Finite Group Databasebeta

← #6470760

Subgroups — #6470760

Order 512.

2380 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3199C212yesmaximal, complemented
2#37297C211yesmaximal, characteristic, complemented
3#34265C211yesmaximal, characteristic, complemented
4#33116C212yesmaximal, complemented
5#32902C212yesmaximal, complemented
6#29758C212yesmaximal, complemented
7#29721C212yesmaximal, complemented
8#25218C214yesmaximal, complemented
9#25207C214yesmaximal, complemented
10#25097C214yesmaximal, complemented
11#25076C214yesmaximal, complemented
12#21937C211yesmaximal, characteristic, complemented
13#20914C212yesmaximal, complemented
14C2×#535C2212yes—
15C2×#535C2211yescharacteristic
16C2×#520C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic, complemented
18#3199—216no—
19C2×#482C2214yescomplemented
20C2×#482C2214yescomplemented
21C2×#469C2214yes—
22C2×#466C2214yes—
23C2×#459C2214yes—
24C2×#447C2214yescomplemented
25C2×#447C2214yescomplemented
26C2×#418C2212yescomplemented
27C22×#102C2212yescomplemented
28C2×#408C2212yes—
29#2856C2212yes—
30#2733—22no—
31#2679—22no—
32#2669—22no—
33#2669—22no—
34#2619C2212yes—
35#2500C2214yescomplemented
36#2433C2214yescomplemented
37#2367—22no—
38#2366—22no—
39#2363C2214yescomplemented
40#2361C2214yescomplemented
41#2321—22no—
42#2309C2211yescharacteristic, complemented
43C4×#120—22no—
44#2297C2211yescharacteristic
45#2296C2211yescharacteristic, complemented
46#2274C2211yescharacteristic, complemented
47#2084C2212yescomplemented
48#2083C2212yescomplemented
49#2073C2212yescomplemented
50#2073C2212yescomplemented
51#2067C2214yescomplemented
52#2065C2214yescomplemented
53#2054C2214yescomplemented
54#2054C2214yescomplemented
55#2041C2212yescomplemented
56#2041C2212yescomplemented
57#2011C2212yes—
58#2011C2212yes—
59#1982C2214yescomplemented
60#1949C2214yescomplemented
61#1926C2214yescomplemented
62#1925C2214yescomplemented
63#1879C2214yescomplemented
64#1879C2214yescomplemented
65#1875C2214yescomplemented
66#1875C2214yescomplemented
67#1874C2214yescomplemented
68#1874C2214yescomplemented
69#1851C2214yescomplemented
70#1850C2214yescomplemented
71#1802C2214yescomplemented
72#1798C2212yescomplemented
73#1797C2212yescomplemented
74#1736C2212yescomplemented
75#1735C2212yescomplemented
76C22×#41C2311yescharacteristic
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C22×D8C2314yes—
81C22×D8C2314yes—
82C22×M16C2312yes—
83C23×C8C2312yes—
84#535—24no—
85#535—28no—
86#535—28no—
87#535D418yescomplemented
88C2×#127—21no—
89C2×#127—22no—
90C2×#127C2312yes—
91C2×#127C2312yes—
92C2×#124—21no—
93C2×#124—22no—
94C2×#121—22no—
95C2×#121—22no—
96C2×#121C2314yes—
97C2×#121C2314yes—
98C2×#121C2314yes—
99C2×#120—22no—
100C2×#120—22no—
101C2×#120—22no—
102C2×#120—24no—
103C2×#120—24no—
104#520—44no—
105C2×C4×Q8—21no—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4—24no—
109C2×C4×D4—22no—
110C2×C4×D4—44no—
111C2×C4×D4C2314yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—21no—
120C22×C42C2311yescharacteristic
121#482—216no—
122#482—216no—
123#469—216no—
124#466—216no—
125#459—48no—
126#447—216no—
127#447—216no—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132#418—44no—
133C2×#102—44no—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142#408—28no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145#394—24no—
146#394—24no—
147#394—24no—
148#392—24no—
149#392—24no—
150#392—24no—
