Finite Group Databasebeta

← #6500746

Subgroups — #6500746

Order 512.

2998 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#35222C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic, complemented
3#27594C212yesmaximal, complemented
4#27591C212yesmaximal, complemented
5#24873C214yesmaximal, complemented
6#24869C214yesmaximal, complemented
7#24865C214yesmaximal, complemented
8#24861C214yesmaximal, complemented
9#24015C212yesmaximal, complemented
10#24013C212yesmaximal, complemented
11#21941C212yesmaximal, complemented
12#21739C211yesmaximal, characteristic, complemented
13C2×#1786C212yesmaximal, complemented
14C24×D4C2211yescharacteristic
15C2×#535C2211yescharacteristic
16C2×#535C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#481C2212yescomplemented
19C2×#482C2214yescomplemented
20C2×#480C2212yescomplemented
21C22×#102C2214yes—
22C22×#102C2214yes—
23C22×C22⋊C8C2212yes—
24#2846—22no—
25#2613—22no—
26#2435—22no—
27#2433—22no—
28#2408C2214yes—
29#2408C2214yes—
30#2376—22no—
31#2358—22no—
32#2352C2214yescomplemented
33D4×C4⋊C4C2214yescomplemented
34#2329—22no—
35#2325—22no—
36#2314C2212yescomplemented
37#2311C2212yescomplemented
38#2309C2211yescharacteristic
39#2309C2211yescharacteristic
40#2280C2211yescharacteristic
41C2×#380C2212yes—
42#2094C2212yescomplemented
43#2093C2212yescomplemented
44#2092C2214yescomplemented
45#2073C2212yescomplemented
46#2073C2212yescomplemented
47#2071C2214yescomplemented
48#2034C2214yescomplemented
49#2034C2214yescomplemented
50#2033C2214yescomplemented
51#2033C2214yescomplemented
52#1926C2214yescomplemented
53#1926C2214yescomplemented
54#1925C2214yescomplemented
55#1925C2214yescomplemented
56#1875C2212yescomplemented
57#1875C2212yescomplemented
58#1874C2214yescomplemented
59#1869C2214yescomplemented
60#1869C2214yescomplemented
61#1854C2214yescomplemented
62#1854C2214yescomplemented
63#1853C2212yescomplemented
64#1853C2212yescomplemented
65#1852C2212yescomplemented
66#1852C2212yescomplemented
67#1818C2212yescomplemented
68#1808C2214yescomplemented
69#1807C2214yescomplemented
70#1786C2214yes—
71#1786C2214yes—
72#1786C2214yes—
73#1786C2214yes—
74#1780C2212yescomplemented
75C26—21no—
76C22×#41C2311yescharacteristic
77C23×D4—21no—
78C23×D4—22no—
79C23×D4—28no—
80C23×D4—44no—
81C23×D4—44no—
82C23×D4C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C23×D4C2314yes—
85C24×C4C2311yescharacteristic
86C23×C8C2314yes—
87#535—44no—
88#535—44no—
89C2×#127—21no—
90C2×#127—21no—
91C2×#124—21no—
92C2×#124—21no—
93#525—28no—
94C2×#123D412yes—
95C2×C22⋊Q8D412yes—
96C2×#121D412yes—
97C2×#121C2312yes—
98C2×#121C2314yes—
99C2×#121C2312yes—
100C2×#120—22no—
101C2×#120—22no—
102C2×#120—24no—
103C2×#120—24no—
104C2×#120D412yes—
105C2×C4×D4—24no—
106C2×C4×D4C2314yes—
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2312yescomplemented
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yescomplemented
114C22×C22⋊C4—22no—
115#482—48no—
116#481—44no—
117#480—44no—
118C2×#107C2314yescomplemented
119C2×#107C2314yescomplemented
120C2×#106C2314yescomplemented
121C2×#106C2314yescomplemented
122C2×#102—48no—
123C2×#102—48no—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×C22⋊C8—28no—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yescomplemented
137C2×C22⋊C8C2314yescomplemented
138C2×C22⋊C8C2312yescomplemented
139#394—24no—
140#394—24no—
141#392—24no—
142#392—24no—
143#391—22no—
144#390—22no—
145#390—22no—
146#390—22no—
147#390C2314yes—
148#390C2312yes—
149#390C2312yes—
150#389—24no—
151#388—24no—
152#388—24no—
153#387—24no—
154#387—24no—
155#386—22no—
156#386—22no—
157#386—24no—
158#385—24no—
159#385—24no—
160#384C2312yescomplemented
161#382C2312yescomplemented
162#381—21no—
163#380—22no—
164#380—22no—
165#380C2314yes—
