Finite Group Databasebeta

← #6795011

Subgroups — #6795011

Order 512.

1695 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic
3#32953C211yesmaximal, characteristic
4#32435C214yesmaximal, complemented
5#32351C214yesmaximal, complemented
6#29913C212yesmaximal, complemented
7#28529C212yesmaximal, complemented
8#27728C212yesmaximal, complemented
9#27588C212yesmaximal
10#25811C211yesmaximal, characteristic, complemented
11#25452C212yesmaximal, complemented
12#24679C212yesmaximal, complemented
13#24501C212yesmaximal
14#24302C211yesmaximal, characteristic, complemented
15#22599C211yesmaximal, characteristic, complemented
16#21983C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#498C2212yescomplemented
21C2×#477C2212yescomplemented
22C2×#474C2212yes—
23C2×#450C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#431C2212yes—
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2571C2211yescharacteristic, complemented
29#2568C2211yescharacteristic, complemented
30#2563—21no—
31#2563—21no—
32#2563—22no—
33#2563C2212yes—
34#2563C2214yes—
35#2563C2212yes—
36#2533—22no—
37#2514—21no—
38#2514C2212yescomplemented
39#2509C2212yescomplemented
40#2503—21no—
41#2493—22no—
42C2×#393—22no—
43#2365C2211yescharacteristic, complemented
44#2360—21no—
45#2356—21no—
46#2355C2211yescharacteristic, complemented
47C4×C22⋊Q8C2211yescharacteristic
48C2×#390C2212yes—
49#2283C2212yes—
50C2×#382C2212yes—
51C2×#380—21no—
52C2×#380—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yescomplemented
55#2097C2214yescomplemented
56#2092C2214yes—
57#2085C2214yescomplemented
58#2074C2214yescomplemented
59#2073C2214yes—
60#2065C2214yes—
61#2058C2214yescomplemented
62#2057C2212yescomplemented
63#2055C2214yes—
64#2024C2212yescomplemented
65#1990C2212yescomplemented
66#1987C2212yescomplemented
67#1968C2214yes—
68#1949C2214yescomplemented
69#1949C2214yescomplemented
70#1939C2212yescomplemented
71#1934C2214yes—
72#1926C2214yes—
73#1925C2214yes—
74#1916C2214yescomplemented
75#1877C2212yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1864C2212yes—
79#1862C2214yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1852C2212yes—
84#1852C2212yes—
85#1846C2214yescomplemented
86#1835C2214yescomplemented
87#1817C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90#1786C2212yes—
91#1786C2211yescharacteristic
92C8×C4⋊C4C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—44no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—22no—
121C22×C22⋊C4—22no—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#498—44no—
125#477—44no—
126#474—44no—
127#450—24no—
128C4×SD16—44no—
129#431—28no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×C4×C8C2314yes—
148C2×C4×C8C2312yes—
149#401—22no—
150#400—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#400—22no—
155#393—24no—
156#393—24no—
157#393—24no—
158#393—24no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391C2312yes—
164#391C2312yes—
165#391C2311yescharacteristic
166#390—22no—
167#390—22no—
168#390C2314yes—
169#390C2314yes—
170#389—21no—
171#389—22no—
172#389—21no—
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—21no—
181#387—22no—
182#387—22no—
183#387—22no—
184#385—22no—
185#385—21no—
186#384C2311yescharacteristic
187#383—22no—
188#383—22no—
189#382—21no—
190#382—21no—
191#382—22no—
192#382—22no—
193#382C2314yes—
194#382C2314yes—
195#382C2312yes—
196#382C2312yes—
197#381—22no—
198#381—22no—
199#380—21no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—24no—
205#380—24no—
206#380C2312yes—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221#376—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C25—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—82no—
242C22×Q8—82no—
243C22×Q8C2411yescharacteristic
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—42no—
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—42no—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C22⋊Q8—22no—
275C22⋊Q8—22no—
276C22⋊Q8—44no—
277C4×Q8—24no—
278C4×Q8—44no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—44no—
284C4×D4—44no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—84no—
327C2×C22⋊C4—84no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343#107—44no—
344#107—44no—
345#106—42no—
346#106—44no—
347#106—44no—
348C4⋊C8—28no—
349C4⋊C8—24no—
350#103—42no—
351#103—44no—
352#102—28no—
353#102—24no—
354#102—24no—
355#102—42no—
356#102—44no—
357C22⋊C8—24no—
358C22⋊C8—22no—
359C22⋊C8#4114yes—
360C4×C8—28no—
361C4×C8—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—24no—
390C24—22no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24C22×D411yescharacteristic
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—816no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D42−1+414yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—816no—
457C22×C4#13514yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4112yes—
468C22×C42−1+412yes—
469C22×C4C2511yescharacteristic
470SD16—48no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8C2×#4114yes—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—81no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—44no—
528C42—42no—
529C42—42no—
530C42—82no—
531C42H(5,𝔽2)14yes—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—24no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23#54712yes—
551C23C2×#13512yes—
552C23#57712yes—
553C23C23×D411yescharacteristic
554Q8—42no—
555Q8—42no—
556Q8—88no—
557Q8—88no—
558Q8—84no—
559D4—24no—
560D4—24no—
561D4—48no—
562D4—42no—
563D4—42no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—88no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—88no—
618C2×C4—88no—
619C2×C4#53512yes—
620C2×C4#53512yes—
621C2×C4#54712yes—
622C2×C4C2×#13512yes—
623C2×C4C2×#13512yes—
624C2×C4#57512yes—
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4C2×2−1+412yes—
628C2×C4C2×2−1+411yescharacteristic
629C8—24no—
630C8—44no—
631C8—42no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—816no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22#334014yes—
647C22C2×#53511yescharacteristic
648C22C2×#54711yescharacteristic
649C22#348611yescharacteristic
650C22C22×#13511yescharacteristic
651C22C2×#57511yescharacteristic
652C22C2×#57711yescharacteristic
653C22#358511yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—81no—
663C4—81no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—81no—
668C4—84no—
669C4—81no—
670C4#321114yes—
671C4#352012yes—
672C4#353812yes—
673C2—24no—
674C2—84no—
675C2#3891711yescharacteristic
676C2C2×#321111yescharacteristic
677C2#6081211yescharacteristic
678C2C2×#334011yescharacteristic
679C2#6125611yescharacteristic
680C2#6181911yescharacteristic
681C2#6184111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #36177C2semidirect
4. #32435C2semidirect
5. #32351C2semidirect
6. #29913C2semidirect
7. #28529C2semidirect
8. #27728C2semidirect
10. #25811C2semidirect
11. #25452C2semidirect
12. #24679C2semidirect
14. #24302C2semidirect
15. #22599C2semidirect
16. #21983C2semidirect
20. C2×#498C22semidirect
21. C2×#477C22semidirect
28. #2571C22semidirect
29. #2568C22semidirect
38. #2514C22semidirect
39. #2509C22semidirect
43. #2365C22semidirect
46. #2355C22semidirect
54. #2101C22semidirect
55. #2097C22semidirect
57. #2085C22semidirect
58. #2074C22semidirect
61. #2058C22semidirect
62. #2057C22semidirect
64. #2024C22semidirect
65. #1990C22semidirect
66. #1987C22semidirect
68. #1949C22semidirect
69. #1949C22semidirect
70. #1939C22semidirect
74. #1916C22semidirect
85. #1846C22semidirect
86. #1835C22semidirect
92. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#361772
C2×#25632
#329532
#324352
#323512
#299132
#285292
#277282
#275882
#258112
#254522
#246792
#245012
#243022
#225992
#219832
C4×#4502