Finite Group Databasebeta

← #6471249

Subgroups — #6471249

Order 512.

2170 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic, complemented
2C2×#2366C211yesmaximal, characteristic
3#33099C212yesmaximal, complemented
4#32953C211yesmaximal, characteristic, complemented
5#32905C211yesmaximal, characteristic, complemented
6#29950C211yesmaximal, characteristic, complemented
7#29714C211yesmaximal, characteristic, complemented
8#29625C212yesmaximal
9#27081C214yesmaximal, complemented
10#26912C214yesmaximal, complemented
11#26771C214yesmaximal, complemented
12#26603C214yesmaximal, complemented
13#22049C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21453C212yesmaximal, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4—21no—
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#474C2212yescomplemented
21C2×#461C2212yescomplemented
22C2×#450C2212yes—
23C2×#450C2212yes—
24C2×#440C2214yescomplemented
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2646—21no—
29#2627—21no—
30#2590—22no—
31#2588—22no—
32#2511C2212yescomplemented
33#2507C2212yescomplemented
34#2451C2212yescomplemented
35#2446C2212yescomplemented
36#2366—22no—
37#2366—22no—
38#2366C2214yes—
39#2366C2214yes—
40#2329—21no—
41#2313—21no—
42C4×C22⋊Q8C2212yes—
43C2×#390—21no—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#386—21no—
47C2×#386—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2093C2212yescomplemented
51#2071C2212yescomplemented
52#2067C2214yescomplemented
53#2055C2214yescomplemented
54#2041C2214yes—
55#2041C2214yes—
56#2023C2212yescomplemented
57#2014C2214yes—
58#2014C2214yes—
59#2013C2212yescomplemented
60#1973C2214yescomplemented
61#1936C2214yescomplemented
62#1928C2214yescomplemented
63#1928C2214yescomplemented
64#1925C2214yescomplemented
65#1921C2214yescomplemented
66#1921C2214yescomplemented
67#1917C2214yescomplemented
68#1917C2214yescomplemented
69#1880C2214yescomplemented
70#1877C2214yescomplemented
71#1877C2214yescomplemented
72#1872C2212yes—
73#1872C2212yes—
74#1862C2214yescomplemented
75#1862C2214yescomplemented
76#1852C2212yes—
77#1852C2212yes—
78#1817C2212yes—
79#1802C2212yescomplemented
80#1802C2212yescomplemented
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1786C2212yes—
84#1786C2212yes—
85#1762C2214yescomplemented
86#1749C2212yes—
87C23×D4—21no—
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C24×C4—21no—
93C22×SD16C2314yes—
94C22×SD16C2314yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127—24no—
98C2×#127—24no—
99C2×#127—22no—
100C2×#127C2314yes—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—44no—
110C2×C4×D4—44no—
111C2×C4×D4—48no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2314yes—
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×C22⋊C4—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#474—44no—
130#461—44no—
131#450—28no—
132#450—28no—
133#440—48no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×#97C2314yes—
149#400—22no—
150#400—22no—
151#400—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#398—22no—
156#395—22no—
157#391—21no—
158#391—22no—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#390—24no—
164#390—24no—
165#390—22no—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#390C2314yes—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#386—24no—
189#386—24no—
190#386—24no—
191#386—24no—
192#385—22no—
193#385—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#381—22no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C25—21no—
224C22×Q8—21no—
225C22×Q8C2412yes—
226C22×D4—28no—
227C22×D4—28no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—42no—
235C22×D4—42no—
236C22×D4—42no—
237C22×D4—42no—
238C22×D4—44no—
239C22×D4—44no—
240C22×D4—82no—
241C22×D4—82no—
242C22×D4#4114yes—
243C22×D4#4114yes—
244C22×D4C2412yes—
245C22×D4C2412yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—24no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4—82no—
256C23×C4—82no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2411yescharacteristic
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—44no—
267C2×SD16—44no—
268C2×SD16—44no—
269C2×SD16—44no—
270C22×C8C2414yes—
271C22×C8C2414yes—
272C4⋊Q8—42no—
273C4⋊Q8—42no—
274#127—216no—
275#127—28no—
276#127—28no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—44no—
280C4×Q8—44no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4—42no—
286C4×D4—42no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—48no—
