Finite Group Databasebeta

← #6487874

Subgroups — #6487874

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37643C211yesmaximal, characteristic, complemented
2#34181C211yesmaximal, characteristic
3#32966C214yesmaximal
4#29978C214yesmaximal, complemented
5#26124C214yesmaximal, complemented
6#26110C212yesmaximal, complemented
7#26106C212yesmaximal, complemented
8#26059C214yesmaximal, complemented
9#26045C212yesmaximal, complemented
10#26041C212yesmaximal, complemented
11#21988C211yesmaximal, characteristic, complemented
12#21984C211yesmaximal, characteristic, complemented
13#21954C211yesmaximal, characteristic, complemented
14#20738C211yesmaximal, characteristic
15#20732C211yesmaximal, characteristic
16C2×#518C2212yes—
17C22×C4×D4C2211yescharacteristic
18C2×#478C2214yes—
19C2×#478C2214yes—
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2845—22no—
23#2629—22no—
24#2629—22no—
25#2629C2214yes—
26#2552—21no—
27#2552—21no—
28#2552C2212yescomplemented
29#2552C2212yescomplemented
30#2552C2212yescomplemented
31#2552C2212yescomplemented
32#2349—22no—
33#2349C2214yes—
34D4×C4⋊C4—21no—
35D4×C4⋊C4—21no—
36D4×C4⋊C4C2212yes—
37D4×C4⋊C4C2212yes—
38#2316C2212yes—
39#2315—22no—
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42#2285C2211yescharacteristic
43#2285C2211yescharacteristic
44C2×#380—22no—
45#2100C2214yescomplemented
46#2099C2214yescomplemented
47#2088C2214yescomplemented
48#2088C2214yescomplemented
49#2087C2214yescomplemented
50#2086C2214yescomplemented
51#2059C2214yescomplemented
52#2059C2214yescomplemented
53#1949C2214yescomplemented
54#1948C2212yescomplemented
55#1947C2212yescomplemented
56#1940C2214yescomplemented
57#1939C2212yescomplemented
58#1939C2212yescomplemented
59#1864C2214yes—
60#1864C2214yes—
61#1862C2214yes—
62#1862C2214yes—
63#1854C2214yes—
64#1854C2214yes—
65#1847C2212yescomplemented
66#1845C2211yescharacteristic, complemented
67#1844C2211yescharacteristic, complemented
68#1836C2212yescomplemented
69#1834C2211yescharacteristic, complemented
70#1834C2211yescharacteristic, complemented
71#1824C2212yes—
72#1823C2212yes—
73#1802C2212yes—
74#1802C2212yes—
75#1802C2212yes—
76#1802C2214yes—
77#1802C2214yes—
78#1802C2212yes—
79#1798C2214yes—
80#1786C2212yes—
81#1786C2212yes—
82#1762C2214yes—
83C4×#102C2212yes—
84C4×#102C2212yes—
85C23×D4C2311yescharacteristic
86C24×C4—21no—
87C2×#125—22no—
88C2×#125—22no—
89C2×#125C2314yes—
90#518—48no—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—24no—
96C2×C4×D4Q818yes—
97C2×C4×D4C2314yes—
98C2×C4×D4C2314yes—
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×#117—21no—
102C2×#117—21no—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—21no—
109C22×C22⋊C4—24no—
110C22×C42C2311yescharacteristic
111#478—48no—
112#478—48no—
113#432—24no—
114#431—24no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2314yes—
122C2×C4⋊C8C2314yes—
123C2×#102C2312yes—
124C2×#102C2314yes—
125C2×#102C2312yes—
126C2×#102C2312yes—
127C2×#102C2314yes—
128C2×#102C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×#97C2312yes—
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134#398—24no—
135#398—24no—
136#398—24no—
137#398—24no—
138#398—24no—
139#398—24no—
140#396—24no—
141#396—24no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#385—24no—
153#385—24no—
154#385—24no—
155#385—24no—
156#384—22no—
157#384—22no—
158#384D411yescharacteristic
159#384D411yescharacteristic
160#384C2311yescharacteristic
161#384C2311yescharacteristic
162#384C2314yes—
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#384C2311yescharacteristic
166#384C2311yescharacteristic
167#382—21no—
168#382—21no—
169#382D412yes—
170#382C2312yes—
171#380—21no—
172#380—21no—
173#380—21no—
174#380—21no—
175#380—22no—
176#380—22no—
177#380—24no—
178#380—24no—
179#380D412yes—
180#380D412yes—
181#380C2312yes—
182#380C2312yes—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C2×#95—24no—
192#340C2312yes—
193#340C2312yes—
194#340C2314yes—
195#340C2312yes—
196#340C2312yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2314yes—
203#336C2314yes—
204C25—21no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4C2×Q814yes—
210C22×D4C2412yes—
211C23×C4—21no—
212C23×C4—22no—
213C23×C4—28no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—82no—
218C23×C4—82no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2411yescharacteristic
222C22×C8C2412yes—
223C22×C8C2412yes—
224C22×C8C2414yes—
225#125—44no—
226#125—44no—
227#125—84no—
228#125—84no—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—44no—
236C4×D4—44no—
237C4×D4—48no—
238C4×D4—48no—
