Finite Group Databasebeta

← #6480108

Subgroups — #6480108

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33134C214yesmaximal, complemented
4#30291C211yesmaximal, characteristic, complemented
5#30240C211yesmaximal, characteristic
6#30187C211yesmaximal, characteristic, complemented
7#30185C211yesmaximal, characteristic
8#29581C212yesmaximal, complemented
9#29580C212yesmaximal, complemented
10#29561C212yesmaximal, complemented
11#29560C212yesmaximal, complemented
12#29521C212yesmaximal, complemented
13#29520C212yesmaximal, complemented
14#29500C212yesmaximal, complemented
15#29498C212yesmaximal, complemented
16#22185C214yesmaximal, complemented
17C2×#1745C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#465C2214yes—
20C22×#97C2212yes—
21#2864—21no—
22#2863—21no—
23#2863C2212yes—
24#2862C2211yescharacteristic
25#2861—21no—
26#2860—21no—
27#2860C2212yes—
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2855—21no—
36#2855C2212yes—
37#2854—21no—
38#2854C2212yes—
39#2851—21no—
40#2850—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#125—21no—
46C4×#124—21no—
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8C2212yes—
49C4×#121—21no—
50C4×#121C2212yes—
51#2295—21no—
52#2295C2212yes—
53#2293—21no—
54#2293C2212yes—
55#2283—21no—
56C2×C43C2211yescharacteristic
57#2100C2212yes—
58#2099C2212yes—
59#2096C2212yes—
60#2094C2212yes—
61#2083C2212yes—
62#2083C2212yes—
63#2072C2212yes—
64#2071C2212yes—
65#2057C2212yes—
66#2057C2212yes—
67#2043C2212yes—
68#2041C2212yes—
69#2024C2212yes—
70#2023C2212yes—
71#2014C2212yes—
72#2013C2212yes—
73#1973C2214yes—
74#1973C2214yes—
75#1943C2214yes—
76#1943C2214yes—
77#1936C2214yes—
78#1936C2214yes—
79#1935C2214yes—
80#1935C2214yes—
81#1928C2214yes—
82#1925C2214yes—
83#1920C2214yes—
84#1919C2214yes—
85#1893C2214yes—
86#1892C2214yes—
87#1880C2214yes—
88#1877C2214yes—
89#1863C2214yes—
90#1862C2214yes—
91#1808C2214yes—
92#1745C2212yes—
93#1745C2212yes—
94#1745C2212yes—
95#1745C2212yes—
96C2×#336C2211yescharacteristic
97C2×#336C2212yes—
98C2×#336C2211yescharacteristic
99C22×SD16C2314yes—
100C23×C8C2312yes—
101C23×C8C2312yes—
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×Q8C2312yes—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×#117—22no—
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#465—216no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#97—28no—
145C2×#97C2314yes—
146C2×#97C2314yes—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
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#389C2312yes—
180#389C2312yes—
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#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#386—22no—
204#386—22no—
205#385—22no—
206#385—22no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#383—22no—
212#382—21no—
213#382—21no—
214#382—21no—
215#382—21no—
216#381—22no—
217#380—21no—
218#380—21no—
219#380—21no—
220#380—21no—
221#380C2312yes—
222#380C2312yes—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234#376—22no—
235C43D411yescharacteristic
236C43D411yescharacteristic
237C43D411yescharacteristic
238C43D411yescharacteristic
239C43C2311yescharacteristic
240C43C2311yescharacteristic
241C43C2311yescharacteristic
242C43C2311yescharacteristic
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2312yes—
250#336C2312yes—
251#336C2312yes—
252#336C2312yes—
253#336C2314yes—
254#336C2314yes—
255C22×Q8—21no—
256C22×Q8—21no—
257C22×Q8C2412yes—
258C22×D4—22no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4C2412yes—
262C23×C4—22no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×SD16—28no—
271C2×SD16—44no—
272C2×SD16—44no—
273C22×C8—24no—
274C22×C8—42no—
275C22×C8—42no—
276C22×C8—42no—
277C22×C8—42no—
278C22×C8#4118yes—
279C22×C8C2414yes—
280C22×C8C2414yes—
281C4⋊Q8—42no—
282C4⋊Q8—42no—
283C4⋊Q8—42no—
284#127—42no—
285#126—48no—
286#125—44no—
287#125—44no—
288#124—44no—
289#124—44no—
290#123—48no—
291C22⋊Q8—28no—
292C22⋊Q8—44no—
293C22⋊Q8—44no—
294#121—28no—
295#121—44no—
296#121—44no—
297C4×Q8—42no—
298C4×Q8—42no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×D4—42no—
302C4×D4—42no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—48no—
306#117—48no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
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×C42—24no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42—41no—
357C2×C42—41no—
358C2×C42—42no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—44no—
388#107—44no—
389#106—28no—
390#103—28no—
391#103—44no—
392#103—44no—
393#102—28no—
394#102—44no—
395#102—44no—
396#97—28no—
397#97—28no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
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—
418C24—22no—
419C24C22×D411yescharacteristic
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—42no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
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—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4AGL(1,ℤ/8)14yes—
473C22×C4#13714yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
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×C4C2×#4114yes—
488C22×C4C2511yescharacteristic
489SD16—88no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—48no—
496C2×C8—48no—
497C2×C8C2×#4118yes—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
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—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
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—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—22no—
562C42—22no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—44no—
572C42—44no—
573C42—44no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23#53212yes—
586C23C2×AGL(1,ℤ/8)12yes—
587C23C2×#13712yes—
588C23C23×D411yescharacteristic
589Q8—44no—
590Q8—44no—
591Q8—84no—
592Q8—84no—
593D4—44no—
594D4—44no—
595D4—84no—
596D4—84no—
597D4—88no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
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—24no—
613C2×C4—24no—
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—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
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#53212yes—
661C2×C4C2×AGL(1,ℤ/8)12yes—
662C2×C4C2×AGL(1,ℤ/8)12yes—
663C2×C4C2×#13712yes—
664C2×C4C2×#13712yes—
665C2×C4#57512yes—
666C2×C4#57512yes—
667C2×C4C22×#4112yes—
668C2×C4#58511yescharacteristic
669C2×C4#58512yes—
670C2×C4#58511yescharacteristic
671C8—28no—
672C8—44no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22#318214yes—
687C22C2×#53211yescharacteristic
688C22#344911yescharacteristic
689C22#344911yescharacteristic
690C22C22×AGL(1,ℤ/8)11yescharacteristic
691C22C22×#13711yescharacteristic
692C22C2×#57511yescharacteristic
693C22C2×#57511yescharacteristic
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4#349112yes—
715C4#349312yes—
716C2—24no—
717C2—162no—
718C2#3511911yescharacteristic
719C2#6067611yescharacteristic
720C2C2×#318211yescharacteristic
721C2#6095811yescharacteristic
722C2#6096011yescharacteristic
723C2#6096211yescharacteristic
724C2#6096311yescharacteristic
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
1. #36635C2semidirect
3. #33134C2semidirect
4. #30291C2semidirect
6. #30187C2semidirect
8. #29581C2semidirect
9. #29580C2semidirect
10. #29561C2semidirect
11. #29560C2semidirect
12. #29521C2semidirect
13. #29520C2semidirect
14. #29500C2semidirect
15. #29498C2semidirect
16. #22185C2semidirect
17. C2×#1745C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#366312
#331342
#302912
#302402
#301872
#301852
#295812
#295802
#295612
#295602
#295212
#295202
#295002
#294982
#221852
C2×#17452