Finite Group Databasebeta

← #6615448

Subgroups — #6615448

Order 512.

1584 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#36114C211yesmaximal, characteristic
3#33294C211yesmaximal, characteristic, complemented
4#33217C212yesmaximal, complemented
5#28854C214yesmaximal, complemented
6#28852C214yesmaximal, complemented
7#28693C211yesmaximal, characteristic, complemented
8#28519C211yesmaximal, characteristic, complemented
9#27881C212yesmaximal, complemented
10#25678C212yesmaximal, complemented
11#25340C212yesmaximal, complemented
12#25339C212yesmaximal
13#22971C212yesmaximal, complemented
14#22969C212yesmaximal, complemented
15C4×#489C211yesmaximal, characteristic, complemented
16#21406C212yesmaximal, complemented
17D4×#97C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C2×#501C2212yes—
20C2×#489C2211yescharacteristic
21C2×#477C2212yes—
22C2×#476C2212yes—
23C2×C8×D4C2212yes—
24C22×#97C2212yes—
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2212yes—
28#2851C2212yes—
29#2571—21no—
30#2563C2212yes—
31#2538—21no—
32#2538C2211yescharacteristic
33#2537C2211yescharacteristic
34#2497C2214yes—
35#2496—21no—
36#2492—22no—
37#2470—22no—
38#2466—22no—
39#2439—22no—
40#2422—21no—
41#2421C2212yes—
42#2420C2212yes—
43#2369C2212yes—
44#2361—21no—
45#2355—21no—
46#2354—21no—
47Q8×C4⋊C4C2211yescharacteristic
48D4×C4⋊C4C2211yescharacteristic
49#2344—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#124C2211yescharacteristic
52#2285C2212yes—
53C42×D4C2211yescharacteristic
54#2096C2212yes—
55#2093C2212yes—
56#2085C2214yes—
57#2077C2214yes—
58#2076C2214yes—
59#2036C2214yes—
60#2034C2214yes—
61#1993C2211yescharacteristic
62#1992C2212yes—
63#1992C2212yes—
64#1992C2212yes—
65#1990C2212yes—
66#1990C2212yes—
67#1959C2212yes—
68#1959C2211yescharacteristic
69#1948C2212yes—
70#1947C2212yes—
71#1944C2214yes—
72#1943C2214yes—
73#1936C2212yes—
74#1935C2212yes—
75#1920C2214yes—
76#1918C2214yes—
77#1893C2214yes—
78#1892C2214yes—
79#1865C2214yes—
80#1862C2214yes—
81#1847C2212yes—
82#1836C2214yes—
83#1836C2211yescharacteristic
84#1808C2214yes—
85#1807C2214yes—
86#1770C2212yes—
87#1767C2211yescharacteristic
88#1763C2214yes—
89#1762C2214yes—
90C4×#103C2212yes—
91C4×#102C2212yes—
92#1752C2214yes—
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95C23×C8C2314yes—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100#529—28no—
101#529—44no—
102C2×#126—22no—
103C2×#125C2312yes—
104C2×#124C2311yescharacteristic
105C2×#123—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×#117—22no—
120C2×#117—22no—
121C22×C4⋊C4—21no—
122C22×C42C2312yes—
123#501—44no—
124#489—24no—
125#477—28no—
126#476—28no—
127C8×D4—28no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2314yes—
140C2×#97—44no—
141C2×#97C2312yes—
142C2×#97C2311yescharacteristic
143C2×#97C2311yescharacteristic
144C2×#97C2312yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147#400—24no—
148#400—22no—
149#397—22no—
150#397—22no—
151#396—24no—
152#396—22no—
153#394—22no—
154#394—22no—
155#393—24no—
156#391—21no—
157#391—22no—
158#391—24no—
159#391C2311yescharacteristic
160#390—22no—
161#390—21no—
162#390C2311yescharacteristic
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2314yes—
176#388—24no—
177#388—22no—
178#388—21no—
179#388—21no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—24no—
184#386—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#382—22no—
197#382—22no—
198#382C2311yescharacteristic
199#382C2312yes—
200#381—22no—
201#380—22no—
202#380—22no—
203#380—21no—
204#380—21no—
205#380C2314yes—
206#380C2314yes—
207#380C2312yes—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C43C2311yescharacteristic
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2411yescharacteristic
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C22×C8—28no—
248C22×C8—28no—
249C22×C8—44no—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—82no—
258C4⋊Q8—82no—
259#126—48no—
260#125—24no—
261#125—24no—
262#124—22no—
263#124#4114yes—
264#123—48no—
265C22⋊Q8—24no—
266C22⋊Q8—42no—
267C22⋊Q8—84no—
268#121—24no—
269#121—42no—
270#121—44no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—42no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—42no—
280C4×D4—42no—
281C4×D4—48no—
282#117—48no—
283#117—84no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—24no—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—28no—
348#107—44no—
349#106—28no—
350#106—44no—
351C4⋊C8#4118yes—
352#103—24no—
353#103—28no—
354#103—24no—
355#102—24no—
356#102—28no—
357#102—24no—
358C22⋊C8—28no—
359#97—24no—
360#97—22no—
361#97—22no—
362#97—24no—
363#97—44no—
364C4×C8—24no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C24C22×D412yes—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8C2×#4112yes—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—82no—
404C2×D4#13714yes—
405C2×D4C2×#4112yes—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—82no—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2511yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—416no—
448C2×C8C2×#4118yes—
449C2×C82−1+414yes—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—84no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—82no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—41no—
509C42—41no—
510C42—44no—
511C42—42no—
512C42—44no—
513C42—84no—
514C42C2×#4114yes—
515C42C2×#4112yes—
516C42H(5,𝔽2)12yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23C2×#13714yes—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—42no—
534Q8—84no—
535Q8—164no—
536D4—28no—
537D4—24no—
538D4—42no—
539D4—88no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—48no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—88no—
590C2×C4—84no—
591C2×C4#53212yes—
592C2×C4#53212yes—
593C2×C4#54612yes—
594C2×C4#54612yes—
595C2×C4C2×#13712yes—
596C2×C4#57512yes—
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4112yes—
599C2×C4C2×H(5,𝔽2)11yescharacteristic
600C2×C4C2×2−1+412yes—
601C8—24no—
602C8—28no—
603C8—42no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22C2×#53211yescharacteristic
619C22C2×#54611yescharacteristic
620C22#349911yescharacteristic
621C22C22×#13711yescharacteristic
622C22C2×#57511yescharacteristic
623C22#358511yescharacteristic
624C22#358511yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—82no—
636C4—81no—
637C4—84no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4#318114yes—
642C4#348212yes—
643C4#352012yes—
644C2—44no—
645C2—82no—
646C2#3640611yescharacteristic
647C2C2×#318111yescharacteristic
648C2#6072511yescharacteristic
649C2#6116211yescharacteristic
650C2#6116611yescharacteristic
651C2#6250411yescharacteristic
652C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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. #33294C2semidirect
4. #33217C2semidirect
5. #28854C2semidirect
6. #28852C2semidirect
7. #28693C2semidirect
8. #28519C2semidirect
9. #27881C2semidirect
10. #25678C2semidirect
11. #25340C2semidirect
13. #22971C2semidirect
14. #22969C2semidirect
15. C4×#489C2semidirect
16. #21406C2semidirect
17. D4×#97C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361142
#332942
#332172
#288542
#288522
#286932
#285192
#278812
#256782
#253402
#253392
#229712
#229692
C4×#4892
#214062
D4×#972