Finite Group Databasebeta

← #6615449

Subgroups — #6615449

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#36114C211yesmaximal, characteristic
3#33294C211yesmaximal, characteristic, complemented
4#33217C212yesmaximal, complemented
5#28856C214yesmaximal, complemented
6#28850C214yesmaximal, complemented
7#28693C211yesmaximal, characteristic, complemented
8#28520C211yesmaximal, characteristic, complemented
9#27880C212yesmaximal, complemented
10#25679C212yesmaximal, complemented
11#25341C212yesmaximal, complemented
12#25338C212yesmaximal
13#22970C212yesmaximal, complemented
14#22968C212yesmaximal, complemented
15C4×#489C211yesmaximal, characteristic, complemented
16#21406C212yesmaximal, complemented
17D4×#97C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#529C2212yes—
20C2×#501C2212yes—
21C2×#489C2211yescharacteristic
22C2×#475C2212yes—
23C2×#474C2212yes—
24C2×C8×D4C2212yes—
25C22×#97C2212yes—
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2857C2212yes—
29#2851C2212yes—
30#2568—21no—
31#2547—21no—
32#2547C2212yes—
33#2538C2211yescharacteristic
34#2537C2211yescharacteristic
35#2499—22no—
36#2496—21no—
37#2496—22no—
38#2496C2214yes—
39#2470—22no—
40#2466—22no—
41#2422—21no—
42#2421C2212yes—
43#2420C2212yes—
44#2365—21no—
45#2365C2212yes—
46#2361—21no—
47Q8×C4⋊C4C2211yescharacteristic
48D4×C4⋊C4C2211yescharacteristic
49#2344—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#124C2211yescharacteristic
52#2285C2212yes—
53C42×D4C2211yescharacteristic
54#2095C2212yes—
55#2094C2212yes—
56#2085C2214yes—
57#2077C2214yes—
58#2075C2214yes—
59#2035C2214yes—
60#2033C2214yes—
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#1921C2214yes—
76#1919C2214yes—
77#1893C2214yes—
78#1892C2214yes—
79#1864C2214yes—
80#1863C2214yes—
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—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101#529—28no—
102#529—44no—
103C2×#126—24no—
104C2×#126—22no—
105C2×#125C2312yes—
106C2×#124—22no—
107C2×#124C2311yescharacteristic
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×D4—22no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—22no—
125C22×C4⋊C4—21no—
126C22×C42C2312yes—
127#501—44no—
128#489—24no—
129#475—28no—
130#474—28no—
131C8×D4—28no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2314yes—
144C2×#97—44no—
145C2×#97C2312yes—
146C2×#97C2311yescharacteristic
147C2×#97C2311yescharacteristic
148C2×#97C2312yes—
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#400—24no—
152#400—22no—
153#397—22no—
154#397—24no—
155#397—24no—
156#397—22no—
157#396—22no—
158#394—22no—
159#394—22no—
160#391—22no—
161#391—22no—
162#391—21no—
163#391—22no—
164#391C2311yescharacteristic
165#390—21no—
166#390C2311yescharacteristic
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2314yes—
179#388—22no—
180#388—21no—
181#388—21no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—24no—
186#387—24no—
187#386—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#382—22no—
200#382C2311yescharacteristic
201#382C2312yes—
202#381—22no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380—21no—
207#380C2312yes—
208#380C2314yes—
209#380C2312yes—
210#380C2314yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224C43C2311yescharacteristic
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#336C2312yes—
236#336C2314yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2411yescharacteristic
240C22×D4—22no—
241C22×D4C2×D411yescharacteristic
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—22no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C22×C8—28no—
249C22×C8—28no—
250C22×C8—44no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260#126—48no—
261#126—416no—
262#125—24no—
263#125—24no—
264#124—22no—
265#124—48no—
266#124#4114yes—
267#123—48no—
268C22⋊Q8—24no—
269C22⋊Q8—42no—
270C22⋊Q8—48no—
271#121—24no—
272#121—42no—
273#121—44no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—42no—
277C4×Q8—44no—
278C4×Q8—48no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—48no—
286#117—48no—
287#117—48no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—28no—
352#107—44no—
353#106—28no—
354#106—44no—
355C4⋊C8#4118yes—
356#103—24no—
357#103—28no—
358#103—24no—
359#102—24no—
360#102—28no—
361#102—24no—
362C22⋊C8—28no—
363#97—24no—
364#97—22no—
365#97—22no—
366#97—24no—
367#97—44no—
368C4×C8—24no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—22no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—21no—
389C24C22×D412yes—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8C2×#4112yes—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D4AGL(1,ℤ/8)14yes—
409C2×D4C2×#4112yes—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—82no—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—416no—
452C2×C8C2×#4118yes—
453C2×C82−1+414yes—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—42no—
502C22⋊C4—81no—
503C22⋊C4—81no—
504C22⋊C4—82no—
505C42—24no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—41no—
513C42—41no—
514C42—44no—
515C42—48no—
516C42—42no—
517C42—44no—
518C42C2×#4114yes—
519C42C2×#4112yes—
520C42H(5,𝔽2)12yes—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23C2×AGL(1,ℤ/8)14yes—
535C23C23×D411yescharacteristic
536Q8—24no—
537Q8—42no—
538Q8—84no—
539Q8—88no—
540D4—28no—
541D4—24no—
542D4—42no—
543D4—88no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—48no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—84no—
593C2×C4—88no—
594C2×C4—84no—
595C2×C4#53212yes—
596C2×C4#53212yes—
597C2×C4#54612yes—
598C2×C4#54612yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4#57512yes—
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4112yes—
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×2−1+412yes—
605C8—24no—
606C8—28no—
607C8—42no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22C2×#53211yescharacteristic
623C22C2×#54611yescharacteristic
624C22#349911yescharacteristic
625C22C22×AGL(1,ℤ/8)11yescharacteristic
626C22C2×#57511yescharacteristic
627C22#358611yescharacteristic
628C22#358611yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—81no—
639C4—84no—
640C4—82no—
641C4—81no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4#318014yes—
646C4#348212yes—
647C4#352012yes—
648C2—44no—
649C2—82no—
650C2#3640611yescharacteristic
651C2C2×#318011yescharacteristic
652C2#6072611yescharacteristic
653C2#6116111yescharacteristic
654C2#6116511yescharacteristic
655C2#6250311yescharacteristic
656C2#6250711yescharacteristic
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. #36188C2semidirect
3. #33294C2semidirect
4. #33217C2semidirect
5. #28856C2semidirect
6. #28850C2semidirect
7. #28693C2semidirect
8. #28520C2semidirect
9. #27880C2semidirect
10. #25679C2semidirect
11. #25341C2semidirect
13. #22970C2semidirect
14. #22968C2semidirect
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
#361882
#361142
#332942
#332172
#288562
#288502
#286932
#285202
#278802
#256792
#253412
#253382
#229702
#229682
C4×#4892
#214062
D4×#972