Finite Group Databasebeta

← #6793567

Subgroups — #6793567

Order 512.

1575 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#31445C214yesmaximal, complemented
5#31369C214yesmaximal, complemented
6#29937C212yesmaximal, complemented
7#29287C211yesmaximal, characteristic, complemented
8#29219C211yesmaximal, characteristic, complemented
9#28800C212yesmaximal, complemented
10#27822C212yesmaximal, complemented
11#26301C212yesmaximal, complemented
12#24966C212yesmaximal
13#24709C212yesmaximal, complemented
14#24300C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic, complemented
16#21839C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#529—21no—
20C2×#476C2211yescharacteristic
21C2×#474C2212yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2553—22no—
26#2552C2211yescharacteristic
27#2552C2212yes—
28#2547C2211yescharacteristic
29#2542—21no—
30#2542C2211yescharacteristic
31#2538—21no—
32#2537C2211yescharacteristic
33#2496C2212yes—
34#2489—21no—
35#2441C2214yes—
36#2429—22no—
37#2427—21no—
38#2423—22no—
39#2422C2212yes—
40#2395—22no—
41#2365—21no—
42#2354C2212yes—
43Q8×C4⋊C4—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8—21no—
47C4×#121C2212yes—
48#2285C2212yes—
49C42×D4C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2100C2214yes—
52#2099C2214yes—
53#2088C2214yes—
54#2084C2214yes—
55#2077C2214yes—
56#2077C2214yes—
57#2076C2214yes—
58#2059C2214yes—
59#2034C2214yes—
60#2025C2214yes—
61#1992C2214yes—
62#1992C2214yes—
63#1990C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1988C2212yes—
67#1988C2212yes—
68#1987C2211yescharacteristic
69#1987C2212yes—
70#1954C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2211yescharacteristic
74#1947C2212yes—
75#1947C2212yes—
76#1944C2214yes—
77#1936C2214yes—
78#1924C2214yes—
79#1920C2212yes—
80#1875C2214yes—
81#1864C2212yes—
82#1854C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1825C2214yes—
86#1825C2214yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1802C2212yes—
90#1797C2212yes—
91C4×#106C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#533—28no—
96#533—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102#529—48no—
103C2×#126—22no—
104C2×#125—21no—
105C2×#125C2314yes—
106C2×#124—21no—
107C2×#123—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×#121—21no—
114C2×#121C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×D4—22no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C2×#117C2314yes—
128C22×C42—21no—
129C22×C42C2312yes—
130#476—24no—
131#474—44no—
132#432—44no—
133C2×#107C2314yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#399—24no—
151#399—22no—
152#398—22no—
153#398—24no—
154#397—22no—
155#397—22no—
156#397—22no—
157#396—24no—
158#394—22no—
159#394—22no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—22no—
164#390—22no—
165#390C2312yes—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—24no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2314yes—
176#388—24no—
177#388—22no—
178#387—24no—
179#387—22no—
180#386—22no—
181#385—22no—
182#384—22no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#380—22no—
193#380—22no—
194#380—21no—
195#380C2312yes—
196#380C2314yes—
197#380C2312yes—
198#380C2312yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×D4—22no—
238C22×D4C2×D411yescharacteristic
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—44no—
255C4⋊Q8—82no—
256C4⋊Q8#4114yes—
257#126—48no—
258#125—28no—
259#125—44no—
260#125—44no—
261#124—44no—
262#123—48no—
263C22⋊Q8—48no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C22⋊Q8—48no—
267C22⋊Q8—84no—
268#121—24no—
269#121—42no—
270#121—44no—
271C4×Q8—48no—
272C4×Q8—48no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×D4—22no—
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—42no—
284C4×D4—41no—
285C4×D4—48no—
286#117—28no—
287#117—44no—
288#117—48no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—44no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—44no—
355#106—24no—
356#106—42no—
357#106—44no—
358C4⋊C8—22no—
359C4⋊C8—22no—
360C4⋊C8—24no—
361C4⋊C8—24no—
362C4⋊C8—42no—
363#102—24no—
364#102—24no—
365#102—44no—
366C22⋊C8—44no—
367C22⋊C8—44no—
368#97—44no—
369C4×C8—24no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—21no—
387C24C22×D412yes—
388C2×Q8—24no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D42−1+412yes—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—81no—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2511yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4C2×#4114yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—48no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—84no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—21no—
519C42—41no—
520C42—48no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—44no—
526C42AGL(1,ℤ/8)14yes—
527C42C2×#4112yes—
528C42H(5,𝔽2)12yes—
529C422−1+412yes—
530C422−1+412yes—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C23×D411yescharacteristic
546Q8—48no—
547Q8—88no—
548Q8—84no—
549D4—22no—
550D4—22no—
551D4—44no—
552D4—41no—
553D4—41no—
554D4—88no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4#53612yes—
607C2×C4#53612yes—
608C2×C4#54612yes—
609C2×C4#54612yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4#57512yes—
614C2×C4#57712yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58511yescharacteristic
621C8—42no—
622C8—42no—
623C8—44no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22C2×#53611yescharacteristic
639C22C2×#54611yescharacteristic
640C22#348911yescharacteristic
641C22C22×AGL(1,ℤ/8)11yescharacteristic
642C22C2×#57511yescharacteristic
643C22C2×#57711yescharacteristic
644C22#358611yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—81no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4—82no—
658C4—81no—
659C4—84no—
660C4#323514yes—
661C4#333214yes—
662C4#351811yescharacteristic
663C4#352311yescharacteristic
664C4#355211yescharacteristic
665C4#355411yescharacteristic
666C2—82no—
667C2—82no—
668C2#3891111yescharacteristic
669C2C2×#323511yescharacteristic
670C2#6088911yescharacteristic
671C2C2×#333211yescharacteristic
672C2#6115911yescharacteristic
673C2#6210911yescharacteristic
674C2#6213411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36134C2semidirect
3. #33195C2semidirect
4. #31445C2semidirect
5. #31369C2semidirect
6. #29937C2semidirect
7. #29287C2semidirect
8. #29219C2semidirect
9. #28800C2semidirect
10. #27822C2semidirect
11. #26301C2semidirect
13. #24709C2semidirect
14. #24300C2semidirect
15. #22799C2semidirect
16. #21839C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#349322
#331952
#314452
#313692
#299372
#292872
#292192
#288002
#278222
#263012
#249662
#247092
#243002
#227992
#218392
C4×#4762