Finite Group Databasebeta

← #6568259

Subgroups — #6568259

Order 512.

1819 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34310C211yesmaximal, characteristic
3#30172C212yesmaximal, complemented
4#29868C211yesmaximal, characteristic, complemented
5#29254C211yesmaximal, characteristic, complemented
6#29109C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic, complemented
8#27582C212yesmaximal, complemented
9#27541C214yesmaximal, complemented
10#27113C214yesmaximal, complemented
11#26302C212yesmaximal, complemented
12#25881C212yesmaximal
13#23774C211yesmaximal, characteristic, complemented
14#23761C212yesmaximal, complemented
15#23223C212yesmaximal, complemented
16#23067C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×D42C2211yescharacteristic
19C2×#518—21no—
20C2×#486C2212yes—
21C2×#476C2212yes—
22#2868C2211yescharacteristic, complemented
23#2857C2212yescomplemented
24#2571—21no—
25#2563C2212yes—
26#2538—21no—
27#2497C2214yescomplemented
28#2492—22no—
29#2439—22no—
30#2369C2212yescomplemented
31#2366C2211yescharacteristic
32#2363—21no—
33#2361C2212yes—
34#2358—22no—
35#2355—21no—
36#2354—21no—
37#2354C2211yescharacteristic
38D4×C4⋊C4C2211yescharacteristic
39D4×C22⋊C4—21no—
40D4×C22⋊C4C2212yes—
41#2324—21no—
42#2321—22no—
43C4×C4⋊Q8C2211yescharacteristic, complemented
44C4×#127C2211yescharacteristic
45C4×#121C2212yes—
46#2293C2211yescharacteristic
47C2×#390C2212yes—
48#2285C2212yescomplemented
49C42×D4C2211yescharacteristic
50#2099C2211yescharacteristic, complemented
51#2093C2211yescharacteristic
52#2093C2212yescomplemented
53#2086C2212yescomplemented
54#2084C2212yescomplemented
55#2084C2212yescomplemented
56#2076C2214yescomplemented
57#2076C2214yescomplemented
58#2071C2212yes—
59#2071C2212yes—
60#2071C2212yescomplemented
61#2071C2212yescomplemented
62#2067C2212yes—
63#2057C2211yescharacteristic, complemented
64#2034C2214yes—
65#2034C2214yes—
66#2023C2211yescharacteristic
67#2023C2212yescomplemented
68#1990C2212yescomplemented
69#1987C2212yescomplemented
70#1982C2212yes—
71#1954C2212yes—
72#1947C2212yescomplemented
73#1936C2214yes—
74#1936C2214yescomplemented
75#1925C2214yes—
76#1925C2214yescomplemented
77#1924C2214yescomplemented
78#1919C2214yescomplemented
79#1862C2214yescomplemented
80#1862C2214yescomplemented
81#1852C2214yescomplemented
82#1831C2211yescharacteristic, complemented
83#1824C2214yescomplemented
84#1817C2212yescomplemented
85#1802C2214yescomplemented
86#1786C2214yes—
87#1786C2214yes—
88#1776C2214yescomplemented
89C4×#107C2212yescomplemented
90#1770C2212yescomplemented
91C4×C4⋊C8C2211yescharacteristic, complemented
92#1762C2214yescomplemented
93C4×#102C2212yes—
94#1749C2212yescomplemented
95C23×D4—21no—
96C23×D4C2312yes—
97D42—24no—
98D42—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104C2×C22⋊Q8—22no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×#120—22no—
108#518—48no—
109C2×C4×Q8—21no—
110C2×C4×D4—21no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×#117—22no—
123C2×#117—21no—
124C2×#117—21no—
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—22no—
129C22×C42C2312yes—
130#486—28no—
131#476—28no—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—24no—
150#396—24no—
151#393—24no—
152#391—21no—
153#391—22no—
154#391—24no—
155#390—22no—
156#390—21no—
157#390—21no—
158#390—21no—
159#390—22no—
160#390—22no—
161#390C2312yes—
162#390C2311yescharacteristic
163#390C2311yescharacteristic
164#390C2312yes—
165#390C2312yes—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389C2312yes—
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#389C2314yes—
174#388—24no—
175#388—21no—
176#388—22no—
177#388—21no—
178#388—22no—
179#388—22no—
180#387—24no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—22no—
195#382—22no—
196#380—22no—
197#380—22no—
198#380—21no—
199#380C2314yes—
200#380C2314yes—
201#380C2312yes—
202#379—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C25—22no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×D4—28no—
237C22×D4—22no—
238C22×D4—24no—
239C22×D4—24no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—44no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4—44no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2412yes—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—42no—
263C4⋊Q8—82no—
264C4⋊Q8—82no—
265#127—22no—
266#127#4114yes—
267C22⋊Q8—84no—
268#121—28no—
269#121—24no—
270#121—24no—
271#120—216no—
272C4×Q8—44no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4—48no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—82no—
286C4×D4#4114yes—
287#117—44no—
288#117—44no—
