Finite Group Databasebeta

← #6570676

Subgroups — #6570676

Order 512.

1906 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#36259C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29937C211yesmaximal, characteristic, complemented
4#29907C212yesmaximal, complemented
5#27961C211yesmaximal, characteristic, complemented
6#27857C212yesmaximal, complemented
7#27678C211yesmaximal, characteristic, complemented
8#27613C212yesmaximal, complemented
9#26278C212yesmaximal, complemented
10#26202C214yesmaximal, complemented
11#24992C212yesmaximal, complemented
12#24937C214yesmaximal, complemented
13#23755C211yesmaximal, characteristic, complemented
14C2×#2100C212yesmaximal, complemented
15#22703C212yesmaximal, complemented
16#22485C212yesmaximal, complemented
17#22469C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#547C2211yescharacteristic
20C2×#520—21no—
21C2×C4×#41C2211yescharacteristic
22C2×C8×D4C2212yes—
23#2857C2211yescharacteristic
24#2854C2212yes—
25#2546C2211yescharacteristic
26#2538C2211yescharacteristic
27#2486C2212yes—
28#2429—21no—
29#2425—22no—
30#2416C2212yes—
31#2415—21no—
32#2413—22no—
33Q8×C4⋊C4—21no—
34Q8×C4⋊C4C2212yes—
35D4×C4⋊C4—21no—
36D4×C4⋊C4—22no—
37D4×C4⋊C4C2214yes—
38D4×C4⋊C4C2212yes—
39D4×C4⋊C4C2211yescharacteristic
40#2339—21no—
41#2317—22no—
42#2317—21no—
43C4×#121C2211yescharacteristic
44C2×#389C2212yes—
45#2285C2211yescharacteristic
46#2285C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49#2100C2212yes—
50#2100C2212yes—
51#2100C2212yes—
52#2100C2214yes—
53#2100C2214yes—
54#2100C2211yescharacteristic
55#2100C2212yes—
56#2094C2212yes—
57#2094C2211yescharacteristic
58#2084C2214yes—
59#2084C2211yescharacteristic
60#2084C2212yes—
61#2077C2214yes—
62#2077C2212yes—
63#2071C2214yes—
64#2071C2211yescharacteristic
65#2034C2212yes—
66#1990C2212yes—
67#1988C2212yes—
68#1984C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1926C2214yes—
72#1919C2214yes—
73#1919C2214yes—
74#1919C2214yes—
75#1893C2214yes—
76#1874C2212yes—
77#1874C2214yes—
78#1869C2214yes—
79#1853C2212yes—
80#1853C2214yes—
81#1853C2214yes—
82#1834C2212yes—
83#1824C2214yes—
84#1823C2214yes—
85#1817C2212yes—
86#1807C2214yes—
87#1802C2212yes—
88#1797C2211yescharacteristic
89C4×#106C2212yes—
90C4×#102C2212yes—
91C4×C22⋊C8C2211yescharacteristic
92C2×#340C2214yes—
93C2×#340C2214yes—
94C2×#336C2212yes—
95C22×#41—21no—
96C22×#41D411yescharacteristic
97C23×C8C2314yes—
98#547—28no—
99#547—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#123—22no—
103C2×#123—22no—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×#121—22no—
109C2×#121C2312yes—
110C2×#121C2311yescharacteristic
111#520—48no—
112C4×#41—28no—
113C2×C4×Q8—22no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—24no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D412yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C2×#117—21no—
127C2×#117D412yes—
128C2×#117D411yescharacteristic
129C22×C4⋊C4—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2314yes—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C8×D4—28no—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C4×C8C2312yes—
153#398—22no—
154#398—22no—
155#397—22no—
156#396—22no—
157#396—22no—
158#394—22no—
159#391—21no—
160#391—21no—
161#390C2312yes—
162#390C2311yescharacteristic
163#389—22no—
164#389—24no—
165#389—24no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389C2312yes—
172#389C2312yes—
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#388—21no—
179#386—22no—
180#386—21no—
181#386—22no—
182#385—24no—
183#385—24no—
184#385—24no—
185#385—22no—
186#384—21no—
187#384—22no—
188#384—21no—
189#384—22no—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2314yes—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2311yescharacteristic
196#382—21no—
197#382—22no—
198#381—22no—
199#380—21no—
200#380—22no—
201#380—21no—
202#380—21no—
203#380C2312yes—
204#380C2314yes—
205#380C2312yes—
206#380C2312yes—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C2×#95—22no—
215#340C2314yes—
216#340C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#336C2312yes—
229#336C2314yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—44no—
236C2×#41—44no—
237C2×#41—44no—
238C2×#41SD1618yes—
239C22×Q8—21no—
240C22×Q8C2×D411yescharacteristic
241C22×D4—22no—
242C22×D4—21no—
243C22×D4C2×D412yes—
244C22×D4C2×D411yescharacteristic
245C22×D4C2411yescharacteristic
