Finite Group Databasebeta

← #6489469

Subgroups — #6489469

Order 512.

1628 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33217C212yesmaximal, complemented
4#33052C212yesmaximal
5#30384C212yesmaximal, complemented
6#30383C212yesmaximal, complemented
7#29223C211yesmaximal, characteristic, complemented
8#29183C212yesmaximal, complemented
9#28722C211yesmaximal, characteristic, complemented
10#28692C211yesmaximal, characteristic, complemented
11#28661C212yesmaximal, complemented
12#28516C211yesmaximal, characteristic, complemented
13#27857C212yesmaximal, complemented
14#27856C212yesmaximal, complemented
15#24839C214yesmaximal, complemented
16#22120C214yesmaximal, complemented
17C2×#1767C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#501C2212yes—
20C2×#490C2212yes—
21C2×#478C2214yes—
22C22×C4⋊C8C2212yes—
23C22×#97C2212yes—
24#2864—21no—
25#2864—21no—
26#2861—21no—
27#2860—21no—
28#2860C2212yes—
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2859C2211yescharacteristic
34#2858C2211yescharacteristic
35#2857C2211yescharacteristic
36#2856C2211yescharacteristic
37#2854—21no—
38#2854C2212yes—
39#2851—21no—
40#2851—21no—
41#2851C2212yes—
42#2851C2212yes—
43#2850—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125—21no—
47C4×#125—21no—
48C4×#125C2212yes—
49C4×#125C2212yes—
50C4×#124—21no—
51C4×#121—21no—
52C4×#121C2212yes—
53#2296—21no—
54#2296—21no—
55#2293—21no—
56#2293C2212yes—
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59C2×C43C2211yescharacteristic
60#2100C2212yes—
61#2099C2212yes—
62#2094C2212yes—
63#2093C2212yes—
64#2088C2214yes—
65#2087C2214yes—
66#2084C2212yes—
67#2083C2212yes—
68#2071C2212yes—
69#2071C2212yes—
70#2067C2214yes—
71#2065C2214yes—
72#1993C2212yes—
73#1992C2212yes—
74#1992C2212yes—
75#1989C2214yes—
76#1961C2212yes—
77#1949C2214yes—
78#1944C2214yes—
79#1943C2214yes—
80#1936C2214yes—
81#1936C2214yes—
82#1926C2214yes—
83#1925C2214yes—
84#1919C2214yes—
85#1918C2214yes—
86#1893C2214yes—
87#1893C2214yes—
88#1847C2212yes—
89#1836C2212yes—
90#1808C2214yes—
91#1807C2214yes—
92#1804C2214yes—
93#1767C2211yescharacteristic
94#1767C2211yescharacteristic
95#1767C2212yes—
96#1767C2212yes—
97#1767C2212yes—
98C2×#336C2212yes—
99C23×C8C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127C2311yescharacteristic
102#529—48no—
103C2×#126—22no—
104C2×#125—21no—
105C2×#125—21no—
106C2×#125—21no—
107C2×#125C2312yes—
108C2×#125C2312yes—
109C2×#124—21no—
110C2×#124—21no—
111C2×#123—22no—
112C2×C22⋊Q8—21no—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2312yes—
121C2×#117—21no—
122C2×#117—21no—
123C2×#117—22no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#501—44no—
132#490—44no—
133#478—216no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8—28no—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#97—44no—
147C2×#97C2312yes—
148C2×#97C2312yes—
149C2×#97C2314yes—
150#391—21no—
151#391—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390C2312yes—
159#390C2312yes—
160#390C2312yes—
161#390C2312yes—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
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#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#386—22no—
203#386—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#385—22no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#382—21no—
215#382—21no—
216#382—21no—
217#382—21no—
218#382—21no—
219#382C2312yes—
220#382C2312yes—
221#380—21no—
222#380—21no—
223#380—21no—
224#380C2312yes—
225#380C2312yes—
226#380C2312yes—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4C2312yes—
235C4×C4⋊C4C2312yes—
236C4×C4⋊C4C2312yes—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#336C2312yes—
256#336C2312yes—
257#336C2314yes—
258C22×Q8—21no—
259C22×D4—22no—
260C22×D4—21no—
261C22×D4—21no—
262C22×D4C2412yes—
263C23×C4—22no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2411yescharacteristic
