Finite Group Databasebeta

← #6971072

Subgroups — #6971072

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36289C211yesmaximal, characteristic
2#36151C211yesmaximal, characteristic, complemented
3#32772C212yesmaximal, complemented
4#31943C214yesmaximal, complemented
5#30996C214yesmaximal, complemented
6#30213C211yesmaximal, characteristic, complemented
7#29952C212yesmaximal, complemented
8#29668C212yesmaximal, complemented
9#29142C211yesmaximal, characteristic, complemented
10#28374C211yesmaximal, characteristic, complemented
11#24828C212yesmaximal, complemented
12#24747C212yesmaximal, complemented
13#24252C212yesmaximal
14#24134C212yesmaximal, complemented
15#23933C212yesmaximal, complemented
16#23807C211yesmaximal, characteristic, complemented
17#22010C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C2×D42C2211yescharacteristic
20C2×#533C2212yescomplemented
21C2×#475C2212yescomplemented
22C2×#464C2214yescomplemented
23C2×#460C2212yes—
24C2×#450C2214yes—
25C2×#441C2212yescomplemented
26C2×#440C2212yes—
27#2862C2211yescharacteristic, complemented
28#2858C2211yescharacteristic
29#2857C2212yescomplemented
30#2850C2212yes—
31#2563C2211yescharacteristic
32#2548C2211yescharacteristic
33#2536C2212yes—
34#2504C2212yes—
35#2490—22no—
36#2489—22no—
37#2472—22no—
38#2472—22no—
39#2453—22no—
40#2435C2212yes—
41#2421C2214yescomplemented
42#2415—22no—
43#2362—22no—
44#2351C2212yescomplemented
45D4×C22⋊C4—22no—
46C4×C4⋊Q8C2211yescharacteristic, complemented
47C4×#124C2211yescharacteristic
48C2×#390C2212yes—
49C42×D4C2211yescharacteristic
50#2097C2212yescomplemented
51#2095C2211yescharacteristic, complemented
52#2094C2211yescharacteristic
53#2094C2212yescomplemented
54#2085C2214yescomplemented
55#2073C2214yescomplemented
56#2065C2214yes—
57#2065C2214yescomplemented
58#2042C2214yescomplemented
59#2035C2214yescomplemented
60#2033C2214yes—
61#2024C2212yescomplemented
62#2023C2212yes—
63#2023C2212yescomplemented
64#2023C2212yescomplemented
65#2016C2211yescharacteristic, complemented
66#2015C2212yes—
67#2013C2211yescharacteristic
68#2013C2212yescomplemented
69#2001C2214yescomplemented
70#2000C2214yes—
71#1993C2212yescomplemented
72#1968C2214yescomplemented
73#1961C2212yes—
74#1958C2212yes—
75#1955C2212yes—
76#1940C2212yescomplemented
77#1923C2214yescomplemented
78#1880C2214yes—
79#1880C2214yescomplemented
80#1872C2214yescomplemented
81#1853C2214yescomplemented
82#1835C2212yescomplemented
83#1817C2212yescomplemented
84#1805C2212yescomplemented
85#1805C2214yescomplemented
86#1804C2212yes—
87#1797C2212yescomplemented
88#1786C2214yes—
89C4×C4⋊C8C2211yescharacteristic, complemented
90#1745C2211yescharacteristic, complemented
91C23×D4—21no—
92C23×D4C2312yes—
93C22×Q16C2312yes—
94C22×SD16C2312yes—
95C22×SD16C2314yes—
96#554—44no—
97D42—44no—
98#533—48no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#125—21no—
104C2×#125—22no—
105C2×#124—22no—
106C2×#124C2311yescharacteristic
107C2×#124C2312yes—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2312yes—
113C2×#121—21no—
114C2×#121C2312yes—
115C2×#120—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2314yes—
120C2×C4×D4—21no—
121C2×C4×D4D412yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C42C2312yes—
129#475—44no—
130#464—48no—
131#460—44no—
132#450—48no—
133#441—44no—
134#440—44no—
135C2×#107C2312yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97C2312yes—
153C2×C4×C8C2312yes—
154#400—24no—
155#398—22no—
156#397—24no—
157#397—22no—
158#397—22no—
159#394—22no—
160#394—22no—
161#394—22no—
162#394—21no—
163#394—21no—
164#393—21no—
165#393—21no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391C2312yes—
170#391C2311yescharacteristic
171#390—22no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#390C2311yescharacteristic
180#389—24no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#388—24no—
185#388—24no—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—24no—
190#386—22no—
191#386—22no—
192#386—22no—
193#386—22no—
194#384C2311yescharacteristic
195#382—22no—
196#382C2312yes—
197#382C2312yes—
198#382C2312yes—
199#382C2311yescharacteristic
200#380—22no—
201#380—22no—
202#380C2314yes—
203#379—22no—
204#379—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—22no—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4C2312yes—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2311yescharacteristic
219#336C2312yes—
220#336C2311yescharacteristic
221#336C2314yes—
222#336C2312yes—
223C25—22no—
224C22×Q8—22no—
225C22×Q8—21no—
226C22×Q8C2411yescharacteristic
227C22×Q8C2412yes—
228C22×D4—24no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—44no—
234C22×D4—42no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—48no—
238C22×D4C2×D411yescharacteristic
239C22×D4C2412yes—