289#117—84no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—48no—
334C2×C22⋊C4—48no—
335C2×C22⋊C4—84no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—44no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—42no—
355#107—44no—
356#107—42no—
357#106—28no—
358C4⋊C8—24no—
359C4⋊C8—42no—
360#102—28no—
361#102—28no—
362#102—28no—
363#102—44no—
364#102—44no—
365#102—42no—
366#102—42no—
367C22⋊C8—44no—
368C22⋊C8—44no—
369#97—44no—
370C4×C8—24no—
371#95—21no—
372#95—22no—
373#95—21no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—28no—
386C24—21no—
387C24—22no—
388C24—44no—
389C24—44no—
390C24—82no—
391C24—82no—
392C24—82no—
393C24—82no—
394C24C22×D412yes—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—432no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—88no—
423C2×D4—88no—
424C2×D4#13714yes—
425C2×D4C2×#4112yes—
426C2×D4C2×#4114yes—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2511yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—82no—
499C4⋊C4—81no—
500C4⋊C4—84no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4C2×#4112yes—
504C22⋊C4—28no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—81no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—42no—
530C42—42no—
531C42—41no—
532C42—44no—
533C42—42no—
534C42—44no—
535C42—84no—
536C42AGL(1,ℤ/8)14yes—
537C42C2×#4112yes—
538C42C2×#4112yes—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—416no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23—88no—
561C23—88no—
562C23—164no—
563C23—164no—
564C23—164no—
565C23—164no—
566C23C2×#13714yes—
567C23C23×D411yescharacteristic
568Q8—84no—
569Q8—164no—
570D4—28no—
571D4—24no—
572D4—28no—
573D4—416no—
574D4—48no—
575D4—44no—
576D4—42no—
577D4—84no—
578D4—88no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—48no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—84no—
625C2×C4—164no—
626C2×C4—164no—
627C2×C4—164no—
628C2×C4#52512yes—
629C2×C4#52512yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4C2×AGL(1,ℤ/8)12yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×#13712yes—
634C2×C4#57612yes—
635C2×C4C22×#4111yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4#58512yes—
638C2×C4#58512yes—
639C8—44no—
640C8—42no—
641C8—42no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—82no—
656C22—81no—
657C22—82no—
658C22—81no—
659C22—82no—
660C22—81no—
661C22—82no—
662C22—81no—
663C22—1616no—
664C22C2×#52511yescharacteristic
665C22#348511yescharacteristic
666C22#350011yescharacteristic
667C22C22×AGL(1,ℤ/8)11yescharacteristic
668C22C22×#13711yescharacteristic
669C22C2×#57611yescharacteristic
670C22#358511yescharacteristic
671C4—22no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—42no—
676C4—42no—
677C4—41no—
678C4—42no—
679C4—42no—
680C4—81no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—81no—
685C4—84no—
686C4#311014yes—
687C4#351012yes—
688C4#352112yes—
689C2—44no—
690C2—84no—
691C2—82no—
692C2#3603311yescharacteristic
693C2C2×#311011yescharacteristic
694C2#6059011yescharacteristic
695C2#6172011yescharacteristic
696C2#6172611yescharacteristic
697C2#6253511yescharacteristic
698C2#6258211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #30172C2semidirect
4. #29868C2semidirect
5. #29254C2semidirect
6. #29109C2semidirect
7. #28716C2semidirect
8. #27582C2semidirect
9. #27541C2semidirect
10. #27113C2semidirect
11. #26302C2semidirect
13. #23774C2semidirect
14. #23761C2semidirect
15. #23223C2semidirect
16. #23067C2semidirect
17. #22839C2semidirect
22. #2868C22semidirect
23. #2857C22semidirect
27. #2497C22semidirect
30. #2369C22semidirect
43. C4×C4⋊Q8C22semidirect
48. #2285C22semidirect
50. #2099C22semidirect
52. #2093C22semidirect
53. #2086C22semidirect
54. #2084C22semidirect
55. #2084C22semidirect
56. #2076C22semidirect
57. #2076C22semidirect
60. #2071C22semidirect
61. #2071C22semidirect
63. #2057C22semidirect
67. #2023C22semidirect
68. #1990C22semidirect
69. #1987C22semidirect
72. #1947C22semidirect
74. #1936C22semidirect
76. #1925C22semidirect
77. #1924C22semidirect
78. #1919C22semidirect
79. #1862C22semidirect
80. #1862C22semidirect
81. #1852C22semidirect
82. #1831C22semidirect
83. #1824C22semidirect
84. #1817C22semidirect
85. #1802C22semidirect
88. #1776C22semidirect
89. C4×#107C22semidirect
90. #1770C22semidirect
91. C4×C4⋊C8C22semidirect
92. #1762C22semidirect
94. #1749C22semidirect

Maximal subgroups

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

subgroupindex
#361942
#343102
#301722
#298682
#292542
#291092
#287162
#275822
#275412
#271132
#263022
#258812
#237742
#237612
#232232
#230672
#228392