Finite Group Databasebeta

← #6563544

Subgroups — #6563544

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36259C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29936C211yesmaximal, characteristic, complemented
4#29910C212yesmaximal, complemented
5#27948C211yesmaximal, characteristic, complemented
6#27862C212yesmaximal, complemented
7#27645C211yesmaximal, characteristic, complemented
8#27611C212yesmaximal, complemented
9#26283C212yesmaximal, complemented
10#26143C214yesmaximal, complemented
11#25059C212yesmaximal, complemented
12#24950C214yesmaximal, complemented
13#23754C211yesmaximal, characteristic, complemented
14#23657C212yesmaximal, complemented
15#22663C212yesmaximal, complemented
16#22500C212yesmaximal, complemented
17#22473C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#547C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#481C2212yes—
22C2×#436C2212yes—
23C2×#428C2214yes—
24#2857C2211yescharacteristic
25#2854C2212yes—
26#2546C2211yescharacteristic
27#2538C2211yescharacteristic
28#2486C2212yes—
29#2429—21no—
30#2425—22no—
31#2416C2212yes—
32#2415—21no—
33#2413—22no—
34#2355—21no—
35#2355C2212yes—
36#2354—21no—
37#2354C2212yes—
38#2352—22no—
39#2352C2214yes—
40D4×C4⋊C4C2211yescharacteristic
41#2339—21no—
42#2314—22no—
43#2311—21no—
44C4×#121C2211yescharacteristic
45C2×#389C2212yes—
46#2285C2211yescharacteristic
47#2285C2212yes—
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51#2101C2214yes—
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2093C2211yescharacteristic
55#2092C2214yes—
56#2085C2214yes—
57#2085C2214yes—
58#2084C2212yes—
59#2084C2212yes—
60#2084C2212yes—
61#2083C2211yescharacteristic
62#2083C2212yes—
63#2075C2214yes—
64#2075C2212yes—
65#2071C2211yescharacteristic
66#2071C2212yes—
67#2033C2212yes—
68#1990C2212yes—
69#1988C2212yes—
70#1984C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1926C2214yes—
74#1921C2214yes—
75#1920C2214yes—
76#1920C2214yes—
77#1895C2214yes—
78#1875C2214yes—
79#1870C2214yes—
80#1854C2214yes—
81#1853C2214yes—
82#1853C2214yes—
83#1844C2212yes—
84#1824C2212yes—
85#1824C2214yes—
86#1817C2212yes—
87#1809C2214yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91C4×#102C2212yes—
92C4×C22⋊C8C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C22×#41—21no—
96C22×#41D411yescharacteristic
97C22×M16C2314yes—
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
111C4×#41—28no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D412yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×#117—21no—
122C2×#117—22no—
123C2×#117D412yes—
124C2×#117D411yescharacteristic
125C2×#117C2314yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#481—44no—
135#436—28no—
136#428—48no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C4×C8C2312yes—
151#398—22no—
152#398—22no—
153#397—22no—
154#396—22no—
155#396—22no—
156#394—22no—
157#391—22no—
158#391—21no—
159#391—21no—
160#390—22no—
161#390C2311yescharacteristic
162#390C2312yes—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2311yescharacteristic
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#388—24no—
173#388—24no—
174#388—22no—
175#388—22no—
176#388—21no—
177#387—24no—
178#387—24no—
179#386—24no—
180#386—22no—
181#386—21no—
182#386—22no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#382—21no—
192#382—22no—
193#382C2314yes—
194#381—22no—
195#380—22no—
196#380—22no—
197#380—21no—
198#380—21no—
199#380—22no—
200#380—21no—
201#380C2312yes—
202#380C2314yes—
203#380C2312yes—
204#380C2311yescharacteristic
205#380C2312yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—24no—
216C4×C22⋊C4—21no—
217C2×#95—22no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—42no—
237C2×#41—44no—
238C2×#41—44no—
239C2×#41—44no—
240C22×Q8—21no—
241C22×Q8C2×D411yescharacteristic
242C22×D4—22no—
243C22×D4—21no—
244C22×D4C2×D412yes—
245C22×D4C2×D411yescharacteristic
246C22×D4C2411yescharacteristic