239C4×D4C2×Q818yes—
240C4×D4C2×Q818yes—
241#117—44no—
242#117—44no—
243#117—82no—
244#117—82no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—84no—
268C2×C4⋊C4—84no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×Q814yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—416no—
292C2×C22⋊C4C2×D414yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×Q814yes—
299C2×C42#4114yes—
300C2×C42C2414yes—
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303#107—44no—
304#107—44no—
305#106—44no—
306#106—44no—
307C4⋊C8—24no—
308C4⋊C8—24no—
309C4⋊C8—44no—
310C4⋊C8—44no—
311#102—24no—
312#102—28no—
313#102—24no—
314#102—24no—
315#102—28no—
316#102—24no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319#97—24no—
320C4×C8—22no—
321C4×C8—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95C2×D414yes—
333#95C2×D414yes—
334C24—24no—
335C24—24no—
336C24—22no—
337C24—24no—
338C24—81no—
339C24—81no—
340C24—81no—
341C24—81no—
342C24C22×D411yescharacteristic
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—216no—
346C2×D4—216no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4C22×Q814yes—
352C2×D4H(5,𝔽2)14yes—
353C2×D42−1+414yes—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—816no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4C22⋊Q814yes—
384C22×C4C22⋊Q814yes—
385C22×C4C22×D414yes—
386C22×C4C22×D414yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×Q812yes—
393C22×C4C2×#4112yes—
394C22×C4C2511yescharacteristic
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4C22×Q814yes—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—44no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—44no—
447C22⋊C4—82no—
448C22⋊C4—82no—
449C42—24no—
450C42—24no—
451C42—44no—
452C42—44no—
453C42—44no—
454C42C22⋊Q814yes—
455C42C22⋊Q814yes—
456C42C22×Q812yes—
457C42C22×Q812yes—
458C42C2×#4112yes—
459C42C2×#4112yes—
460C23—216no—
461C23—21no—
462C23—21no—
463C23—24no—
464C23—21no—
465C23—24no—
466C23—21no—
467C23—24no—
468C23—21no—
469C23—21no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—84no—
475C23—84no—
476C23—84no—
477C23—84no—
478C23—84no—
479C23—84no—
480C23C2×C22⋊Q812yes—
481C23C23×D411yescharacteristic
482D4—216no—
483D4—24no—
484D4—24no—
485D4—24no—
486D4#54418yes—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—416no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—88no—
528C2×C4—84no—
529C2×C4—168no—
530C2×C4—168no—
531C2×C4C2×C22⋊Q812yes—
532C2×C4C2×C22⋊Q812yes—
533C2×C4C2×C22⋊Q812yes—
534C2×C4C2×C22⋊Q812yes—
535C2×C4C2×C22⋊Q812yes—
536C2×C4#57612yes—
537C2×C4#57712yes—
538C2×C4C23×Q811yescharacteristic
539C2×C4C22×#4111yescharacteristic
540C2×C4C2×H(5,𝔽2)12yes—
541C2×C4C2×2−1+412yes—
542C8—42no—
543C8—42no—
544C8—44no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—28no—
549C22—44no—
550C22—44no—
551C22—41no—
552C22—44no—
553C22—41no—
554C22—44no—
555C22—44no—
556C22—41no—
557C22—44no—
558C22—41no—
559C22—816no—
560C22C22×C22⋊Q811yescharacteristic
561C22#348411yescharacteristic
562C22#348611yescharacteristic
563C22C2×#57611yescharacteristic
564C22C2×#57711yescharacteristic
565C22#358511yescharacteristic
566C22#358611yescharacteristic
567C4—22no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—44no—
572C4—44no—
573C4—44no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4#308114yes—
581C4C2×#54412yes—
582C4#349212yes—
583C2—42no—
584C2—48no—
585C2#3522411yescharacteristic
586C2C2×#308111yescharacteristic
587C2#6049311yescharacteristic
588C2#6168411yescharacteristic
589C2#6168511yescharacteristic
590C2#6184711yescharacteristic
591C2#6184811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37643C2semidirect
4. #29978C2semidirect
5. #26124C2semidirect
6. #26110C2semidirect
7. #26106C2semidirect
8. #26059C2semidirect
9. #26045C2semidirect
10. #26041C2semidirect
11. #21988C2semidirect
12. #21984C2semidirect
13. #21954C2semidirect
28. #2552C22semidirect
29. #2552C22semidirect
30. #2552C22semidirect
31. #2552C22semidirect
45. #2100C22semidirect
46. #2099C22semidirect
47. #2088C22semidirect
48. #2088C22semidirect
49. #2087C22semidirect
50. #2086C22semidirect
51. #2059C22semidirect
52. #2059C22semidirect
53. #1949C22semidirect
54. #1948C22semidirect
55. #1947C22semidirect
56. #1940C22semidirect
57. #1939C22semidirect
58. #1939C22semidirect
65. #1847C22semidirect
66. #1845C22semidirect
67. #1844C22semidirect
68. #1836C22semidirect
69. #1834C22semidirect
70. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#376432
#341812
#329662
#299782
#261242
#261102
#261062
#260592
#260452
#260412
#219882
#219842
#219542
#207382
#207322