Finite Group Databasebeta

← #6471068

Subgroups — #6471068

Order 512.

1932 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic, complemented
2C2×#2363C211yesmaximal, characteristic
3#33147C211yesmaximal, characteristic, complemented
4#33109C211yesmaximal, characteristic, complemented
5#32985C212yesmaximal, complemented
6#29950C211yesmaximal, characteristic, complemented
7#29714C211yesmaximal, characteristic, complemented
8#29598C212yesmaximal
9#27041C214yesmaximal, complemented
10#26731C214yesmaximal, complemented
11#26627C214yesmaximal, complemented
12#26557C214yesmaximal, complemented
13#22079C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic, complemented
15C4×#450C212yesmaximal, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#484C2212yes—
20C2×#485C2214yescomplemented
21C2×#450C2212yes—
22C2×C4×SD16C2214yescomplemented
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2848C2212yes—
26#2646—21no—
27#2626—21no—
28#2589—22no—
29#2587—22no—
30#2497C2212yescomplemented
31#2492C2212yescomplemented
32#2429C2212yescomplemented
33#2423C2212yescomplemented
34#2363—22no—
35#2363—22no—
36#2363C2214yes—
37#2363C2214yes—
38#2329—21no—
39#2312—21no—
40C4×C22⋊Q8C2212yes—
41C2×#390—21no—
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#386—21no—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4—21no—
48C2×C4×C22⋊C4C2211yescharacteristic
49#2093C2212yescomplemented
50#2071C2212yescomplemented
51#2067C2214yescomplemented
52#2055C2214yescomplemented
53#2033C2214yes—
54#2033C2214yes—
55#2023C2212yescomplemented
56#2013C2212yescomplemented
57#2000C2214yes—
58#2000C2214yes—
59#1988C2212yescomplemented
60#1981C2212yescomplemented
61#1948C2212yescomplemented
62#1947C2212yescomplemented
63#1942C2214yescomplemented
64#1933C2214yescomplemented
65#1928C2214yescomplemented
66#1923C2214yescomplemented
67#1923C2214yescomplemented
68#1921C2214yescomplemented
69#1919C2214yescomplemented
70#1877C2214yescomplemented
71#1873C2214yescomplemented
72#1872C2212yes—
73#1872C2212yes—
74#1864C2214yescomplemented
75#1863C2214yescomplemented
76#1854C2214yescomplemented
77#1853C2212yes—
78#1852C2212yes—
79#1825C2214yescomplemented
80#1824C2212yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1786C2212yes—
84#1779C2212yes—
85C4×#102C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×SD16C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×#121—24no—
99C2×#121—22no—
100C2×#121—24no—
101C2×#121C2314yes—
102C2×C4×Q8C2312yes—
103C2×C4×D4—22no—
104C2×C4×D4—44no—
105C2×C4×D4C2314yes—
106C2×#117—22no—
107C2×#117—44no—
108C2×#117C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4D411yescharacteristic
119C22×C22⋊C4D411yescharacteristic
120C22×C42—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#485—48no—
125#484—28no—
126#450—28no—
127C4×SD16—48no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C4×C8C2314yes—
144#400—22no—
145#400—22no—
146#399—22no—
147#398—22no—
148#398—22no—
149#396—22no—
150#396—22no—
151#393—22no—
152#391—21no—
153#391—22no—
154#391C2312yes—
155#391C2312yes—
156#390—24no—
157#390—24no—
158#390—21no—
159#390—21no—
160#390C2312yes—
161#390C2312yes—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2314yes—
169#389C2312yes—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—24no—
175#388—24no—
176#387—22no—
177#387—22no—
178#387—22no—
179#386—24no—
180#386—24no—
181#385—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#380—21no—
188#380—21no—
189#380—21no—
190#380C2312yes—
191#380C2312yes—
192#380C2312yes—
193#379—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—24no—
202C4×C22⋊C4—24no—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4D412yes—
205C4×C22⋊C4C2312yes—
206C4×C22⋊C4C2312yes—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25—21no—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8C2412yes—
224C22×D4—28no—
225C22×D4—24no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—42no—
229C22×D4—42no—
230C22×D4—82no—
231C22×D4—82no—
232C22×D4#4114yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—82no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C2×SD16—28no—
254C2×SD16—44no—
255C2×SD16—44no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265#121—216no—
266#121—28no—
267#121—28no—
268C4×Q8—44no—