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2412yes—
257C2×M16—28no—
258C2×M16—28no—
259C2×M16—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
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×D4—24no—
278C4×D4—28no—
279C4×D4—28no—
280C4×D4—24no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4#4118yes—
284#117—24no—
285#117—28no—
286#117—44no—
287#117—44no—
288#117—44no—
289#117—84no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
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—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—28no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—24no—
359#107—24no—
360#107—44no—
361#106—28no—
362#106—44no—
363#106—44no—
364C4⋊C8—24no—
365#102—24no—
366#102—42no—
367#102—44no—
368C22⋊C8—28no—
369C22⋊C8—24no—
370C22⋊C8—42no—
371C4×C8—24no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—21no—
379#95—24no—
380#95—22no—
381#95—24no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387C24—21no—
388C24C22×D412yes—
389C24C22×D411yescharacteristic
390#41—416no—
391#41—816no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—81no—
413C2×D4#12314yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4AGL(1,ℤ/8)14yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4118yes—
462C22×C4C2511yescharacteristic
463M16—416no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C82−1+414yes—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4#12314yes—
508C4⋊C4H(5,𝔽2)12yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—82no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4#12318yes—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—22no—
530C42—24no—
531C42—44no—
532C42—42no—
533C42—84no—
534C42—84no—
535C42C2×#4114yes—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—21no—
545C23—22no—
546C23—22no—
547C23—21no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×#12314yes—
556C23C2×AGL(1,ℤ/8)12yes—
557C23C23×D411yescharacteristic
558Q8—44no—
559Q8—84no—
560D4—28no—
561D4—48no—
562D4—42no—
563D4—42no—
564D4—84no—
565D4—88no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—88no—
616C2×C4—88no—
617C2×C4—84no—
618C2×C4—164no—
619C2×C4C2×#12314yes—
620C2×C4C2×#12312yes—
621C2×C4C2×#12312yes—
622C2×C4C2×AGL(1,ℤ/8)12yes—
623C2×C4C2×AGL(1,ℤ/8)12yes—
624C2×C4#57512yes—
625C2×C4#57712yes—
626C2×C4C22×#4112yes—
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+412yes—
629C2×C4#58511yescharacteristic
630C8—24no—
631C8—42no—
632C8—44no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—41no—
642C22—42no—
643C22—42no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22C22×#12311yescharacteristic
655C22#348811yescharacteristic
656C22#349911yescharacteristic
657C22C22×AGL(1,ℤ/8)11yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57711yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—24no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—41no—
669C4—82no—
670C4—82no—
671C4—81no—
672C4—82no—
673C4—84no—
674C4—81no—
675C4—82no—
676C4#309414yes—
677C4#351512yes—
678C4#352012yes—
679C2—42no—
680C2—44no—
681C2—161no—
682C2#3599811yescharacteristic
683C2C2×#309411yescharacteristic
684C2#6052211yescharacteristic
685C2#6196211yescharacteristic
686C2#6199711yescharacteristic
687C2#6247711yescharacteristic
688C2#6251111yescharacteristic
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. #34188C2semidirect
3. #29936C2semidirect
4. #29910C2semidirect
5. #27948C2semidirect
6. #27862C2semidirect
7. #27645C2semidirect
8. #27611C2semidirect
9. #26283C2semidirect
10. #26143C2semidirect
11. #25059C2semidirect
12. #24950C2semidirect
13. #23754C2semidirect
14. #23657C2semidirect
15. #22663C2semidirect
16. #22500C2semidirect
17. #22473C2semidirect

Maximal subgroups

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

subgroupindex
#362592
#341882
#299362
#299102
#279482
#278622
#276452
#276112
#262832
#261432
#250592
#249502
#237542
#236572
#226632
#225002
#224732