Finite Group Databasebeta

← #6480288

Subgroups — #6480288

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35285C211yesmaximal, characteristic
2#34281C211yesmaximal, characteristic, complemented
3#33039C214yesmaximal, complemented
4#30066C212yesmaximal, complemented
5#30041C212yesmaximal, complemented
6#29561C212yesmaximal, complemented
7#29555C212yesmaximal, complemented
8#29360C212yesmaximal, complemented
9#28392C212yesmaximal, complemented
10#27591C212yesmaximal, complemented
11#24306C212yesmaximal, complemented
12#24211C212yesmaximal
13#23942C212yesmaximal, complemented
14#22052C214yesmaximal, complemented
15#20785C211yesmaximal, characteristic, complemented
16C2×#541C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C22×#127C2211yescharacteristic
19C2×#479C2214yescomplemented
20C2×#460C2214yes—
21C2×#450C2214yes—
22#2858C2212yes—
23#2858C2212yes—
24#2641C2212yes—
25#2641C2212yes—
26#2548—22no—
27#2548—22no—
28#2435—22no—
29#2435—22no—
30#2373—22no—
31#2373—22no—
32#2371—22no—
33#2370—22no—
34#2330C2211yescharacteristic, complemented
35#2327C2211yescharacteristic, complemented
36#2323C2212yes—
37#2314C2211yescharacteristic
38#2309C2212yes—
39#2309C2212yes—
40C4×C4⋊Q8C2211yescharacteristic, complemented
41C4×#125C2211yescharacteristic, complemented
42C4×C22⋊Q8C2212yes—
43#2295C2212yescomplemented
44#2276C2212yescomplemented
45C4×#117C2211yescharacteristic
46#2092C2214yescomplemented
47#2071C2214yescomplemented
48#2058C2212yescomplemented
49#2057C2212yescomplemented
50#2041C2212yes—
51#2041C2212yes—
52#2025C2214yescomplemented
53#2024C2212yescomplemented
54#2024C2212yescomplemented
55#2014C2212yes—
56#2014C2212yes—
57#2013C2214yescomplemented
58#1989C2214yescomplemented
59#1982C2214yes—
60#1973C2214yescomplemented
61#1968C2214yescomplemented
62#1936C2214yescomplemented
63#1934C2214yescomplemented
64#1926C2214yescomplemented
65#1926C2214yescomplemented
66#1925C2214yescomplemented
67#1925C2214yescomplemented
68#1880C2214yescomplemented
69#1880C2214yescomplemented
70#1880C2214yescomplemented
71#1880C2214yescomplemented
72#1873C2214yes—
73#1854C2214yes—
74#1839C2212yescomplemented
75#1832C2212yescomplemented
76#1818C2212yes—
77#1797C2212yes—
78#1786C2214yes—
79#1768C2212yescomplemented
80#1750C2212yes—
81#1745C2211yescharacteristic, complemented
82#1744C2211yescharacteristic, complemented
83C23×D4—22no—
84C23×D4C2312yes—
85C22×SD16C2314yes—
86C22×SD16C2314yes—
87#541—28no—
88#533—44no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×#127—22no—
92C2×#127—44no—
93C2×#127C2312yes—
94C2×#127C2312yes—
95C2×#126—22no—
96C2×#125—22no—
97C2×#125C2311yescharacteristic
98C2×#125C2312yes—
99C2×#124—22no—
100C2×#124—22no—
101C2×#124—22no—
102C2×#124—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8C2312yes—
105C2×C4×Q8—22no—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×#117C2312yes—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#479—48no—
114#460—48no—
115#450—216no—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×C22⋊C8C2314yes—
128C2×#97C2312yes—
129C2×C4×C8C2312yes—
130#394—24no—
131#394—24no—
132#394—24no—
133#394—24no—
134#391C2312yes—
135#391C2312yes—
136#390—24no—
137#390—22no—
138#390—22no—
139#390—22no—
140#390—22no—
141#390—22no—
142#390—22no—
143#390C2314yes—
144#390C2314yes—
145#390C2314yes—
146#390C2314yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389C2312yes—
151#389C2312yes—
152#388—22no—
153#388—22no—
154#388—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#385—22no—
165#384C2311yescharacteristic
166#382C2311yescharacteristic
167#380C2312yes—
168#379—24no—
169#379—22no—
170#379—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4D412yes—
179C4×C4⋊C4D412yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C22⋊C4—22no—
184C4×C22⋊C4D412yes—
185C4×C22⋊C4D412yes—
186#376—22no—
187#376—22no—
188#376C2311yescharacteristic
189#376C2311yescharacteristic
190C43C2311yescharacteristic
191C43C2311yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202C25—22no—
203C22×Q8—22no—
204C22×Q8C2412yes—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—28no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—44no—
213C22×D4—48no—
214C22×D4—84no—
215C22×D4—84no—
216C22×D4C2414yes—
217C23×C4C2×D412yes—
218C23×C4C2411yescharacteristic
219C2×SD16—28no—
220C2×SD16—28no—
221C2×SD16—44no—
