Finite Group Databasebeta

← #6479678

Subgroups — #6479678

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35287C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic, complemented
3#32966C212yesmaximal, complemented
4#32911C212yesmaximal, complemented
5#29910C212yesmaximal, complemented
6#29622C212yesmaximal, complemented
7#29499C212yesmaximal, complemented
8#29493C212yesmaximal, complemented
9#29376C214yesmaximal, complemented
10#29136C212yesmaximal, complemented
11#28387C212yesmaximal, complemented
12#28207C212yesmaximal, complemented
13#28156C212yesmaximal, complemented
14#22056C214yesmaximal, complemented
15#20780C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C2×#518C2212yes—
18C2×#478C2214yescomplemented
19C2×#459C2214yescomplemented
20C2×#447C2214yes—
21#2860C2212yescomplemented
22#2854C2212yes—
23#2647C2211yescharacteristic, complemented
24#2641C2211yescharacteristic, complemented
25#2629C2211yescharacteristic
26#2619C2211yescharacteristic
27#2593—22no—
28#2554—22no—
29#2468—22no—
30#2457—22no—
31#2366—22no—
32#2366—22no—
33D4×C22⋊C4—22no—
34D4×C22⋊C4—22no—
35#2328C2212yes—
36#2328C2212yes—
37#2324C2212yes—
38#2319C2212yescomplemented
39#2309C2212yes—
40C4×#127C2212yes—
41C4×#127C2212yes—
42C4×#127C2211yescharacteristic, complemented
43C4×#125C2211yescharacteristic, complemented
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2101C2214yescomplemented
47#2092C2214yescomplemented
48#2084C2212yescomplemented
49#2083C2212yescomplemented
50#2071C2212yescomplemented
51#2071C2212yescomplemented
52#2041C2212yes—
53#2041C2212yes—
54#2041C2214yes—
55#2011C2212yes—
56#2011C2212yes—
57#2011C2214yes—
58#1972C2214yescomplemented
59#1970C2214yescomplemented
60#1936C2214yescomplemented
61#1934C2214yescomplemented
62#1926C2214yescomplemented
63#1925C2214yescomplemented
64#1879C2214yescomplemented
65#1879C2214yescomplemented
66#1879C2214yescomplemented
67#1879C2214yescomplemented
68#1875C2214yes—
69#1864C2214yescomplemented
70#1862C2214yescomplemented
71#1854C2214yes—
72#1840C2212yescomplemented
73#1831C2212yescomplemented
74#1817C2212yes—
75#1804C2214yescomplemented
76#1802C2214yescomplemented
77#1798C2212yes—
78#1786C2214yes—
79#1768C2212yescomplemented
80#1750C2212yes—
81#1745C2211yescharacteristic, complemented
82#1744C2211yescharacteristic, complemented
83C23×D4—22no—
84C23×D4C2312yes—
85C22×D8C2314yes—
86C22×D8C2314yes—
87C2×#127—28no—
88C2×#127—22no—
89C2×#127C2312yes—
90C2×#127C2311yescharacteristic
91C2×#127C2312yes—
92C2×#126—22no—
93C2×#125C2311yescharacteristic
94C2×#123—22no—
95C2×#121C2312yes—
96#518—48no—
97C2×C4×D4—24no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4C2314yes—
104C2×C4×D4C2314yes—
105C2×C4×D4C2312yes—
106C2×#117C2312yes—
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C22⋊C4—24no—
110C22×C42C2311yescharacteristic
111#478—48no—
112#459—48no—
113#447—216no—
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—22no—
129#398—22no—
130#398—22no—
131#396—22no—
132#394—22no—
133#394—22no—
134#394—22no—
135#392—22no—
136#390—22no—
137#390—22no—
138#390—22no—
139#390—22no—
140#390—22no—
141#390—22no—
142#390—22no—
143#390—22no—
144#390C2314yes—
145#390C2314yes—
146#390C2312yes—
147#390C2312yes—
148#390C2312yes—
149#390C2312yes—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#387—22no—
163#386—22no—
164#386—22no—
165#386—24no—
166#386—24no—
167#386—24no—
168#386—24no—
169#385—22no—
170#384C2312yes—
171#384C2311yescharacteristic
172#382C2311yescharacteristic
173#382C2312yes—
174#381—22no—
175#380—22no—
176#380—22no—
177#380C2312yes—
178#379—22no—
179#379—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2312yes—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187#376C2311yescharacteristic
188#376C2311yescharacteristic
189C43C2311yescharacteristic
190C43C2311yescharacteristic
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#336C2312yes—
196#336C2312yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200#336C2314yes—
201C25—22no—
202C22×D4—216no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—84no—
213C22×D4—84no—
214C22×D4#4118yes—
215C22×D4C2412yes—
216C22×D4C2414yes—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4C2×D412yes—
221C23×C4C2411yescharacteristic
222C2×D8—28no—
223C2×D8—28no—
224C2×D8—44no—
225C2×D8—44no—
226C2×D8—44no—
227C2×D8—44no—
228C22×C8C2414yes—
