Finite Group Databasebeta

← #7487463

Subgroups — #7487463

Order 512.

1673 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46147C211yesmaximal, characteristic, complemented
2#38957C211yesmaximal, characteristic
3#32646C212yesmaximal, complemented
4#32027C212yesmaximal, complemented
5#31306C214yesmaximal, complemented
6#31132C214yesmaximal, complemented
7#30227C212yesmaximal, complemented
8#29580C212yesmaximal, complemented
9#29054C214yesmaximal, complemented
10#27262C214yesmaximal, complemented
11#25661C212yesmaximal, complemented
12#24032C212yesmaximal
13#23273C211yesmaximal, characteristic, complemented
14C2×#563—21no—
15C2×#556C2211yescharacteristic
16C2×D42C2211yescharacteristic
17C2×#489C2212yes—
18C2×#480C2212yes—
19C2×#477C2212yescomplemented
20C22×#102C2212yes—
21#2859C2212yescomplemented
22#2858C2211yescharacteristic
23#2822—21no—
24#2734—21no—
25#2724—21no—
26#2698—24no—
27#2685—24no—
28#2620—21no—
29#2548C2211yescharacteristic
30#2546C2212yescomplemented
31#2542C2211yescharacteristic
32#2500—21no—
33#2496C2214yescomplemented
34#2486C2212yes—
35#2440C2214yescomplemented
36#2435—21no—
37#2435C2214yes—
38#2426C2214yes—
39#2309—21no—
40C4×#124C2211yescharacteristic
41#2293C2211yescharacteristic
42#2100C2212yescomplemented
43#2094C2212yes—
44#2083C2212yescomplemented
45#2071C2212yes—
46#2065C2214yescomplemented
47#2043C2214yescomplemented
48#2042C2214yes—
49#2042C2214yescomplemented
50#2042C2214yescomplemented
51#2041C2214yes—
52#2035C2214yescomplemented
53#2035C2214yescomplemented
54#2033C2214yes—
55#2023C2214yescomplemented
56#1985C2212yescomplemented
57#1983C2212yes—
58#1959C2214yescomplemented
59#1959C2214yescomplemented
60#1943C2214yescomplemented
61#1936C2214yes—
62#1936C2214yescomplemented
63#1926C2214yescomplemented
64#1924C2214yescomplemented
65#1919C2214yescomplemented
66#1893C2212yes—
67#1892C2214yescomplemented
68#1872C2214yescomplemented
69#1869C2212yes—
70#1863C2214yescomplemented
71#1862C2214yes—
72#1786C2212yes—
73#1786C2214yes—
74#1763C2212yescomplemented
75#1752C2214yescomplemented
76#1745C2212yescomplemented
77C23×D4—21no—
78C23×D4C2312yes—
79C23×C8C2312yes—
80#563—48no—
81#556—42no—
82#556—42no—
83D42—42no—
84D42—42no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89C2×#126—22no—
90C2×#126—24no—
91C2×#125—22no—
92C2×#124—21no—
93C2×#124D411yescharacteristic
94C2×#124C2312yes—
95C2×#124C2314yes—
96C2×#123—22no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8C2314yes—
100C2×#121—21no—
101C2×#121C2312yes—
102C2×#120—22no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2311yescharacteristic
105C2×C4×D4C2311yescharacteristic
106C2×C4×D4C2311yescharacteristic
107C2×#117—21no—
108C2×#117—21no—
109C22×C4⋊C4C2312yes—
110#489—44no—
111#480—44no—
112#477—44no—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2312yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#102—44no—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2312yes—
130C2×#97C2314yes—
131#400—24no—
132#399—22no—
133#398—22no—
134#398—22no—
135#398—22no—
136#398—24no—
137#397—22no—
138#397—24no—
139#396—24no—
140#395—24no—
141#394—22no—
142#394—24no—
143#394—24no—
144#394—24no—
145#391—22no—
146#391D411yescharacteristic
147#390—21no—
148#390—21no—
149#390—24no—
150#390—21no—
151#390—22no—
152#390—21no—
153#390D411yescharacteristic
154#390C2314yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389C2314yes—
159#389C2312yes—
160#389C2314yes—
161#389C2314yes—
162#388—22no—
163#388—24no—
164#388D414yes—
165#387—24no—
166#385—24no—
167#384—21no—
168#384C2312yes—
169#384C2312yes—
170#382—22no—
171#382D411yescharacteristic
172#381—22no—
173#380C2314yes—
174#380C2312yes—
175#379—24no—
176#379—24no—
177C4×C4⋊C4C2311yescharacteristic
178C4×C4⋊C4C2312yes—
179C4×C22⋊C4—21no—
180C4×C22⋊C4C2314yes—
181C43C2311yescharacteristic
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2312yes—
187#336C2312yes—
188#336C2314yes—
189#336C2314yes—
190C25—22no—
191C22×Q8—22no—
192C22×Q8C2×D411yescharacteristic
193C22×Q8C2412yes—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4—22no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—82no—
205C22×D4C2×D411yescharacteristic
206C22×D4C2412yes—
207C22×D4C2411yescharacteristic
208C23×C4—21no—
209C23×C4—22no—
210C23×C4C2412yes—
211C22×C8—44no—
212C22×C8—42no—
213C22×C8C2414yes—
214C22×C8C2414yes—
215C4⋊Q8—42no—
216C4⋊Q8—42no—
217C4⋊Q8—41no—
218C4⋊Q8—41no—
219C4⋊Q8—48no—
220#127—41no—
221#127—41no—
222#126—48no—
223#126—416no—
224#125—48no—
225#124—24no—
226#124—24no—
227#124—42no—
228#124—44no—
