Finite Group Databasebeta

← #6503186

Subgroups — #6503186

Order 512.

1880 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27860C214yesmaximal, complemented
4#26210C214yesmaximal, complemented
5#26138C214yesmaximal, complemented
6#24943C214yesmaximal, complemented
7#24871C214yesmaximal, complemented
8#24554C214yesmaximal, complemented
9#21940C211yesmaximal, characteristic, complemented
10#21759C211yesmaximal, characteristic, complemented
11C2×#1892C211yesmaximal, characteristic, complemented
12C2×#1869C211yesmaximal, characteristic
13C2×#1807C211yesmaximal, characteristic, complemented
14C24×C8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#485C2214yescomplemented
18C22×#106C2212yescomplemented
19C22×#103C2212yescomplemented
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2212yes—
22#2855C2212yes—
23#2848C2212yes—
24#2627—21no—
25#2626—21no—
26#2579—22no—
27#2564—22no—
28#2487C2212yes—
29#2417C2214yes—
30#2409C2212yes—
31#2353—22no—
32#2353—22no—
33#2352C2214yes—
34#2352C2214yes—
35#2315—21no—
36#2313—21no—
37#2312—21no—
38#2308—21no—
39#2308—21no—
40#2283C2212yes—
41#2280C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45#2273—21no—
46#2101C2214yescomplemented
47#2097C2214yescomplemented
48#2085C2214yescomplemented
49#2077C2214yescomplemented
50#2075C2214yescomplemented
51#2074C2214yescomplemented
52#2036C2214yescomplemented
53#2035C2214yescomplemented
54#1928C2214yescomplemented
55#1927C2214yescomplemented
56#1919C2214yescomplemented
57#1918C2214yescomplemented
58#1892C2214yes—
59#1892C2214yes—
60#1892C2212yescomplemented
61#1892C2212yescomplemented
62#1873C2214yescomplemented
63#1872C2214yescomplemented
64#1869C2212yes—
65#1869C2214yes—
66#1869C2212yes—
67#1854C2214yescomplemented
68#1852C2214yescomplemented
69#1852C2214yescomplemented
70#1825C2214yescomplemented
71#1824C2212yescomplemented
72#1808C2214yes—
73#1807C2214yes—
74#1807C2214yescomplemented
75#1787C2212yescomplemented
76C2×#340C2212yescomplemented
77C2×#340C2212yescomplemented
78C2×#336C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4D411yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C23×C8—24no—
84C23×C8—44no—
85C23×C8C2311yescharacteristic
86C23×C8C2312yes—
87C23×C8C2312yes—
88C23×C8C2312yes—
89C23×C8C2311yescharacteristic
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8C2312yes—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2314yes—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4D412yes—
100C2×C4×D4D412yes—
101C2×C4×D4C2312yes—
102C2×C4×D4C2312yes—
103C2×#117C2314yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42C2311yescharacteristic
117#485—48no—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#106—44no—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×#103—44no—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×C22⋊C8—24no—
129C2×C22⋊C8—44no—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2314yes—
134#400—22no—
135#399—22no—
136#398—24no—
137#396—24no—
138#395—22no—
139#393—22no—
140#391C2314yes—
141#389C2314yes—
142#388—24no—
143#388—24no—
144#387—24no—
145#387—24no—
146#387—24no—
147#386—24no—
148#386—22no—
149#386—22no—
150#386—24no—
151#386—24no—
152#385—24no—
153#385—22no—
154#385—22no—
155#385—22no—
156#385—22no—
157#383—22no—
158#382C2314yes—
159#381—22no—
160#381—22no—
161#381—22no—
162#380—21no—
163#380—21no—
164#380—21no—
165#380—21no—
166#380—21no—
167#380—21no—
168#380—24no—
169#380C2312yes—
170#380C2312yes—
171#380C2312yes—
172#380C2312yes—
173#380C2312yes—
174#380C2314yes—
175#380C2314yes—
176#380C2312yes—
177#379—22no—
178#379—22no—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—24no—
184C4×C22⋊C4—24no—
185#376—22no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C25C2×D411yescharacteristic
200C25C2×D411yescharacteristic
201C22×Q8—21no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8C2412yes—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4C2×D412yes—
211C22×D4C2×D412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4#4114yes—
231C23×C4#4114yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C22×C8—22no—
237C22×C8—24no—
238C22×C8—24no—
239C22×C8—24no—
240C22×C8—22no—
241C22×C8—24no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8—42no—