166#380C2312yes—
167#380C2314yes—
168#380C2312yes—
169#380C2314yescomplemented
170#379—21no—
171#379—22no—
172#379—22no—
173#379—24no—
174#379—24no—
175C4×C4⋊C4C2314yescomplemented
176C4×C22⋊C4—22no—
177C4×C22⋊C4—24no—
178C2×#95—22no—
179#340C2314yescomplemented
180#340C2314yescomplemented
181#340C2314yescomplemented
182#340C2314yescomplemented
183#336C2314yescomplemented
184#336C2312yescomplemented
185#336C2312yescomplemented
186C25—21no—
187C25—21no—
188C25—24no—
189C25—44no—
190C25—44no—
191C25—44no—
192C25—84no—
193C25—84no—
194C25C2×D411yescharacteristic
195C2×#41—24no—
196C2×#41—42no—
197C2×#41—42no—
198C22×Q8C2×D411yescharacteristic
199C22×D4—28no—
200C22×D4—28no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—24no—
209C22×D4—28no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—28no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4—464no—
218C22×D4—48no—
219C22×D4—48no—
220C22×D4—44no—
221C22×D4—44no—
222C22×D4—48no—
223C22×D4—44no—
224C22×D4C2×D414yes—
225C22×D4C2×D412yes—
226C22×D4C2×D411yescharacteristic
227C22×D4C2412yes—
228C22×D4C2412yes—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4#4118yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C22×C8—28no—
241C22×C8—44no—
242C22×C8—44no—
243C22×C8C2414yes—
244C22×C8C2414yes—
245#127—44no—
246#127—44no—
247#124—44no—
248#124—44no—
249#123—28no—
250C22⋊Q8—28no—
251#121—28no—
252#121—24no—
253#121—24no—
254#121—28no—
255#121—42no—
256#121—44no—
257#121—42no—
258#120—28no—
259#120—48no—
260#120—48no—
261#120—88no—
262#120—88no—
263C4×D4—28no—
264C4×D4—28no—
265C4×D4—88no—
266#117—24no—
267#117—42no—
268#117—42no—
269#117—44no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2411yescharacteristic
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2411yescharacteristic
285C2×C22⋊C4—24no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—84no—
301C2×C22⋊C4C2×D412yes—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4C2×D414yes—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42C2412yes—
309#107—44no—
310#107—44no—
311#106—44no—
312#106—44no—
313#102—28no—
314#102—28no—
315#102—28no—
316#102—28no—
317#102—44no—
318#102—44no—
319#102—44no—
320#102—44no—
321C22⋊C8—28no—
322C22⋊C8—24no—
323C22⋊C8—24no—
324C22⋊C8—42no—
325C22⋊C8—42no—
326C22⋊C8—44no—
327#95—24no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—41no—
341C24—41no—
342C24—48no—
343C24—48no—
344C24—42no—
345C24—42no—
346C24—432no—
347C24—48no—
348C24—42no—
349C24—42no—
350C24—48no—
351C24—48no—
352C24—84no—
353C24—84no—
354C24—82no—
355C24—82no—
356C24—82no—
357C24—82no—
358C24—832no—
359C24—832no—
360C24—816no—
361C24—816no—
362C24#12314yes—
363C24C22×D412yes—
364C24C22×D411yescharacteristic
365C24C22×D412yes—
366#41—88no—
367C2×Q8—22no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×D4—232no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—416no—
385C2×D4—416no—
386C2×D4—416no—
387C2×D4—416no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—416no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—464no—
398C2×D4—432no—
399C2×D4—48no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—84no—
406C2×D4—84no—
407C2×D4—84no—
408C2×D4—84no—
409C2×D4H(5,𝔽2)14yes—
410C2×D4H(5,𝔽2)14yes—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—416no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—84no—
437C22×C4#12318yes—
438C22×C4#12314yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C22×C4C2511yescharacteristic
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—88no—
454C4⋊C4—24no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—84no—
467C22⋊C4—24no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—48no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—82no—
