Finite Group Databasebeta

← #6486065

Subgroups — #6486065

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33280C212yesmaximal, complemented
4#33259C212yesmaximal, complemented
5#30360C214yesmaximal, complemented
6#29523C214yesmaximal, complemented
7#29224C211yesmaximal, characteristic, complemented
8#29212C211yesmaximal, characteristic, complemented
9#29149C211yesmaximal, characteristic, complemented
10#28521C211yesmaximal, characteristic, complemented
11#27897C212yesmaximal
12#27882C212yesmaximal, complemented
13#24709C212yesmaximal, complemented
14#24708C212yesmaximal, complemented
15#22111C212yesmaximal, complemented
16#22110C212yesmaximal, complemented
17C4×#432C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#476C2212yes—
20C2×#474C2212yes—
21C2×#432C2212yes—
22#2864C2212yes—
23#2863—21no—
24#2863—21no—
25#2862C2211yescharacteristic
26#2861—21no—
27#2860—21no—
28#2860C2212yes—
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2856C2211yescharacteristic
36#2855—21no—
37#2855—21no—
38#2854—21no—
39#2854C2212yes—
40#2851C2212yes—
41#2850—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125C2212yes—
47C4×#124—21no—
48C4×C22⋊Q8—21no—
49C4×C22⋊Q8—21no—
50C4×#121—21no—
51C4×#121C2212yes—
52#2295—21no—
53#2295—21no—
54#2293—21no—
55#2293C2212yes—
56#2283C2212yes—
57C2×C43C2211yescharacteristic
58#2101C2214yes—
59#2092C2214yes—
60#2077C2214yes—
61#2076C2214yes—
62#2059C2214yes—
63#2034C2214yes—
64#2033C2214yes—
65#2025C2214yes—
66#1993C2212yes—
67#1993C2212yes—
68#1988C2212yes—
69#1987C2212yes—
70#1961C2212yes—
71#1961C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1944C2214yes—
75#1943C2214yes—
76#1942C2214yes—
77#1941C2214yes—
78#1936C2214yes—
79#1936C2214yes—
80#1934C2214yes—
81#1934C2214yes—
82#1926C2214yes—
83#1925C2214yes—
84#1919C2214yes—
85#1918C2214yes—
86#1847C2212yes—
87#1847C2212yes—
88#1836C2212yes—
89#1836C2212yes—
90#1802C2212yes—
91#1802C2212yes—
92#1767C2211yescharacteristic
93#1767C2211yescharacteristic
94C4×C4⋊C8C2212yes—
95#1749C2212yes—
96C4×C22⋊C8C2212yes—
97#1745C2212yes—
98C4×#97C2212yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103#529—48no—
104C2×#126—22no—
105C2×#125—21no—
106C2×#125C2312yes—
107C2×#124—21no—
108C2×#124—21no—
109C2×#123—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4C2312yes—
123C2×#117—22no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#476—28no—
132#474—28no—
133#432—28no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2314yes—
145C2×#97C2312yes—
146C2×#97C2312yes—
147C2×#97C2314yes—
148C2×C4×C8C2314yes—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
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#389—21no—
180#389—21no—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#386—22no—
206#386—22no—
207#385—22no—
208#385—22no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#383—22no—
214#382—21no—
215#382C2312yes—
216#382C2312yes—
217#382C2312yes—
218#381—22no—
219#380—21no—
220#380—21no—
221#380—21no—
222#380C2312yes—
223#380C2312yes—
224#380C2312yes—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232C4×C4⋊C4C2312yes—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236#376—22no—
237C43D411yescharacteristic
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43C2311yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#336C2312yes—
254#336C2312yes—
255#336C2314yes—
256C22×Q8—21no—
257C22×Q8—21no—
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×C8C2414yes—
272C22×C8C2414yes—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277#127—22no—
278#127—22no—
279#126—48no—
280#125—44no—
281#125—44no—
282#124—44no—
283#124—44no—
284#123—48no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288#121—42no—
289#121—42no—
290#121—44no—
291#121—44no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—48no—
300#117—48no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
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⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
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×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—42no—
350C2×C42—42no—
351C2×C42—42no—
352C2×C42—42no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—28no—
382#107—44no—
383#106—28no—
384#106—44no—
385C4⋊C8—24no—
386C4⋊C8#4118yes—
387#102—28no—
388#102—28no—
389#102—44no—
390#102—44no—
391C22⋊C8—28no—
392#97—28no—
393#97—24no—
394#97—24no—
395C4×C8—44no—
396#95—22no—
397#95—22no—
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—
416C24—22no—
417C24C22×D411yescharacteristic
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4—42no—
442C22×C4—22no—
443C22×C4—22no—
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—21no—
456C22×C4—21no—
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—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
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×C4C2511yescharacteristic
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8C2×#4118yes—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
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—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
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—
537C22⋊C4—44no—
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—21no—
551C42—21no—
552C42—22no—
553C42—22no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—44no—
565C42—44no—
566C42—44no—
567C42AGL(1,ℤ/8)14yes—
568C42C2×#4114yes—
569C422−1+412yes—
570C422−1+412yes—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23#57512yes—
585C23C23×D411yescharacteristic
586Q8—84no—
587Q8—84no—
588Q8—84no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—84no—
594D4—84no—
595D4—88no—
596C2×C4—21no—
597C2×C4—21no—
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—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—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
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—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—88no—
656C2×C4#53212yes—
657C2×C4#53212yes—
658C2×C4#53612yes—
659C2×C4#53612yes—
660C2×C4C2×AGL(1,ℤ/8)12yes—
661C2×C4C2×AGL(1,ℤ/8)12yes—
662C2×C4C2×AGL(1,ℤ/8)12yes—
663C2×C4#57512yes—
664C2×C4#57512yes—
665C2×C4C22×#4112yes—
666C2×C4C2×2−1+411yescharacteristic
667C2×C4C2×2−1+411yescharacteristic
668C2×C4#58512yes—
669C8—28no—
670C8—44no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22C2×#53211yescharacteristic
686C22C2×#53611yescharacteristic
687C22#344911yescharacteristic
688C22C22×AGL(1,ℤ/8)11yescharacteristic
689C22C2×#57511yescharacteristic
690C22C2×#57511yescharacteristic
691C22C2×#57511yescharacteristic
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4#317114yes—
712C4#351512yes—
713C4#351812yes—
714C2—42no—
715C2—162no—
716C2#3520111yescharacteristic
717C2C2×#317111yescharacteristic
718C2#6070211yescharacteristic
719C2#6086411yescharacteristic
720C2#6086611yescharacteristic
721C2#6096111yescharacteristic
722C2#6096211yescharacteristic
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. #33280C2semidirect
4. #33259C2semidirect
5. #30360C2semidirect
6. #29523C2semidirect
7. #29224C2semidirect
8. #29212C2semidirect
9. #29149C2semidirect
10. #28521C2semidirect
12. #27882C2semidirect
13. #24709C2semidirect
14. #24708C2semidirect
15. #22111C2semidirect
16. #22110C2semidirect
17. C4×#432C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#366312
#332802
#332592
#303602
#295232
#292242
#292122
#291492
#285212
#278972
#278822
#247092
#247082
#221112
#221102
C4×#4322