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—24no—
250C23×C4—21no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2412yes—
257C22×C8—28no—
258C22×C8—44no—
259C22×C8—44no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265#123—48no—
266#123—48no—
267#123—48no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271#121—28no—
272#121—22no—
273#121—41no—
274#121—48no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—48no—
278C4×D4—24no—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—416no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—44no—
286C4×D4#4118yes—
287#117—24no—
288#117—28no—
289#117—48no—
290#117—44no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4—84no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—28no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#106—24no—
364#106—44no—
365#106—42no—
366#106—44no—
367C4⋊C8—24no—
368#102—42no—
369#102—42no—
370#102—44no—
371C22⋊C8—28no—
372C22⋊C8—42no—
373C22⋊C8—42no—
374C4×C8—24no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388C24—21no—
389C24C22×D411yescharacteristic
390C24C22×D412yes—
391#41—232no—
392#41—816no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8C2×SD1614yes—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—81no—
411C2×D4#12314yes—
412C2×D4C2×SD1618yes—
413C2×D4C2×SD1614yes—
414C2×D4AGL(1,ℤ/8)14yes—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—82no—
451C22×C4—84no—
452C22×C4—82no—
453C22×C4—84no—
454C22×C4—82no—
455C22×C4C2×SD1618yes—
456C22×C4C2×SD1614yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4118yes—
466C22×C4C2511yescharacteristic
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—88no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—48no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—84no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—84no—
514C4⋊C4#12314yes—
515C4⋊C42−1+412yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—82no—
530C22⋊C4#12318yes—
531C42—24no—
532C42—22no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—24no—
537C42—48no—
538C42—42no—
539C42—44no—
540C42C2×#4114yes—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C2×#12314yes—
559C23C22×SD1612yes—
560C23C22×SD1614yes—
561C23C2×AGL(1,ℤ/8)14yes—
562C23C23×D411yescharacteristic
563Q8—28no—
564Q8—84no—
565D4—216no—
566D4—42no—
567D4—42no—
568D4—84no—
569D4—88no—
570D4#481116yes—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—216no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—88no—
620C2×C4—88no—
621C2×C4—84no—
622C2×C4—168no—
623C2×C4—164no—
624C2×C4C2×#12314yes—
625C2×C4C2×#12312yes—
626C2×C4C2×#12312yes—
627C2×C4C22×SD1612yes—
628C2×C4C22×SD1614yes—
629C2×C4C22×SD1612yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4#57812yes—
632C2×C4C22×#4112yes—
633C2×C4C2×2−1+411yescharacteristic
634C2×C4#58512yes—
635C2×C4#58511yescharacteristic
636C8—42no—
637C8—44no—
638C8—42no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22C2×#48118yes—
660C22C22×#12311yescharacteristic
661C22#348911yescharacteristic
662C22#350011yescharacteristic
663C22C23×SD1611yescharacteristic
664C22C22×AGL(1,ℤ/8)11yescharacteristic
665C22C2×#57811yescharacteristic
666C22#358511yescharacteristic
667C4—22no—
668C4—24no—
669C4—22no—
670C4—22no—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—41no—
675C4—81no—
676C4—81no—
677C4—82no—
678C4—82no—
679C4—84no—
680C4—82no—
681C4—82no—
682C4C2×#48114yes—
683C4#351512yes—
684C4#352112yes—
685C2—28no—
686C2—42no—
687C2—161no—
688C2#3606711yescharacteristic
689C2C22×#48111yescharacteristic
690C2#6052611yescharacteristic
691C2#6209611yescharacteristic
692C2#6213211yescharacteristic
693C2#6252611yescharacteristic
694C2#6257611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36259C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34183C2semidirect
3. #29937C2semidirect
4. #29907C2semidirect
5. #27961C2semidirect
6. #27857C2semidirect
7. #27678C2semidirect
8. #27613C2semidirect
9. #26278C2semidirect
10. #26202C2semidirect
11. #24992C2semidirect
12. #24937C2semidirect
13. #23755C2semidirect
14. C2×#2100C2semidirect
15. #22703C2semidirect
16. #22485C2semidirect
17. #22469C2semidirect

Maximal subgroups

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

subgroupindex
#362592
#341832
#299372
#299072
#279612
#278572
#276782
#276132
#262782
#262022
#249922
#249372
#237552
C2×#21002
#227032
#224852
#224692