245C22×C8—42no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8—416no—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C4×D4—28no—
262C4×D4—28no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—24no—
267#117—44no—
268#117—44no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2414yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321#107—44no—
322#107—44no—
323#106—28no—
324#106—44no—
325#103—28no—
326#103—44no—
327#103—44no—
328#103—44no—
329C22⋊C8—24no—
330C22⋊C8—24no—
331C22⋊C8—28no—
332C22⋊C8—44no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24#12314yes—
355C24#13514yes—
356C24#13714yes—
357C24C22×D412yes—
358C24C22×D411yescharacteristic
359C24C22×D411yescharacteristic
360C24C22×D411yescharacteristic
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—48no—
366C2×Q8—42no—
367C2×Q8—82no—
368C2×Q8—82no—
369C2×Q8—82no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—416no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—84no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4#12314yes—
424C22×C4#12314yes—
425C22×C4#12314yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2511yescharacteristic
439C2×C8—28no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—48no—
449C2×C8—48no—
450C2×C8—44no—
451C2×C8—84no—
452C2×C8H(5,𝔽2)14yes—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—84no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—84no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—84no—
491C42—84no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—44no—
504C23—44no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—44no—
509C23—82no—
510C23—84no—
511C23—82no—
512C23#485116yescomplemented
513C23C2×#12312yes—
514C23C2×#12312yes—
515C23C2×#12312yes—
516C23C2×#13512yes—
517C23C2×#13512yes—
518C23C2×#13512yes—
519C23C2×#13712yes—
520C23C2×#13712yes—
521C23C2×#13712yes—
522C23C23×D411yescharacteristic
523Q8—42no—
524Q8—42no—
525Q8—44no—
526Q8—84no—
527D4—48no—
528D4—48no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—84no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—164no—
574C2×C4—164no—
575C2×C4—164no—
576C2×C4C2×#12312yes—
577C2×C4C2×#12312yes—
578C2×C4C2×#12312yes—
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×H(5,𝔽2)12yes—
582C2×C4#58512yes—
583C8—24no—
584C8—42no—
585C8—44no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—416no—
599C22—42no—
600C22—44no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—88no—
605C22C2×#48514yes—
606C22C2×#48514yes—
607C22C2×#48514yes—
608C22C22×#12311yescharacteristic
609C22#349612yes—
610C22C22×#13511yescharacteristic
611C22C22×#13711yescharacteristic
612C22#358412yes—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—82no—
620C4—84no—
621C4—84no—
622C4—84no—
623C4—82no—
624C4#349011yescharacteristic
625C4#349111yescharacteristic
626C4#350512yes—
627C2—24no—
628C2—24no—
629C2—24no—
630C2—82no—
631C2#3536011yescharacteristic
632C2C22×#48511yescharacteristic
633C2#6051611yescharacteristic
634C2#6224412yes—
635C2#6225412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27860C2semidirect
4. #26210C2semidirect
5. #26138C2semidirect
6. #24943C2semidirect
7. #24871C2semidirect
8. #24554C2semidirect
9. #21940C2semidirect
10. #21759C2semidirect
11. C2×#1892C2semidirect
13. C2×#1807C2semidirect
17. C2×#485C22semidirect
18. C22×#106C22semidirect
19. C22×#103C22semidirect
46. #2101C22semidirect
47. #2097C22semidirect
48. #2085C22semidirect
49. #2077C22semidirect
50. #2075C22semidirect
51. #2074C22semidirect
52. #2036C22semidirect
53. #2035C22semidirect
54. #1928C22semidirect
55. #1927C22semidirect
56. #1919C22semidirect
57. #1918C22semidirect
60. #1892C22semidirect
61. #1892C22semidirect
62. #1873C22semidirect
63. #1872C22semidirect
67. #1854C22semidirect
68. #1852C22semidirect
69. #1852C22semidirect
70. #1825C22semidirect
71. #1824C22semidirect
74. #1807C22semidirect
75. #1787C22semidirect
76. C2×#340C22semidirect
77. C2×#340C22semidirect
512. C23#485semidirect

Maximal subgroups

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

subgroupindex
#350102
#341852
#278602
#262102
#261382
#249432
#248712
#245542
#219402
#217592
C2×#18922
C2×#18692
C2×#18072