Finite Group Databasebeta

← #6471599

Subgroups — #6471599

Order 512.

2088 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic, complemented
2C2×#2349C211yesmaximal, characteristic, complemented
3#33004C212yesmaximal, complemented
4#32953C211yesmaximal, characteristic, complemented
5#32902C211yesmaximal, characteristic
6#29967C211yesmaximal, characteristic, complemented
7#29912C212yesmaximal, complemented
8#29755C211yesmaximal, characteristic, complemented
9#27574C214yesmaximal, complemented
10#27318C214yesmaximal, complemented
11#26388C214yesmaximal, complemented
12#26178C214yesmaximal, complemented
13#22082C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21452C212yesmaximal, complemented
16C2×#539—21no—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#481C2212yes—
21C2×#474C2212yes—
22C2×#459C2212yes—
23C2×#447C2212yes—
24C2×#442C2214yes—
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#2445C2212yescomplemented
36#2349—22no—
37#2349—22no—
38#2349C2214yes—
39#2349C2214yes—
40#2325—21no—
41C4×#121C2212yes—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#385—21no—
45C2×#385—21no—
46C2×#382C2211yescharacteristic
47C2×#380—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C22×#95—21no—
50#2101C2214yescomplemented
51#2099C2212yescomplemented
52#2092C2214yescomplemented
53#2088C2214yescomplemented
54#2085C2214yescomplemented
55#2084C2212yescomplemented
56#2073C2214yescomplemented
57#2065C2214yescomplemented
58#2058C2212yescomplemented
59#2041C2212yescomplemented
60#1944C2214yescomplemented
61#1936C2214yescomplemented
62#1919C2214yescomplemented
63#1917C2214yescomplemented
64#1917C2214yescomplemented
65#1879C2214yes—
66#1879C2214yes—
67#1879C2214yes—
68#1879C2214yes—
69#1874C2212yes—
70#1874C2212yes—
71#1864C2214yes—
72#1864C2214yes—
73#1864C2214yes—
74#1862C2214yes—
75#1862C2214yescomplemented
76#1853C2212yes—
77#1852C2212yes—
78#1824C2212yes—
79#1817C2212yes—
80#1802C2212yes—
81#1802C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1786C2212yes—
85#1762C2214yes—
86#1749C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C24×C4—21no—
90C24×C4—21no—
91C22×D8C2314yes—
92#539—48no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#126—22no—
96C2×#125—22no—
97C2×#123—22no—
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—22no—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×D4—24no—
105C2×C4×D4—22no—
106C2×C4×D4—24no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—44no—
112C2×C4×D4—44no—
113C2×C4×D4C2314yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2314yes—
117C2×C4×D4C2312yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#481—28no—
132#474—44no—
133#459—44no—
134#447—28no—
135#442—48no—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#102C2314yes—
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—22no—
153#396—22no—
154#396—22no—
155#394—22no—
156#394—22no—
157#393—22no—
158#391—21no—
159#390—21no—
160#390—21no—
161#390C2312yes—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#385—24no—
185#385—24no—
186#385—24no—
187#385—24no—
188#385—22no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#382—22no—
192#382C2314yes—
193#380—21no—
194#380—24no—
195#380—24no—
196#380C2312yes—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—24no—
208C2×#95—24no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C22×Q8—21no—
224C22×D4—24no—
225C22×D4—24no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—22no—
231C22×D4#4114yes—
232C22×D4C2412yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—28no—
236C23×C4—24no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4—82no—
249C23×C4—82no—
250C23×C4—82no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×D8—28no—
259C2×D8—28no—
260C2×D8—28no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263C4⋊Q8—42no—
264#127—42no—
265#126—48no—
266#125—48no—
267#123—48no—
268#123—48no—
269C22⋊Q8—48no—
270C22⋊Q8—44no—
271#121—24no—
272#121—24no—
273#121—44no—
274#121—44no—
