Finite Group Databasebeta

← #6800683

Subgroups — #6800683

Order 512.

1475 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36560C211yesmaximal, characteristic, complemented
2#34945C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic
4#32434C214yesmaximal, complemented
5#31642C214yesmaximal
6#29988C212yesmaximal, complemented
7#29140C212yesmaximal
8#28717C211yesmaximal, characteristic, complemented
9#28403C212yesmaximal
10#28255C211yesmaximal, characteristic, complemented
11#28159C212yesmaximal
12#27730C212yesmaximal, complemented
13#25931C212yesmaximal, complemented
14#24328C212yesmaximal
15#22821C211yesmaximal, characteristic, complemented
16#22148C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#487C2212yes—
20C2×#474C2212yes—
21C2×#462C2212yes—
22C2×#462C2212yes—
23C2×#461C2211yescharacteristic
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic
26#2859C2212yes—
27#2573—22no—
28#2571C2211yescharacteristic
29#2568C2211yescharacteristic
30#2568C2212yes—
31#2565C2214yes—
32#2547C2211yescharacteristic
33#2546—21no—
34#2546—21no—
35#2538C2211yescharacteristic
36#2514C2212yes—
37#2506—22no—
38#2506—22no—
39#2494—21no—
40#2447C2212yes—
41#2421—21no—
42#2401—22no—
43#2365—21no—
44#2365—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#124C2212yes—
49C4×C22⋊Q8—21no—
50#2295C2212yes—
51#2283—21no—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2101C2214yes—
55#2097C2214yes—
56#2074C2214yes—
57#2072C2214yes—
58#2059C2214yes—
59#2056C2214yes—
60#2026C2214yes—
61#2026C2214yes—
62#2025C2214yes—
63#2015C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1984C2212yes—
67#1981C2211yescharacteristic
68#1969C2214yes—
69#1958C2214yes—
70#1948C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1933C2214yes—
74#1927C2212yes—
75#1916C2214yes—
76#1880C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1877C2212yes—
80#1863C2214yes—
81#1842C2211yescharacteristic
82#1835C2214yes—
83#1831C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C22×Q16C2314yes—
96C22×SD16C2312yes—
97#557—44no—
98#557—44no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125—24no—
106C2×#125C2314yes—
107C2×#124C2312yes—
108C2×C22⋊Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C22×C42—21no—
120#487—44no—
121#474—44no—
122#462—28no—
123#462—44no—
124#461—24no—
125C4×SD16—44no—
126C2×#107C2314yes—
127C2×#106C2312yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2314yes—
144#401—22no—
145#400—22no—
146#400—22no—
147#400—24no—
148#400—22no—
149#398—24no—
150#398—24no—
151#397—22no—
152#397—22no—
153#397—22no—
154#396—22no—
155#391—21no—
156#391—22no—
157#391—22no—
158#391C2312yes—
159#391C2314yes—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#390C2312yes—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—24no—
184#387—24no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#383—22no—
193#382—21no—
194#382—21no—
195#382C2314yes—
196#382C2312yes—
197#381—22no—
198#380—21no—
199#380—21no—
200#380—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222#376—22no—
223C43C2311yescharacteristic
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C2×Q16—28no—
245C2×Q16—44no—
246C2×Q16—44no—
247C2×SD16—24no—
248C2×SD16—24no—
249C2×SD16—42no—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260C4⋊Q8—44no—
261C4⋊Q8#4114yes—
262#125—44no—
263#125—44no—
264#125—416no—
265#124—24no—
266#124—42no—
267C22⋊Q8—44no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—42no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—48no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—41no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—44no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—44no—
345#106—42no—
346C4⋊C8—22no—
347C4⋊C8—24no—
348C4⋊C8—24no—
349C4⋊C8—22no—
350C4⋊C8—42no—
351#103—24no—
352#103—28no—
353#103—42no—
354#103—42no—
355#103—44no—
356#103—44no—
357#103—44no—
358#102—42no—
359#102—44no—
360C4×C8—24no—
361C4×C8—44no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8C2×#4114yes—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D42−1+412yes—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—81no—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C2511yescharacteristic
439Q16—88no—
440SD16—48no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—82no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—21no—
502C42—22no—
503C42—22no—
504C42—21no—
505C42—41no—
506C42—42no—
507C42—42no—
508C42—48no—
509C42—44no—
510C42—44no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42#13514yes—
515C42C2×#4112yes—
516C42H(5,𝔽2)12yes—
517C42H(5,𝔽2)12yes—
518C422−1+412yes—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C23×D411yescharacteristic
527Q8—24no—
528Q8—24no—
529Q8—44no—
530Q8—42no—
531Q8—42no—
532Q8—42no—
533Q8—42no—
534Q8—42no—
535Q8—42no—
536Q8—88no—
537D4—22no—
538D4—22no—
539D4—41no—
540D4—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4#53512yes—
590C2×C4#53512yes—
591C2×C4#54812yes—
592C2×C4#54812yes—
593C2×C4C2×#13512yes—
594C2×C4C2×#13512yes—
595C2×C4C2×#13512yes—
596C2×C4#57512yes—
597C2×C4#57812yes—
598C2×C4C22×#4111yescharacteristic
599C2×C4C2×H(5,𝔽2)11yescharacteristic
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×2−1+411yescharacteristic
602C2×C4C2×2−1+411yescharacteristic
603C2×C4#58511yescharacteristic
604C8—44no—
605C8—42no—
606C8—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C2×#53511yescharacteristic
615C22C2×#54811yescharacteristic
616C22#348911yescharacteristic
617C22C22×#13511yescharacteristic
618C22C2×#57511yescharacteristic
619C22C2×#57811yescharacteristic
620C22#358611yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—41no—
628C4—42no—
629C4—44no—
630C4—81no—
631C4—84no—
632C4—81no—
633C4—82no—
634C4—81no—
635C4—82no—
636C4—82no—
637C4#321214yes—
638C4#336414yes—
639C4#351411yescharacteristic
640C4#352011yescharacteristic
641C4#354711yescharacteristic
642C4#355211yescharacteristic
643C2—82no—
644C2#3912411yescharacteristic
645C2C2×#321211yescharacteristic
646C2#6083711yescharacteristic
647C2C2×#336411yescharacteristic
648C2#6132011yescharacteristic
649C2#6209711yescharacteristic
650C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36560C2semidirect
4. #32434C2semidirect
6. #29988C2semidirect
8. #28717C2semidirect
10. #28255C2semidirect
12. #27730C2semidirect
13. #25931C2semidirect
15. #22821C2semidirect
16. #22148C2semidirect

Maximal subgroups

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

subgroupindex
#365602
#349452
#331952
#324342
#316422
#299882
#291402
#287172
#284032
#282552
#281592
#277302
#259312
#243282
#228212
#221482
C4×#4612