Finite Group Databasebeta

← #6488641

Subgroups — #6488641

Order 512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35268C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic, complemented
3#33014C214yesmaximal, complemented
4#30077C212yesmaximal, complemented
5#30038C212yesmaximal, complemented
6#29499C212yesmaximal, complemented
7#29493C212yesmaximal, complemented
8#29263C212yesmaximal, complemented
9#29116C212yesmaximal, complemented
10#28319C212yesmaximal, complemented
11#28165C212yesmaximal, complemented
12#27650C212yesmaximal, complemented
13#24208C212yesmaximal, complemented
14#22158C214yesmaximal, complemented
15#20785C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C2×#518C2212yes—
18C2×#482C2214yes—
19C2×#459C2214yes—
20C2×#447C2214yes—
21#2864C2211yescharacteristic, complemented
22#2860C2212yescomplemented
23#2858C2211yescharacteristic, complemented
24#2628C2211yescharacteristic
25#2624C2212yes—
26#2619C2211yescharacteristic
27#2521—22no—
28#2516—22no—
29#2467—22no—
30#2459—22no—
31#2366—22no—
32#2366—22no—
33D4×C22⋊C4—22no—
34D4×C22⋊C4—22no—
35#2331C2211yescharacteristic, complemented
36#2328C2212yes—
37#2328C2212yes—
38#2328C2211yescharacteristic, complemented
39#2319C2212yescomplemented
40#2309C2212yes—
41C4×#127C2212yes—
42C4×#127C2212yes—
43C4×#121C2212yes—
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2092C2214yescomplemented
47#2084C2212yescomplemented
48#2083C2212yescomplemented
49#2073C2214yescomplemented
50#2071C2212yescomplemented
51#2071C2212yescomplemented
52#2041C2212yes—
53#2041C2212yes—
54#2023C2214yescomplemented
55#2011C2212yes—
56#2011C2212yes—
57#2011C2214yescomplemented
58#1982C2214yes—
59#1972C2214yescomplemented
60#1970C2214yescomplemented
61#1949C2214yescomplemented
62#1936C2214yescomplemented
63#1934C2214yescomplemented
64#1879C2214yescomplemented
65#1879C2214yescomplemented
66#1879C2214yescomplemented
67#1879C2214yescomplemented
68#1864C2214yescomplemented
69#1864C2214yescomplemented
70#1862C2214yescomplemented
71#1862C2214yescomplemented
72#1839C2212yescomplemented
73#1832C2212yescomplemented
74#1825C2214yes—
75#1818C2212yes—
76#1804C2214yescomplemented
77#1797C2212yes—
78#1786C2214yes—
79#1768C2212yescomplemented
80#1750C2212yes—
81#1745C2211yescharacteristic, complemented
82#1744C2211yescharacteristic, complemented
83C23×D4—22no—
84C23×D4C2312yes—
85C22×D8C2314yes—
86C22×D8C2314yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#127—28no—
89C2×#127—22no—
90C2×#127C2312yes—
91C2×#127C2312yes—
92C2×#127C2311yescharacteristic
93C2×#126—22no—
94C2×#124—22no—
95C2×#121C2312yes—
96#518—48no—
97C2×C4×D4—22no—
98C2×C4×D4—24no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4C2312yes—
104C2×C4×D4C2314yes—
105C2×C4×D4C2314yes—
106C2×#117C2312yes—
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C22⋊C4—24no—
110C22×C42C2311yescharacteristic
111#482—48no—
112#459—216no—
113#447—48no—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×C22⋊C8C2314yes—
126C2×#97C2312yes—
127C2×C4×C8C2312yes—
128#400—21no—
129#399—22no—
130#399—21no—
131#397—22no—
132#395—21no—
133#394—21no—
134#394—21no—
135#394—22no—
136#394—22no—
137#393—21no—
138#392—21no—
139#392—21no—
140#391—22no—
141#390—22no—
142#390—22no—
143#390—22no—
144#390—22no—
145#390—22no—
146#390—22no—
147#390—22no—
148#390C2314yes—
149#390C2314yes—
150#390C2312yes—
151#390C2312yes—
152#390C2312yes—
153#390C2312yes—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#386—22no—
168#386—24no—
169#386—24no—
170#386—24no—
171#386—24no—
172#386—22no—
173#386—22no—
174#384C2312yes—
175#382C2312yes—
176#382C2311yescharacteristic
177#382C2311yescharacteristic
178#381—22no—
179#380—22no—
180#380—22no—
181#380C2312yes—
182#379—22no—
183#379—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4D412yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4D412yes—
190C4×C22⋊C4D412yes—
191#376C2311yescharacteristic
192#376C2311yescharacteristic
193C43C2311yescharacteristic
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204#336C2314yes—
205C25—22no—
206C22×Q8—21no—
207C22×D4—216no—
208C22×D4—28no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4—28no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4—84no—
220C22×D4—84no—
221C22×D4C2412yes—
222C22×D4C2414yes—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4C2×D412yes—
227C23×C4C2411yescharacteristic
228C2×D8—28no—
229C2×D8—28no—
230C2×D8—44no—
231C2×D8—44no—
232C2×D8—44no—
233C2×D8—44no—
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—42no—
237#127—216no—