290C4×D4—416no—
291C4×D4—416no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353#107—28no—
354#107—44no—
355C4⋊C8—42no—
356C4⋊C8—42no—
357#103—28no—
358#103—44no—
359#102—28no—
360#102—28no—
361#102—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367#97—44no—
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#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—24no—
388C24—24no—
389C24—22no—
390C24—22no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24C22×D411yescharacteristic
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—42no—
409C2×Q8—81no—
410C2×Q8—81no—
411C2×D4—216no—
412C2×D4—216no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—48no—
428C2×D4—432no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4—816no—
437C2×D4C2×#4114yes—
438C2×D4C2×#4114yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
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—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—84no—
478C22×C4—816no—
479C22×C4—816no—
480C22×C4AGL(1,ℤ/8)14yes—
481C22×C4#13714yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2511yescharacteristic
494SD16—416no—
495SD16—88no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—81no—
525C4⋊C4—81no—
526C4⋊C4—81no—
527C4⋊C4H(5,𝔽2)14yes—
528C22⋊C4—48no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—84no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—24no—
543C42—22no—
544C42—22no—
545C42—41no—
546C42—41no—
547C42—42no—
548C42—48no—
549C42—82no—
550C42—82no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—24no—
556C23—24no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—48no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—48no—
566C23—48no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23#52512yes—
584C23C2×AGL(1,ℤ/8)12yes—
585C23C2×#13712yes—
586C23C23×D411yescharacteristic
587Q8—44no—
588Q8—44no—
589Q8—162no—
590D4—28no—
591D4—24no—
592D4—24no—
593D4—432no—
594D4—48no—
595D4—48no—
596D4—42no—
597D4—42no—
598D4—44no—
599D4—816no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—48no—
630C2×C4—44no—
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—88no—
649C2×C4—88no—
650C2×C4—82no—
651C2×C4—82no—
652C2×C4—816no—
653C2×C4—84no—
654C2×C4#52512yes—
655C2×C4C2×AGL(1,ℤ/8)12yes—
656C2×C4C2×AGL(1,ℤ/8)12yes—
657C2×C4C2×#13714yes—
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C2×H(5,𝔽2)12yes—
661C2×C4#58512yes—
662C8—44no—
663C8—44no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—816no—
677C22—816no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22#311014yes—
687C22C2×#52511yescharacteristic
688C22#348411yescharacteristic
689C22#348711yescharacteristic
690C22C22×AGL(1,ℤ/8)11yescharacteristic
691C22C22×#13711yescharacteristic
692C22#358511yescharacteristic
693C22#358511yescharacteristic
694C4—24no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—42no—
699C4—44no—
700C4—42no—
701C4—42no—
702C4—41no—
703C4—41no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4#345912yes—
710C4#350512yes—
711C2—24no—
712C2—84no—
713C2—84no—
714C2#3498711yescharacteristic
715C2C2×#311011yescharacteristic
716C2#6056411yescharacteristic
717C2#6157111yescharacteristic
718C2#6158811yescharacteristic
719C2#6189011yescharacteristic
720C2#6189211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. #35015C2semidirect
3. #33099C2semidirect
4. #32953C2semidirect
5. #32905C2semidirect
6. #29950C2semidirect
7. #29714C2semidirect
9. #27081C2semidirect
10. #26912C2semidirect
11. #26771C2semidirect
12. #26603C2semidirect
13. #22049C2semidirect
14. #21916C2semidirect
15. #21453C2semidirect
20. C2×#474C22semidirect
21. C2×#461C22semidirect
24. C2×#440C22semidirect
32. #2511C22semidirect
33. #2507C22semidirect
34. #2451C22semidirect
35. #2446C22semidirect
50. #2093C22semidirect
51. #2071C22semidirect
52. #2067C22semidirect
53. #2055C22semidirect
56. #2023C22semidirect
59. #2013C22semidirect
60. #1973C22semidirect
61. #1936C22semidirect
62. #1928C22semidirect
63. #1928C22semidirect
64. #1925C22semidirect
65. #1921C22semidirect
66. #1921C22semidirect
67. #1917C22semidirect
68. #1917C22semidirect
69. #1880C22semidirect
70. #1877C22semidirect
71. #1877C22semidirect
74. #1862C22semidirect
75. #1862C22semidirect
79. #1802C22semidirect
80. #1802C22semidirect
85. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#23662
#330992
#329532
#329052
#299502
#297142
#296252
#270812
#269122
#267712
#266032
#220492
#219162
#214532