229#124—44no—
230#124—82no—
231#123—48no—
232C22⋊Q8—416no—
233C22⋊Q8—44no—
234C22⋊Q8—44no—
235C22⋊Q8—44no—
236#121—24no—
237#121—42no—
238#121—44no—
239#120—48no—
240C4×Q8—24no—
241C4×Q8—41no—
242C4×Q8—41no—
243C4×Q8—42no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—41no—
247C4×D4—41no—
248#117—44no—
249#117—82no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2411yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4C2×D414yes—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D414yes—
289C2×C42—22no—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D412yes—
295C2×C42C2411yescharacteristic
296C2×C42C2412yes—
297C2×C42C2412yes—
298#107—28no—
299#106—44no—
300C4⋊C8—42no—
301#103—44no—
302#103—44no—
303#103—44no—
304#103—44no—
305#102—28no—
306#102—44no—
307#102—44no—
308#102—44no—
309#102—44no—
310#102—44no—
311C22⋊C8—44no—
312C22⋊C8—42no—
313#97—44no—
314#97—42no—
315#95—22no—
316#95—21no—
317#95—22no—
318#95—24no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95C2×D414yes—
324#95C2×D414yes—
325C24—21no—
326C24—22no—
327C24—44no—
328C24—44no—
329C24—84no—
330C24—82no—
331C24—84no—
332C24C22×D412yes—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—44no—
341C2×Q8—82no—
342C2×Q8—82no—
343C2×D4—21no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—48no—
357C2×D4—44no—
358C2×D4—42no—
359C2×D4—84no—
360C2×D4—82no—
361C2×D4—84no—
362C2×D4—88no—
363C2×D4—816no—
364C2×D4—82no—
365C2×D4AGL(1,ℤ/8)14yes—
366C2×D4H(5,𝔽2)12yes—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D412yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C2511yescharacteristic
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—84no—
397C4⋊C4—21no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—48no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—84no—
419C4⋊C4—82no—
420C4⋊C4H(5,𝔽2)12yes—
421C22⋊C4—24no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C22⋊C4—84no—
432C22⋊C4—82no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—21no—
442C42—24no—
443C42—42no—
444C42—41no—
445C42—44no—
446C42—42no—
447C42—44no—
448C23—22no—
449C23—24no—
450C23—22no—
451C23—24no—
452C23—22no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—42no—
458C23—44no—
459C23—42no—
460C23—44no—
461C23—42no—
462C23—88no—
463C23—88no—
464C23—168no—
465C23—164no—
466C23—168no—
467C23C2×AGL(1,ℤ/8)14yes—
468C23C23×D411yescharacteristic
469Q8—44no—
470Q8—42no—
471Q8—42no—
472Q8—82no—
473Q8—164no—
474D4—24no—
475D4—44no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—44no—
480D4—84no—
481D4—88no—
482D4—88no—
483D4—162no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—48no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—164no—
520C2×C4C2×AGL(1,ℤ/8)12yes—
521C2×C4C2×H(5,𝔽2)11yescharacteristic
522C2×C4C2×H(5,𝔽2)11yescharacteristic
523C2×C4#58514yes—
524C8—44no—
525C8—44no—
526C22—28no—
527C22—24no—
528C22—42no—
529C22—44no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—84no—
538C22—84no—
539C22—82no—
540C22—82no—
541C22—1616no—
542C22#348411yescharacteristic
543C22#350012yes—
544C22C22×AGL(1,ℤ/8)11yescharacteristic
545C22#358511yescharacteristic
546C22#358612yes—
547C4—22no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—41no—
555C4—82no—
556C4—82no—
557C4—84no—
558C4—84no—
559C4#353312yes—
560C4#355112yes—
561C2—44no—
562C2—82no—
563C2—162no—
564C2#5197811yescharacteristic
565C2#6156911yescharacteristic
566C2#6167411yescharacteristic
567C2#6257512yes—
568C2#6258312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #46147C2semidirect
3. #32646C2semidirect
4. #32027C2semidirect
5. #31306C2semidirect
6. #31132C2semidirect
7. #30227C2semidirect
8. #29580C2semidirect
9. #29054C2semidirect
10. #27262C2semidirect
11. #25661C2semidirect
13. #23273C2semidirect
19. C2×#477C22semidirect
21. #2859C22semidirect
30. #2546C22semidirect
33. #2496C22semidirect
35. #2440C22semidirect
42. #2100C22semidirect
44. #2083C22semidirect
46. #2065C22semidirect
47. #2043C22semidirect
49. #2042C22semidirect
50. #2042C22semidirect
52. #2035C22semidirect
53. #2035C22semidirect
55. #2023C22semidirect
56. #1985C22semidirect
58. #1959C22semidirect
59. #1959C22semidirect
60. #1943C22semidirect
62. #1936C22semidirect
63. #1926C22semidirect
64. #1924C22semidirect
65. #1919C22semidirect
67. #1892C22semidirect
68. #1872C22semidirect
70. #1863C22semidirect
74. #1763C22semidirect
75. #1752C22semidirect
76. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#461472
#389572
#326462
#320272
#313062
#311322
#302272
#295802
#290542
#272622
#256612
#240322
#232732