Finite Group Databasebeta

← #6635622

Subgroups — #6635622

Order 512.

1822 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic, complemented
2C2×#2447C211yesmaximal, characteristic, complemented
3#32953C211yesmaximal, characteristic, complemented
4#32902C211yesmaximal, characteristic
5#31990C214yesmaximal, complemented
6#31926C214yesmaximal, complemented
7#31222C214yesmaximal, complemented
8#31170C214yesmaximal, complemented
9#29951C211yesmaximal, characteristic, complemented
10#29713C211yesmaximal, characteristic, complemented
11#29237C212yesmaximal, complemented
12#24509C212yesmaximal, complemented
13#24506C212yesmaximal, complemented
14#24215C212yesmaximal, complemented
15#21955C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×#124C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#474C2212yes—
20C2×#463C2214yes—
21C2×#459C2212yes—
22C2×#448C2212yes—
23C2×#447C2212yes—
24C2×#432C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2639—21no—
29#2618—21no—
30#2560—22no—
31#2558—22no—
32#2512C2212yescomplemented
33#2506C2212yescomplemented
34#2452C2212yescomplemented
35#2447—22no—
36#2447—22no—
37#2447C2214yes—
38#2447C2214yes—
39#2445C2212yescomplemented
40C2×#397—21no—
41C2×#397—21no—
42#2325—21no—
43C4×#121C2212yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#385—21no—
49C2×#381—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2096C2212yescomplemented
52#2087C2214yescomplemented
53#2072C2212yescomplemented
54#2068C2214yescomplemented
55#2066C2214yescomplemented
56#2065C2214yes—
57#2065C2214yescomplemented
58#2055C2214yescomplemented
59#2055C2214yescomplemented
60#2054C2214yes—
61#2042C2214yes—
62#2024C2212yescomplemented
63#2014C2212yescomplemented
64#2012C2214yes—
65#1983C2214yes—
66#1973C2214yescomplemented
67#1972C2214yes—
68#1940C2214yescomplemented
69#1940C2214yescomplemented
70#1936C2214yes—
71#1935C2214yescomplemented
72#1919C2214yescomplemented
73#1879C2214yes—
74#1874C2212yes—
75#1874C2212yes—
76#1862C2214yes—
77#1862C2214yescomplemented
78#1853C2212yes—
79#1852C2212yes—
80#1802C2212yes—
81#1802C2212yes—
82#1797C2211yescharacteristic
83#1797C2212yes—
84#1797C2211yescharacteristic
85#1787C2212yes—
86#1786C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91C22×D8C2314yes—
92#539—48no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#126—22no—
96C2×#125—22no—
97C2×#124—22no—
98C2×#124—44no—
99C2×#124C2314yes—
100C2×#123—22no—
101C2×#123—22no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—22no—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—44no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#474—44no—
128#463—48no—
129#459—44no—
130#448—44no—
131#447—44no—
132#432—44no—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×#97C2314yes—
150#400—22no—
151#398—22no—
152#397—24no—
153#397—24no—
154#397—22no—
155#397—24no—
156#397—24no—
157#396—22no—
158#396—22no—
159#394—22no—
160#394—22no—
161#393—22no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—22no—
166#391C2312yes—
167#391C2314yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—22no—
172#390C2312yes—
173#390C2314yes—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#389—21no—
178#389—22no—
179#389C2312yes—
180#389C2314yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#385—24no—
193#385—24no—
194#385—22no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#381—24no—
198#381—24no—
199#380—21no—
200#380C2312yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25—21no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8C2412yes—
229C22×D4—24no—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—42no—
237C22×D4C2412yes—
238C22×D4C2412yes—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—44no—
256C2×SD16—44no—
257C2×SD16—44no—
258C2×D8—28no—
259C2×D8—44no—
260C2×D8—44no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263C4⋊Q8—42no—
264#127—42no—
265#126—48no—
266#125—48no—
267#124—28no—
268#124—44no—
269#124—48no—
270#123—48no—
271#123—48no—
