
601D4—84no—
602D4—88no—
603C2×C4—28no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—28no—
627C2×C4—21no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—88no—
660C2×C4#52512yes—
661C2×C4#53212yes—
662C2×C4#53212yes—
663C2×C4C2×AGL(1,ℤ/8)12yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4C2×AGL(1,ℤ/8)12yes—
666C2×C4C2×#13712yes—
667C2×C4C2×#13712yes—
668C2×C4C2×#13712yes—
669C2×C4#57512yes—
670C2×C4#57512yes—
671C2×C4C22×#4112yes—
672C2×C4C22×#4112yes—
673C2×C4#58512yes—
674C8—28no—
675C8—44no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22C2×#52511yescharacteristic
691C22C2×#53211yescharacteristic
692C22#344911yescharacteristic
693C22C22×AGL(1,ℤ/8)11yescharacteristic
694C22C22×#13711yescharacteristic
695C22C2×#57511yescharacteristic
696C22C2×#57511yescharacteristic
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—41no—
709C4—44no—
710C4—41no—
711C4—44no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4#347514yes—
718C2—42no—
719C2—162no—
720C2#3487611yescharacteristic
721C2#6056011yescharacteristic
722C2#6056111yescharacteristic
723C2#6070011yescharacteristic
724C2#6070111yescharacteristic
725C2#6096911yescharacteristic
726C2#6097011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#533C2semidirect
3. #29573C2semidirect
4. #29555C2semidirect
5. #29514C2semidirect
6. #29494C2semidirect
7. #29166C2semidirect
9. #27862C2semidirect
10. #25503C2semidirect
11. #23839C2semidirect
12. #23823C2semidirect
13. #23822C2semidirect
14. #23803C2semidirect
15. #21540C2semidirect
16. #21124C2semidirect
17. #20364C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C4×#5332
#295732
#295552
#295142
#294942
#291662
#285802
#278622
#255032
#238392
#238232
#238222
#238032
#215402
#211242
#203642