484C22⋊C4—84no—
485C42—24no—
486C42—42no—
487C42—42no—
488C42—82no—
489C42—84no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—216no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—44no—
507C23—48no—
508C23—44no—
509C23—44no—
510C23—432no—
511C23—48no—
512C23—48no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—816no—
522C23—816no—
523C23—816no—
524C23—816no—
525C23—816no—
526C23—88no—
527C23—88no—
528C23—88no—
529C23—84no—
530C23—84no—
531C23—8128no—
532C23—88no—
533C23—84no—
534C23—84no—
535C23—816no—
536C23C2×#12312yes—
537C23C2×#12314yes—
538C23#57614yes—
539C23#57714yes—
540C23C23×D411yescharacteristic
541Q8—82no—
542D4—28no—
543D4—28no—
544D4—432no—
545D4—48no—
546D4—48no—
547D4—416no—
548D4—44no—
549D4—44no—
550D4—82no—
551D4—84no—
552D4—168no—
553D4—164no—
554D4—164no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—164no—
586C2×C4—164no—
587C2×C4—164no—
588C2×C4C2×#12314yes—
589C2×C4C2×#12312yes—
590C2×C4C22×#4112yes—
591C2×C4C2×H(5,𝔽2)12yes—
592C2×C4C2×H(5,𝔽2)12yes—
593C8—44no—
594C8—44no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—22no—
602C22—24no—
603C22—416no—
604C22—42no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—48no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—44no—
617C22—864no—
618C22—816no—
619C22—816no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—82no—
624C22—82no—
625C22#309618yes—
626C22C22×#12311yescharacteristic
627C22#348411yescharacteristic
628C22#348411yescharacteristic
629C22C2×#57612yes—
630C22C2×#57712yes—
631C4—24no—
632C4—22no—
633C4—42no—
634C4—44no—
635C4—84no—
636C4—82no—
637C4—84no—
638C4—81no—
639C4—81no—
640C4#349014yes—
641C2—28no—
642C2—42no—
643C2—82no—
644C2—84no—
645C2—84no—
646C2#3535011yescharacteristic
647C2C2×#309612yes—
648C2#6162912yes—
649C2#6163011yescharacteristic
650C2#6163111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35222C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34360C2semidirect
3. #27594C2semidirect
4. #27591C2semidirect
5. #24873C2semidirect
6. #24869C2semidirect
7. #24865C2semidirect
8. #24861C2semidirect
9. #24015C2semidirect
10. #24013C2semidirect
11. #21941C2semidirect
12. #21739C2semidirect
13. C2×#1786C2semidirect
18. C2×#481C22semidirect
19. C2×#482C22semidirect
20. C2×#480C22semidirect
32. #2352C22semidirect
33. D4×C4⋊C4C22semidirect
36. #2314C22semidirect
37. #2311C22semidirect
42. #2094C22semidirect
43. #2093C22semidirect
44. #2092C22semidirect
45. #2073C22semidirect
46. #2073C22semidirect
47. #2071C22semidirect
48. #2034C22semidirect
49. #2034C22semidirect
50. #2033C22semidirect
51. #2033C22semidirect
52. #1926C22semidirect
53. #1926C22semidirect
54. #1925C22semidirect
55. #1925C22semidirect
56. #1875C22semidirect
57. #1875C22semidirect
58. #1874C22semidirect
59. #1869C22semidirect
60. #1869C22semidirect
61. #1854C22semidirect
62. #1854C22semidirect
63. #1853C22semidirect
64. #1853C22semidirect
65. #1852C22semidirect
66. #1852C22semidirect
67. #1818C22semidirect
68. #1808C22semidirect
69. #1807C22semidirect
74. #1780C22semidirect
110. C2×#117C23semidirect
113. C22×C4⋊C4C23semidirect
118. C2×#107C23semidirect
119. C2×#107C23semidirect
120. C2×#106C23semidirect
121. C2×#106C23semidirect
136. C2×C22⋊C8C23semidirect
137. C2×C22⋊C8C23semidirect
138. C2×C22⋊C8C23semidirect
160. #384C23semidirect
161. #382C23semidirect
169. #380C23semidirect
175. C4×C4⋊C4C23semidirect
179. #340C23semidirect
180. #340C23semidirect
181. #340C23semidirect
182. #340C23semidirect
183. #336C23semidirect
184. #336C23semidirect
185. #336C23semidirect

Maximal subgroups

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

subgroupindex
#352222
#343602
#275942
#275912
#248732
#248692
#248652
#248612
#240152
#240132
#219412
#217392
C2×#17862