272C22⋊Q8—48no—
273C22⋊Q8—44no—
274#121—24no—
275#121—24no—
276#121—44no—
277#121—44no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—42no—
281C4×D4—42no—
282C4×D4—44no—
283C4×D4—48no—
284C4×D4—44no—
285C4×D4—44no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—82no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2412yes—
347#107—44no—
348#106—44no—
349C4⋊C8—44no—
350C4⋊C8—42no—
351C4⋊C8—44no—
352C4⋊C8—42no—
353#103—44no—
354#103—44no—
355#103—44no—
356#102—28no—
357#102—44no—
358#102—44no—
359#102—44no—
360#102—44no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363#97—44no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24C22×D411yescharacteristic
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×Q8—84no—
402C2×Q8—84no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—24no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—81no—
423C2×D4—81no—
424C2×D4—81no—
425C2×D4—81no—
426C2×D4H(5,𝔽2)14yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—816no—
471C22×C4AGL(1,ℤ/8)14yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C22×C4C2511yescharacteristic
484SD16—88no—
485D8—88no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—84no—
514C4⋊C4—81no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—84no—
531C22⋊C4—82no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—24no—
537C42—22no—
538C42—41no—
539C42—41no—
540C42—42no—
541C42—44no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—24no—
547C23—48no—
548C23—48no—
549C23—48no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23#53712yes—
568C23C2×AGL(1,ℤ/8)12yes—
569C23#57512yes—
570C23C23×D411yescharacteristic
571Q8—44no—
572Q8—44no—
573Q8—84no—
574Q8—162no—
575D4—24no—
576D4—24no—
577D4—48no—
578D4—42no—
579D4—44no—
580D4—42no—
581D4—44no—
582D4—88no—
583D4—162no—
584D4—162no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—88no—
632C2×C4—88no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—164no—
641C2×C4#53712yes—
642C2×C4C2×AGL(1,ℤ/8)12yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×H(5,𝔽2)12yes—
646C2×C4C2×2−1+411yescharacteristic
647C2×C4#58512yes—
648C8—44no—
649C8—44no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—816no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22#325114yes—
672C22C2×#53711yescharacteristic
673C22#348411yescharacteristic
674C22#348711yescharacteristic
675C22C22×AGL(1,ℤ/8)11yescharacteristic
676C22C2×#57511yescharacteristic
677C22#358511yescharacteristic
678C22#358611yescharacteristic
679C4—22no—
680C4—22no—
681C4—22no—
682C4—42no—
683C4—42no—
684C4—41no—
685C4—41no—
686C4—42no—
687C4—42no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4#353412yes—
695C4#353512yes—
696C2—24no—
697C2—84no—
698C2—162no—
699C2#3672211yescharacteristic
700C2C2×#325111yescharacteristic
701C2#6091911yescharacteristic
702C2#6156911yescharacteristic
703C2#6157511yescharacteristic
704C2#6188811yescharacteristic
705C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32902C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35006C2semidirect
2. C2×#2447C2semidirect
3. #32953C2semidirect
5. #31990C2semidirect
6. #31926C2semidirect
7. #31222C2semidirect
8. #31170C2semidirect
9. #29951C2semidirect
10. #29713C2semidirect
11. #29237C2semidirect
12. #24509C2semidirect
13. #24506C2semidirect
14. #24215C2semidirect
15. #21955C2semidirect
32. #2512C22semidirect
33. #2506C22semidirect
34. #2452C22semidirect
39. #2445C22semidirect
51. #2096C22semidirect
52. #2087C22semidirect
53. #2072C22semidirect
54. #2068C22semidirect
55. #2066C22semidirect
57. #2065C22semidirect
58. #2055C22semidirect
59. #2055C22semidirect
62. #2024C22semidirect
63. #2014C22semidirect
66. #1973C22semidirect
68. #1940C22semidirect
69. #1940C22semidirect
71. #1935C22semidirect
72. #1919C22semidirect
77. #1862C22semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#24472
#329532
#329022
#319902
#319262
#312222
#311702
#299512
#297132
#292372
#245092
#245062
#242152
#219552