151#391D412yes—
152#390—22no—
153#390—24no—
154#390—22no—
155#390—22no—
156#390D412yes—
157#390C2312yes—
158#390C2312yes—
159#390C2314yes—
160#390C2314yes—
161#389C2312yes—
162#389C2312yes—
163#388—24no—
164#388—24no—
165#388—24no—
166#388—24no—
167#386—24no—
168#386—24no—
169#386D412yes—
170#386D412yes—
171#385—21no—
172#385—21no—
173#385—24no—
174#384C2312yes—
175#382C2312yes—
176#379—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—24no—
182C4×C22⋊C4C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2312yes—
188#336C2312yes—
189#336C2312yes—
190#336C2312yes—
191C25—21no—
192C25—22no—
193C2×#41—42no—
194C2×#41—42no—
195C2×#41C2×D414yes—
196C2×#41#4114yes—
197C22×Q8C2×D411yescharacteristic
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—21no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—42no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4C2×D414yes—
216C22×D4C2×D412yes—
217C22×D4C2×D411yescharacteristic
218C22×D4C2414yes—
219C22×D4C2412yes—
220C23×C4—21no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—84no—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C2×D8—28no—
230C2×D8—28no—
231C2×D8—44no—
232C2×D8—44no—
233C2×D8—44no—
234C2×D8—44no—
235C2×M16—24no—
236C2×M16—42no—
237C2×M16—42no—
238C22×C8—24no—
239C22×C8—42no—
240C22×C8—42no—
241C22×C8C2414yes—
242C22×C8C2414yes—
243#127—28no—
244#127—44no—
245#127—44no—
246#127—48no—
247#124—28no—
248#124—48no—
249#121—28no—
250#121—28no—
251#121—28no—
252#121—28no—
253#121—28no—
254#121—84no—
255#121—84no—
256#121C2×D4116yes—
257#120—216no—
258#120—48no—
259#120—48no—
260#120—88no—
261#120—88no—
262C4×Q8—44no—
263C4×D4—28no—
264C4×D4—44no—
265C4×D4—48no—
266C4×D4—44no—
267C4×D4—48no—
268C4×D4—88no—
269#117—22no—
270#117—24no—
271#117—24no—
272#117—24no—
273#117—42no—
274#117—42no—
275#117—42no—
276#117C2×D414yes—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2411yescharacteristic
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4C2×D414yes—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D414yes—
310C2×C42—21no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—42no—
316C2×C42C2×D412yes—
317C2×C42C2412yes—
318C2×C42C2412yes—
319#107—28no—
320#106—28no—
321C4⋊C8—44no—
322C4⋊C8—44no—
323#102—28no—
324#102—28no—
325#102—28no—
326#102—28no—
327#102—28no—
328#102—28no—
329#102—44no—
330#102—44no—
331C22⋊C8—28no—
332C22⋊C8—28no—
333#95—22no—
334#95—24no—
335#95—22no—
336#95—24no—
337#95C2×D414yes—
338#95C2×D414yes—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—42no—
343C24—42no—
344C24—44no—
345C24—82no—
346C24—82no—
347C24—82no—
348C24—82no—
349C24—84no—
350C24—84no—
351C24C22×D412yes—
352C24C22×D411yescharacteristic
353#41—48no—
354#41—48no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8#12114yes—
358C2×D4—216no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—48no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—44no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—82no—
387C2×D4—82no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—84no—
401C2×D4#12118yes—
402C2×D4#12114yes—
403C2×D4C22×D414yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—84no—
424C22×C4—82no—
425C22×C4—84no—
426C22×C4—816no—
427C22×C4C22×D414yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4114yes—
436C22×C4C2511yescharacteristic
437D8—88no—
438D8—88no—
439M16—84no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—84no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4C22×D414yes—