222C2×SD16—44no—
223C2×SD16—44no—
224C2×SD16—44no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—24no—
228C4⋊Q8—44no—
229#127—28no—
230#127—44no—
231#127—48no—
232#126—216no—
233#125—216no—
234#125—24no—
235#125—28no—
236#124—84no—
237#124—84no—
238#124—84no—
239#124—84no—
240C22⋊Q8—24no—
241C22⋊Q8—48no—
242C22⋊Q8#4118yes—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—28no—
246C4×Q8—44no—
247C4×Q8—48no—
248#117—24no—
249#117—42no—
250#117—42no—
251#117—42no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
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#103—44no—
307#103—44no—
308#102—28no—
309#102—28no—
310#102—44no—
311#102—44no—
312#102—44no—
313#102—44no—
314C22⋊C8—28no—
315#97—42no—
316C4×C8—42no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95C2×D412yes—
329#95C2×D412yes—
330#95C2×D412yes—
331#95C2×D412yes—
332C24—24no—
333C24—44no—
334C24—44no—
335C24—44no—
336C24—84no—
337C24—84no—
338C24C22×D411yescharacteristic
339C2×Q8—22no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8C2×#4114yes—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—88no—
363C2×D4—88no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—832no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—42no—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C2×#4114yes—
387C22×C4C2511yescharacteristic
388SD16—88no—
389SD16—88no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C4⋊C4—28no—
397C4⋊C4—28no—
398C4⋊C4—28no—
399C4⋊C4—28no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4C2×#4114yes—
416C22⋊C4—28no—
417C22⋊C4—24no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—44no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C22⋊C4—84no—
428C22⋊C4—84no—
429C22⋊C4—82no—
430C42—28no—
431C42—24no—
432C42—24no—
433C42—24no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—42no—
440C42—42no—
441C42—44no—
442C42—44no—
443C42—44no—
444C42—82no—
445C42—82no—
446C42—82no—
447C42H(5,𝔽2)14yes—
448C23—22no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—88no—
460C23—816no—
461C23—168no—
462C23—164no—
463C23—164no—
464C23#52512yes—
465C23C23×D411yescharacteristic
466Q8—28no—
467Q8—44no—
468Q8—88no—
469D4—44no—
470D4—44no—
471D4—44no—
472D4—44no—
473D4—88no—
474D4—88no—
475D4—88no—
476D4—816no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4#52512yes—
515C2×C4#53514yes—
516C2×C4#57714yes—
517C2×C4C22×#4112yes—
518C2×C4C2×H(5,𝔽2)12yes—
519C2×C4#58512yes—
520C8—44no—
521C8—44no—
522C22—22no—
523C22—24no—
524C22—41no—
525C22—41no—
526C22—42no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22—84no—
531C22—84no—
532C22—84no—
533C22—84no—
534C22—1616no—
535C22C2×#52511yescharacteristic
536C22C2×#53511yescharacteristic
537C22#344911yescharacteristic
538C22C2×#57712yes—
539C22#358512yes—
540C4—22no—
541C4—24no—
542C4—24no—
543C4—44no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—42no—
549C4—42no—
550C4—44no—
551C4—82no—
552C4—82no—
553C4—84no—
554C4#351414yes—
555C2—42no—
556C2—164no—
557C2#3512911yescharacteristic
558C2#6059312yes—
559C2#6081212yes—
560C2#6101211yescharacteristic
561C2#6101911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
2. #34281C2semidirect
3. #33039C2semidirect
4. #30066C2semidirect
5. #30041C2semidirect
6. #29561C2semidirect
7. #29555C2semidirect
8. #29360C2semidirect
9. #28392C2semidirect
10. #27591C2semidirect
11. #24306C2semidirect
13. #23942C2semidirect
14. #22052C2semidirect
15. #20785C2semidirect
19. C2×#479C22semidirect
34. #2330C22semidirect
35. #2327C22semidirect
40. C4×C4⋊Q8C22semidirect
41. C4×#125C22semidirect
43. #2295C22semidirect
44. #2276C22semidirect
46. #2092C22semidirect
47. #2071C22semidirect
48. #2058C22semidirect
49. #2057C22semidirect
52. #2025C22semidirect
53. #2024C22semidirect
54. #2024C22semidirect
57. #2013C22semidirect
58. #1989C22semidirect
60. #1973C22semidirect
61. #1968C22semidirect
62. #1936C22semidirect
63. #1934C22semidirect
64. #1926C22semidirect
65. #1926C22semidirect
66. #1925C22semidirect
67. #1925C22semidirect
68. #1880C22semidirect
69. #1880C22semidirect
70. #1880C22semidirect
71. #1880C22semidirect
74. #1839C22semidirect
75. #1832C22semidirect
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
#352852
#342812
#330392
#300662
#300412
#295612
#295552
#293602
#283922
#275912
#243062
#242112
#239422
#220522
#207852