229C22×C8C2414yes—
230#127—216no—
231#127—28no—
232#127—28no—
233#127—24no—
234#126—84no—
235#125—24no—
236#123—84no—
237#121—28no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—44no—
241C4×D4—44no—
242C4×D4—44no—
243C4×D4—416no—
244C4×D4—84no—
245C4×D4—84no—
246C4×D4—84no—
247C4×D4—84no—
248C4×D4—84no—
249#117—24no—
250#117—42no—
251#117—44no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—416no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C22⋊C4C2×D411yescharacteristic
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303#107—44no—
304#106—44no—
305C4⋊C8—28no—
306#102—28no—
307#102—28no—
308#102—44no—
309#102—44no—
310#102—44no—
311#102—44no—
312#102—44no—
313#102—44no—
314C22⋊C8—28no—
315#97—42no—
316C4×C8—42no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95C2×D412yes—
326#95C2×D412yes—
327#95C2×D412yes—
328#95C2×D412yes—
329C24—24no—
330C24—28no—
331C24—22no—
332C24—44no—
333C24—44no—
334C24—84no—
335C24—84no—
336C24C22×D411yescharacteristic
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—216no—
340C2×D4—28no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—416no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—416no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—832no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4C2×#4118yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—42no—
377C22×C4—48no—
378C22×C4—44no—
379C22×C4—48no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—84no—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C2×#4114yes—
396C22×C4C2511yescharacteristic
397D8—88no—
398D8—88no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C22⋊C4—24no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—48no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—42no—
452C42—42no—
453C42—44no—
454C42—82no—
455C42H(5,𝔽2)14yes—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—28no—
461C23—416no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—816no—
476C23—164no—
477C23—164no—
478C23—168no—
479C23#52512yes—
480C23C23×D411yescharacteristic
481D4—28no—
482D4—28no—
483D4—416no—
484D4—416no—
485D4—416no—
486D4—44no—
487D4—44no—
488D4—44no—
489D4—44no—
490D4—88no—
491D4—816no—
492D4—164no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—168no—
528C2×C4—164no—
529C2×C4—168no—
530C2×C4#52512yes—
531C2×C4#53514yes—
532C2×C4#57614yes—
533C2×C4C22×#4112yes—
534C2×C4C2×H(5,𝔽2)12yes—
535C2×C4#58512yes—
536C8—44no—
537C8—44no—
538C22—22no—
539C22—24no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—48no—
544C22—48no—
545C22—48no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—1616no—
558C22C2×#52511yescharacteristic
559C22C2×#53511yescharacteristic
560C22#344911yescharacteristic
561C22C2×#57612yes—
562C22#358512yes—
563C4—22no—
564C4—24no—
565C4—24no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—82no—
573C4—82no—
574C4—84no—
575C4—82no—
576C4#350514yes—
577C2—42no—
578C2—88no—
579C2—162no—
580C2#3510211yescharacteristic
581C2#6057812yes—
582C2#6083212yes—
583C2#6101111yescharacteristic
584C2#6101911yescharacteristic
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. #35287C2semidirect
2. #34210C2semidirect
3. #32966C2semidirect
4. #32911C2semidirect
5. #29910C2semidirect
6. #29622C2semidirect
7. #29499C2semidirect
8. #29493C2semidirect
9. #29376C2semidirect
10. #29136C2semidirect
11. #28387C2semidirect
12. #28207C2semidirect
13. #28156C2semidirect
14. #22056C2semidirect
15. #20780C2semidirect
18. C2×#478C22semidirect
19. C2×#459C22semidirect
21. #2860C22semidirect
23. #2647C22semidirect
24. #2641C22semidirect
38. #2319C22semidirect
42. C4×#127C22semidirect
43. C4×#125C22semidirect
46. #2101C22semidirect
47. #2092C22semidirect
48. #2084C22semidirect
49. #2083C22semidirect
50. #2071C22semidirect
51. #2071C22semidirect
58. #1972C22semidirect
59. #1970C22semidirect
60. #1936C22semidirect
61. #1934C22semidirect
62. #1926C22semidirect
63. #1925C22semidirect
64. #1879C22semidirect
65. #1879C22semidirect
66. #1879C22semidirect
67. #1879C22semidirect
69. #1864C22semidirect
70. #1862C22semidirect
72. #1840C22semidirect
73. #1831C22semidirect
75. #1804C22semidirect
76. #1802C22semidirect
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
#352872
#342102
#329662
#329112
#299102
#296222
#294992
#294932
#293762
#291362
#283872
#282072
#281562
#220562
#207802