461C4⋊C4H(5,𝔽2)14yes—
462C22⋊C4—24no—
463C22⋊C4—216no—
464C22⋊C4—28no—
465C22⋊C4—42no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C42—24no—
480C42—24no—
481C42—24no—
482C42—42no—
483C42—44no—
484C42—42no—
485C42—44no—
486C42—84no—
487C23—24no—
488C23—22no—
489C23—21no—
490C23—22no—
491C23—21no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—24no—
496C23—48no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—88no—
509C23—88no—
510C23—84no—
511C23—88no—
512C23—84no—
513C23—168no—
514C23—168no—
515C23—164no—
516C23—164no—
517C23C2×#12112yes—
518C23C2×#12114yes—
519C23C23×D411yescharacteristic
520Q8—44no—
521D4—28no—
522D4—44no—
523D4—48no—
524D4—48no—
525D4—44no—
526D4—44no—
527D4—44no—
528D4—44no—
529D4—44no—
530D4—88no—
531D4—816no—
532D4—164no—
533D4—164no—
534D4—168no—
535D4—168no—
536D4—168no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—816no—
569C2×C4—84no—
570C2×C4—82no—
571C2×C4—164no—
572C2×C4C2×#12112yes—
573C2×C4C2×#12114yes—
574C2×C4#57614yes—
575C2×C4C23×D411yescharacteristic
576C2×C4C22×#4111yescharacteristic
577C2×C4C2×H(5,𝔽2)12yes—
578C2×C4#58512yes—
579C8—44no—
580C8—44no—
581C22—28no—
582C22—24no—
583C22—22no—
584C22—24no—
585C22—42no—
586C22—44no—
587C22—41no—
588C22—42no—
589C22—41no—
590C22—42no—
591C22—48no—
592C22—44no—
593C22—816no—
594C22—84no—
595C22—82no—
596C22—82no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—1616no—
605C22C22×#12111yescharacteristic
606C22#348411yescharacteristic
607C22#348511yescharacteristic
608C22C2×#57612yes—
609C22#358512yes—
610C4—22no—
611C4—24no—
612C4—24no—
613C4—44no—
614C4—44no—
615C4—44no—
616C4—42no—
617C4—84no—
618C4—81no—
619C4—81no—
620C4#345912yes—
621C4#349612yes—
622C2—42no—
623C2—44no—
624C2—84no—
625C2—164no—
626C2#3498411yescharacteristic
627C2#6042812yes—
628C2#6163011yescharacteristic
629C2#6165111yescharacteristic
630C2#6180212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3199C2semidirect
2. #37297C2semidirect
3. #34265C2semidirect
4. #33116C2semidirect
5. #32902C2semidirect
6. #29758C2semidirect
7. #29721C2semidirect
8. #25218C2semidirect
9. #25207C2semidirect
10. #25097C2semidirect
11. #25076C2semidirect
12. #21937C2semidirect
13. #20914C2semidirect
17. C22×C4×D4C22semidirect
19. C2×#482C22semidirect
20. C2×#482C22semidirect
24. C2×#447C22semidirect
25. C2×#447C22semidirect
26. C2×#418C22semidirect
27. C22×#102C22semidirect
35. #2500C22semidirect
36. #2433C22semidirect
39. #2363C22semidirect
40. #2361C22semidirect
42. #2309C22semidirect
45. #2296C22semidirect
46. #2274C22semidirect
47. #2084C22semidirect
48. #2083C22semidirect
49. #2073C22semidirect
50. #2073C22semidirect
51. #2067C22semidirect
52. #2065C22semidirect
53. #2054C22semidirect
54. #2054C22semidirect
55. #2041C22semidirect
56. #2041C22semidirect
59. #1982C22semidirect
60. #1949C22semidirect
61. #1926C22semidirect
62. #1925C22semidirect
63. #1879C22semidirect
64. #1879C22semidirect
65. #1875C22semidirect
66. #1875C22semidirect
67. #1874C22semidirect
68. #1874C22semidirect
69. #1851C22semidirect
70. #1850C22semidirect
71. #1802C22semidirect
72. #1798C22semidirect
73. #1797C22semidirect
74. #1736C22semidirect
75. #1735C22semidirect
87. #535D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31992
#372972
#342652
#331162
#329022
#297582
#297212
#252182
#252072
#250972
#250762
#219372
#209142