240C22×D4C2411yescharacteristic
241C23×C4—22no—
242C23×C4—21no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C2×Q16—24no—
247C2×Q16—42no—
248C2×Q16—42no—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—28no—
252C2×SD16—42no—
253C2×SD16—44no—
254C2×SD16—44no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261#127—24no—
262#125—48no—
263#125—44no—
264#124—22no—
265#124—41no—
266#124—44no—
267#124—84no—
268C22⋊Q8—28no—
269C22⋊Q8—48no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—48no—
273C22⋊Q8—48no—
274#121—44no—
275#121—44no—
276#120—48no—
277C4×Q8—28no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—44no—
282C4×Q8—48no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—44no—
286C4×D4—44no—
287#117—48no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—84no—
325C2×C22⋊C4—84no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—24no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—44no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—42no—
342#106—42no—
343#106—44no—
344C4⋊C8—24no—
345C4⋊C8—44no—
346#103—42no—
347#103—42no—
348#103—42no—
349#103—44no—
350#103—44no—
351#102—42no—
352#102—44no—
353#102—44no—
354#102—44no—
355#102—44no—
356C22⋊C8—44no—
357C22⋊C8—44no—
358#97—24no—
359C4×C8—42no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—22no—
370#95C2×D411yescharacteristic
371#95C2×D411yescharacteristic
372#95C2×D412yes—
373#95C2×D412yes—
374C24—21no—
375C24—22no—
376C24—44no—
377C24—44no—
378C24—44no—
379C24—44no—
380C24—82no—
381C24—82no—
382C24—82no—
383C24C22×D412yes—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—82no—
398C2×Q8H(5,𝔽2)14yes—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—88no—
419C2×D4—88no—
420C2×D4—816no—
421C2×D42−1+412yes—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C2511yescharacteristic
452Q16—48no—
453SD16—48no—
454SD16—88no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C4⋊C4—28no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—48no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—48no—
513C42—42no—
514C42—44no—
515C42H(5,𝔽2)14yes—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—88no—
535C23—88no—
536C23—88no—
537C23—88no—
538C23—164no—
539C23—164no—
540C23—164no—
541C23#57514yes—
542C23C23×D411yescharacteristic
543Q8—24no—
544Q8—24no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—88no—
550Q8—162no—
551D4—28no—
552D4—24no—
553D4—44no—
554D4—42no—
555D4—44no—
556D4—84no—
557D4—88no—
558D4—88no—
559D4—162no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—164no—
600C2×C4#53512yes—
601C2×C4#53512yes—
602C2×C4#57714yes—
603C2×C4C2×H(5,𝔽2)12yes—
604C2×C4C2×H(5,𝔽2)12yes—
605C2×C4C2×2−1+411yescharacteristic
606C2×C4#58511yescharacteristic
607C8—42no—
608C8—44no—
609C8—42no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—1616no—
630C22C2×#53511yescharacteristic
631C22#348411yescharacteristic
632C22#348911yescharacteristic
633C22C2×#57512yes—
634C22C2×#57712yes—
635C4—22no—
636C4—22no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—82no—
644C4—81no—
645C4—81no—
646C4—82no—
647C4—84no—
648C4—81no—
649C4#353612yes—
650C4#354212yes—
651C2—44no—
652C2—82no—
653C2—162no—
654C2#4247811yescharacteristic
655C2#6079911yescharacteristic
656C2#6081811yescharacteristic
657C2#6159412yes—
658C2#6212612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
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
2. #36151C2semidirect
3. #32772C2semidirect
4. #31943C2semidirect
5. #30996C2semidirect
6. #30213C2semidirect
7. #29952C2semidirect
8. #29668C2semidirect
9. #29142C2semidirect
10. #28374C2semidirect
11. #24828C2semidirect
12. #24747C2semidirect
14. #24134C2semidirect
15. #23933C2semidirect
16. #23807C2semidirect
17. #22010C2semidirect
20. C2×#533C22semidirect
21. C2×#475C22semidirect
22. C2×#464C22semidirect
25. C2×#441C22semidirect
27. #2862C22semidirect
29. #2857C22semidirect
41. #2421C22semidirect
44. #2351C22semidirect
46. C4×C4⋊Q8C22semidirect
50. #2097C22semidirect
51. #2095C22semidirect
53. #2094C22semidirect
54. #2085C22semidirect
55. #2073C22semidirect
57. #2065C22semidirect
58. #2042C22semidirect
59. #2035C22semidirect
61. #2024C22semidirect
63. #2023C22semidirect
64. #2023C22semidirect
65. #2016C22semidirect
68. #2013C22semidirect
69. #2001C22semidirect
71. #1993C22semidirect
72. #1968C22semidirect
76. #1940C22semidirect
77. #1923C22semidirect
79. #1880C22semidirect
80. #1872C22semidirect
81. #1853C22semidirect
82. #1835C22semidirect
83. #1817C22semidirect
84. #1805C22semidirect
85. #1805C22semidirect
87. #1797C22semidirect
89. C4×C4⋊C8C22semidirect
90. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#362892
#361512
#327722
#319432
#309962
#302132
#299522
#296682
#291422
#283742
#248282
#247472
#242522
#241342
#239332
#238072
#220102