269C4×D4—28no—
270C4×D4—44no—
271C4×D4—48no—
272#117—28no—
273#117—44no—
274#117—48no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4#4114yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—44no—
335C2×C42—82no—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—28no—
343#107—44no—
344#106—28no—
345C4⋊C8—42no—
346C4⋊C8—42no—
347#103—28no—
348#103—28no—
349#103—44no—
350#102—28no—
351#102—44no—
352#102—44no—
353C22⋊C8—24no—
354C22⋊C8—24no—
355C22⋊C8—44no—
356C22⋊C8—44no—
357C4×C8—44no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—41no—
381C24—41no—
382C24—41no—
383C24—41no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24C22×D412yes—
391C24C22×D411yescharacteristic
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—42no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×D4—216no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—48no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—48no—
408C2×D4—84no—
409C2×D4—84no—
410C2×D4—816no—
411C2×D4C2×#4114yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—48no—
454C22×C4—84no—
455C22×C4—816no—
456C22×C4#13514yes—
457C22×C4#13714yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2511yescharacteristic
469SD16—416no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4C2×#4114yes—
501C4⋊C4H(5,𝔽2)14yes—
502C22⋊C4—216no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—41no—
523C42—41no—
524C42—42no—
525C42—48no—
526C42—82no—
527C42—82no—
528C42—82no—
529C42—82no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—44no—
543C23—48no—
544C23—44no—
545C23—48no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—48no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—82no—
555C23—82no—
556C23—82no—
557C23—82no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23#52512yes—
563C23C2×#13512yes—
564C23C2×#13712yes—
565C23C23×D411yescharacteristic
566Q8—44no—
567Q8—44no—
568Q8—162no—
569D4—28no—
570D4—48no—
571D4—44no—
572D4—816no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
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—816no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—84no—
628C2×C4#52512yes—
629C2×C4C2×#13512yes—
630C2×C4C2×#13512yes—
631C2×C4C2×#13714yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)12yes—
635C2×C4#58512yes—
636C8—44no—
637C8—44no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—44no—
654C22—88no—
655C22—816no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22#310714yes—
661C22C2×#52511yescharacteristic
662C22#348411yescharacteristic
663C22#348511yescharacteristic
664C22C22×#13511yescharacteristic
665C22C22×#13711yescharacteristic
666C22#358511yescharacteristic
667C22#358511yescharacteristic
668C4—22no—
669C4—22no—
670C4—24no—
671C4—22no—
672C4—41no—
673C4—41no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#345912yes—
684C4#350612yes—
685C2—24no—
686C2—82no—
687C2—84no—
688C2#3498611yescharacteristic
689C2C2×#310711yescharacteristic
690C2#6056411yescharacteristic
691C2#6156611yescharacteristic
692C2#6157011yescharacteristic
693C2#6169911yescharacteristic
694C2#6172111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35014C2semidirect
3. #33147C2semidirect
4. #33109C2semidirect
5. #32985C2semidirect
6. #29950C2semidirect
7. #29714C2semidirect
9. #27041C2semidirect
10. #26731C2semidirect
11. #26627C2semidirect
12. #26557C2semidirect
13. #22079C2semidirect
14. #21902C2semidirect
15. C4×#450C2semidirect
20. C2×#485C22semidirect
22. C2×C4×SD16C22semidirect
30. #2497C22semidirect
31. #2492C22semidirect
32. #2429C22semidirect
33. #2423C22semidirect
49. #2093C22semidirect
50. #2071C22semidirect
51. #2067C22semidirect
52. #2055C22semidirect
55. #2023C22semidirect
56. #2013C22semidirect
59. #1988C22semidirect
60. #1981C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1942C22semidirect
64. #1933C22semidirect
65. #1928C22semidirect
66. #1923C22semidirect
67. #1923C22semidirect
68. #1921C22semidirect
69. #1919C22semidirect
70. #1877C22semidirect
71. #1873C22semidirect
74. #1864C22semidirect
75. #1863C22semidirect
76. #1854C22semidirect
79. #1825C22semidirect
85. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#23632
#331472
#331092
#329852
#299502
#297142
#295982
#270412
#267312
#266272
#265572
#220792
#219022
C4×#4502