271C22×C8—28no—
272C22×C8—28no—
273C22×C8—44no—
274C22×C8C2414yes—
275C22×C8C2414yes—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278#127—22no—
279#127—22no—
280#126—48no—
281#125—28no—
282#125—24no—
283#125—24no—
284#125—44no—
285#125—44no—
286#125—44no—
287#124—44no—
288#124—44no—
289#123—48no—
290C22⋊Q8—44no—
291#121—28no—
292#121—44no—
293#121—44no—
294C4×Q8—44no—
295C4×D4—28no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—48no—
299#117—48no—
300#117—44no—
301#117—44no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—22no—
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×C42—24no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42—42no—
358C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386#107—28no—
387#107—44no—
388#106—28no—
389#106—44no—
390C4⋊C8—28no—
391C4⋊C8—24no—
392C4⋊C8#4118yes—
393#102—28no—
394#102—28no—
395#102—44no—
396#102—44no—
397#97—24no—
398#97—24no—
399#97—44no—
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—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—42no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
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—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4AGL(1,ℤ/8)14yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C42−1+414yes—
481C22×C4C2511yescharacteristic
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—416no—
488C2×C8C2×#4118yes—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊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—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—21no—
553C42—22no—
554C42—22no—
555C42—21no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—44no—
565C42—44no—
566C42—44no—
567C42C2×#4114yes—
568C42H(5,𝔽2)12yes—
569C422−1+412yes—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23#53612yes—
582C23C2×AGL(1,ℤ/8)12yes—
583C23#57512yes—
584C23C23×D411yescharacteristic
585Q8—84no—
586D4—44no—
587D4—44no—
588D4—84no—
589D4—84no—
590D4—88no—
591C2×C4—28no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
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—24no—
612C2×C4—24no—
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—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
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—88no—
651C2×C4#53212yes—
652C2×C4#53212yes—
653C2×C4#53612yes—
654C2×C4#53712yes—
655C2×C4#53712yes—
656C2×C4C2×AGL(1,ℤ/8)12yes—
657C2×C4C2×AGL(1,ℤ/8)12yes—
658C2×C4#57512yes—
659C2×C4#57512yes—
660C2×C4C22×#4112yes—
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×2−1+412yes—
663C2×C4C2×2−1+411yescharacteristic
664C8—28no—
665C8—44no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22#323414yes—
680C22C2×#53211yescharacteristic
681C22C2×#53611yescharacteristic
682C22C2×#53711yescharacteristic
683C22C22×AGL(1,ℤ/8)11yescharacteristic
684C22C2×#57511yescharacteristic
685C22C2×#57511yescharacteristic
686C22C2×#57511yescharacteristic
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4#352012yes—
708C4#352312yes—
709C2—24no—
710C2—162no—
711C2#3524111yescharacteristic
712C2#6068811yescharacteristic
713C2#6070011yescharacteristic
714C2C2×#323411yescharacteristic
715C2#6085411yescharacteristic
716C2#6091611yescharacteristic
717C2#6091711yescharacteristic
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. #36640C2semidirect
3. #33217C2semidirect
5. #30384C2semidirect
6. #30383C2semidirect
7. #29223C2semidirect
8. #29183C2semidirect
9. #28722C2semidirect
10. #28692C2semidirect
11. #28661C2semidirect
12. #28516C2semidirect
13. #27857C2semidirect
14. #27856C2semidirect
15. #24839C2semidirect
16. #22120C2semidirect
17. C2×#1767C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#366312
#332172
#330522
#303842
#303832
#292232
#291832
#287222
#286922
#286612
#285162
#278572
#278562
#248392
#221202
C2×#17672