275C4×D4—216no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—44no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—48no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4#4114yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
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—44no—
339C2×C22⋊C4—44no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353#107—44no—
354#106—28no—
355#106—44no—
356C4⋊C8—42no—
357C4⋊C8—42no—
358#102—28no—
359#102—28no—
360#102—28no—
361#102—28no—
362#102—44no—
363#102—44no—
364C22⋊C8—24no—
365C22⋊C8—24no—
366#97—44no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—24no—
383C24—24no—
384C24—24no—
385C24—22no—
386C24—22no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C22×D411yescharacteristic
392C2×Q8—42no—
393C2×Q8—81no—
394C2×Q8—81no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—216no—
398C2×D4—216no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4C2×#4114yes—
417C2×D4H(5,𝔽2)14yes—
418C2×D42−1+414yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—216no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—816no—
469C22×C4—816no—
470C22×C4—816no—
471C22×C4AGL(1,ℤ/8)14yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2511yescharacteristic
484D8—416no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—84no—
516C4⋊C4—82no—
517C4⋊C4—81no—
518C4⋊C4C2×#4114yes—
519C22⋊C4—216no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—41no—
541C42—41no—
542C42—42no—
543C42—44no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—216no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—48no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—42no—
558C23—42no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23#52512yes—
570C23C2×AGL(1,ℤ/8)12yes—
571C23#57512yes—
572C23C23×D411yescharacteristic
573Q8—162no—
574D4—216no—
575D4—24no—
576D4—24no—
577D4—24no—
578D4—44no—
579D4—44no—
580D4—88no—
581D4—162no—
582D4—162no—
583D4#53918yescomplemented
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
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—42no—
632C2×C4—42no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—816no—
638C2×C4—816no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—164no—
642C2×C4#52512yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)12yes—
648C2×C4C2×2−1+412yes—
649C8—44no—
650C8—44no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—28no—
658C22—432no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22#311514yes—
673C22C2×#52511yescharacteristic
674C22#348411yescharacteristic
675C22#348611yescharacteristic
676C22C22×AGL(1,ℤ/8)11yescharacteristic
677C22C2×#57511yescharacteristic
678C22#358511yescharacteristic
679C22#358511yescharacteristic
680C4—22no—
681C4—22no—
682C4—22no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—41no—
687C4—41no—
688C4—42no—
689C4—44no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4C2×#53912yes—
696C4#349212yes—
697C2—24no—
698C2—48no—
699C2—162no—
700C2#3498911yescharacteristic
701C2C2×#311511yescharacteristic
702C2#6056411yescharacteristic
703C2#6156911yescharacteristic
704C2#6158111yescharacteristic
705C2#6182511yescharacteristic
706C2#6183111yescharacteristic
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×#2349C2semidirect
3. #33004C2semidirect
4. #32953C2semidirect
6. #29967C2semidirect
7. #29912C2semidirect
8. #29755C2semidirect
9. #27574C2semidirect
10. #27318C2semidirect
11. #26388C2semidirect
12. #26178C2semidirect
13. #22082C2semidirect
14. #21916C2semidirect
15. #21452C2semidirect
32. #2512C22semidirect
33. #2506C22semidirect
34. #2452C22semidirect
35. #2445C22semidirect
50. #2101C22semidirect
51. #2099C22semidirect
52. #2092C22semidirect
53. #2088C22semidirect
54. #2085C22semidirect
55. #2084C22semidirect
56. #2073C22semidirect
57. #2065C22semidirect
58. #2058C22semidirect
59. #2041C22semidirect
60. #1944C22semidirect
61. #1936C22semidirect
62. #1919C22semidirect
63. #1917C22semidirect
64. #1917C22semidirect
75. #1862C22semidirect
583. D4#539semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#23492
#330042
#329532
#329022
#299672
#299122
#297552
#275742
#273182
#263882
#261782
#220822
#219162
#214522