238#127—28no—
239#127—24no—
240#127—42no—
241#127#4118yes—
242#126—84no—
243#124—84no—
244#121—24no—
245#121—24no—
246C4×D4—28no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—44no—
250C4×D4—44no—
251C4×D4—416no—
252C4×D4—84no—
253C4×D4—84no—
254C4×D4—84no—
255C4×D4—84no—
256C4×D4—84no—
257#117—24no—
258#117—28no—
259#117—42no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D411yescharacteristic
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—416no—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D411yescharacteristic
295C2×C22⋊C4C2×D411yescharacteristic
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310#107—44no—
311#106—44no—
312C4⋊C8—28no—
313#102—28no—
314#102—28no—
315#102—44no—
316#102—44no—
317#102—44no—
318#102—44no—
319#102—44no—
320#102—44no—
321C22⋊C8—28no—
322#97—42no—
323C4×C8—42no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—22no—
335#95C2×D412yes—
336#95C2×D412yes—
337#95C2×D412yes—
338#95C2×D412yes—
339C24—28no—
340C24—24no—
341C24—22no—
342C24—44no—
343C24—44no—
344C24—84no—
345C24—84no—
346C24C22×D411yescharacteristic
347C2×Q8—42no—
348C2×Q8—82no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—216no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—416no—
363C2×D4—416no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—832no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4C2×#4118yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—42no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—82no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—82no—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C4C2511yescharacteristic
410D8—88no—
411D8—88no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C22⋊C4—24no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C42—24no—
456C42—24no—
457C42—24no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—42no—
464C42—42no—
465C42—44no—
466C42—82no—
467C42C2×#4114yes—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—28no—
473C23—416no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—816no—
488C23—168no—
489C23—164no—
490C23—164no—
491C23#53512yes—
492C23C23×D411yescharacteristic
493Q8—162no—
494D4—28no—
495D4—28no—
496D4—416no—
497D4—416no—
498D4—416no—
499D4—44no—
500D4—44no—
501D4—44no—
502D4—44no—
503D4—816no—
504D4—88no—
505D4—162no—
506D4—162no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—48no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—164no—
542C2×C4—168no—
543C2×C4—168no—
544C2×C4#52514yes—
545C2×C4#53512yes—
546C2×C4#57614yes—
547C2×C4C22×#4112yes—
548C2×C4C2×H(5,𝔽2)12yes—
549C2×C4#58512yes—
550C8—44no—
551C8—44no—
552C22—22no—
553C22—24no—
554C22—41no—
555C22—41no—
556C22—42no—
557C22—48no—
558C22—48no—
559C22—48no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—1616no—
572C22C2×#52511yescharacteristic
573C22C2×#53511yescharacteristic
574C22#344911yescharacteristic
575C22C2×#57612yes—
576C22#358512yes—
577C4—22no—
578C4—24no—
579C4—24no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—42no—
586C4—82no—
587C4—84no—
588C4—82no—
589C4—82no—
590C4#351314yes—
591C2—42no—
592C2—88no—
593C2—162no—
594C2#3523211yescharacteristic
595C2#6057812yes—
596C2#6083212yes—
597C2#6101111yescharacteristic
598C2#6101711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35268C2semidirect
2. #34210C2semidirect
3. #33014C2semidirect
4. #30077C2semidirect
5. #30038C2semidirect
6. #29499C2semidirect
7. #29493C2semidirect
8. #29263C2semidirect
9. #29116C2semidirect
10. #28319C2semidirect
11. #28165C2semidirect
12. #27650C2semidirect
13. #24208C2semidirect
14. #22158C2semidirect
15. #20785C2semidirect
21. #2864C22semidirect
22. #2860C22semidirect
23. #2858C22semidirect
35. #2331C22semidirect
38. #2328C22semidirect
39. #2319C22semidirect
46. #2092C22semidirect
47. #2084C22semidirect
48. #2083C22semidirect
49. #2073C22semidirect
50. #2071C22semidirect
51. #2071C22semidirect
54. #2023C22semidirect
57. #2011C22semidirect
59. #1972C22semidirect
60. #1970C22semidirect
61. #1949C22semidirect
62. #1936C22semidirect
63. #1934C22semidirect
64. #1879C22semidirect
65. #1879C22semidirect
66. #1879C22semidirect
67. #1879C22semidirect
68. #1864C22semidirect
69. #1864C22semidirect
70. #1862C22semidirect
71. #1862C22semidirect
72. #1839C22semidirect
73. #1832C22semidirect
76. #1804C22semidirect
79. #1768C22semidirect
81. #1745C22semidirect
82. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#352682
#342102
#330142
#300772
#300382
#294992
#294932
#292632
#291162
#283192
#281652
#276502
#242082
#221582
#207852