Finite Group Databasebeta

← #6506439

Subgroups — #6506439

Order 512.

2311 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37481C211yesmaximal, characteristic, complemented
2#35149C211yesmaximal, characteristic
3#29117C212yesmaximal, complemented
4#29115C212yesmaximal, complemented
5#27267C214yesmaximal, complemented
6#27263C214yesmaximal, complemented
7#27259C214yesmaximal, complemented
8#27255C214yesmaximal, complemented
9#25868C212yesmaximal, complemented
10#25867C212yesmaximal, complemented
11#21760C212yesmaximal, complemented
12#20597C212yesmaximal, complemented
13#20528C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#535C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#491C2212yescomplemented
19C2×#490C2212yescomplemented
20C2×#488C2212yescomplemented
21C2×#488C2212yescomplemented
22C2×#416C2214yes—
23C2×#416C2214yes—
24C22×C22⋊C8C2212yes—
25#2849—22no—
26#2842—22no—
27#2667—24no—
28#2629C2212yes—
29#2629C2212yes—
30#2613—22no—
31#2608—22no—
32#2441C2214yescomplemented
33#2441C2214yescomplemented
34#2432C2214yes—
35#2431C2214yes—
36#2407—24no—
37#2310C2211yescharacteristic
38#2309C2211yescharacteristic
39#2280C2212yes—
40#2280C2211yescharacteristic
41#2072C2214yescomplemented
42#2071C2214yescomplemented
43#2043C2214yescomplemented
44#2042C2214yescomplemented
45#2042C2214yescomplemented
46#2041C2214yescomplemented
47#2026C2214yescomplemented
48#2025C2214yescomplemented
49#1959C2212yescomplemented
50#1959C2212yescomplemented
51#1957C2212yescomplemented
52#1956C2212yescomplemented
53#1936C2214yescomplemented
54#1935C2214yescomplemented
55#1934C2214yescomplemented
56#1933C2214yescomplemented
57#1893C2214yescomplemented
58#1892C2214yescomplemented
59#1875C2212yescomplemented
60#1875C2212yescomplemented
61#1873C2212yescomplemented
62#1873C2212yescomplemented
63#1825C2214yes—
64#1788C2214yes—
65#1788C2214yes—
66#1788C2214yes—
67#1787C2214yes—
68#1786C2214yes—
69#1752C2214yescomplemented
70C8×C22⋊C4C2214yescomplemented
71#1750C2212yes—
72#1750C2212yes—
73C22×#41—22no—
74C22×#41—28no—
75C22×#41—44no—
76C22×#41—44no—
77C22×#41C2314yes—
78C22×#41C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4C2311yescharacteristic
84C23×C8C2314yes—
85#536—44no—
86#535—44no—
87C2×C4⋊Q8C2312yes—
88C2×#127C2312yes—
89C2×#125C2312yes—
90C2×#125C2312yes—
91C2×#124C2312yes—
92C2×#124C2312yes—
93#525—28no—
94C2×#123—21no—
95C2×#123—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8C2312yes—
98C2×#121—22no—
99C2×#121C2312yes—
100C2×#120—21no—
101C2×#120—22no—
102C2×C4×D4—22no—
103C2×#117C2311yescharacteristic
104C2×#117C2312yes—
105C2×#117C2312yes—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108#491—44no—
109#490—44no—
110#488—44no—
111#488—44no—
112#416—48no—
113#416—48no—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×C22⋊C8—28no—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×#97C2312yes—
131C2×#97C2312yes—
132C2×C4×C8C2312yes—
133C2×C4×C8C2312yes—
134#399—24no—
135#399—24no—
136#398—24no—
137#398—24no—
138#396—24no—
139#396—24no—
140#396—24no—
141#396—24no—
142#394—24no—
143#394—24no—
144#392—24no—
145#392—24no—
146#391—22no—
147#391C2314yes—
148#390—22no—
149#390C2314yes—
150#389—24no—
151#389—24no—
152#389C2314yes—
153#389C2314yes—
154#387—24no—
155#387—24no—
156#386—24no—
157#386—24no—
158#385—24no—
159#385—24no—
160#384C2312yes—
161#384C2312yes—
162#381—22no—
163#381—22no—
164#381D412yes—
165#380D414yes—
166#380C2314yes—
167#380C2314yes—
168#379—22no—
169#379—22no—
170#379D412yes—
171C4×C22⋊C4—22no—
172C4×C22⋊C4—22no—
173C4×C22⋊C4C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#336C2314yes—
179#336C2314yes—
180C25—21no—
181C25C2×D411yescharacteristic
182C2×#41—28no—
183C2×#41—28no—
184C2×#41—28no—
185C2×#41—28no—
186C2×#41—24no—
187C2×#41—44no—
188C2×#41—44no—
189C2×#41—42no—
190C2×#41—42no—
191C2×#41—48no—
192C2×#41—48no—
193C2×#41—464no—
194C2×#41—48no—
195C22×Q8—24no—
196C22×Q8—21no—
197C22×Q8—24no—
198C22×Q8—42no—
199C22×Q8C2411yescharacteristic
200C22×Q8C2412yes—
201C22×D4—24no—
202C22×D4—21no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—21no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—42no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4C2411yescharacteristic
213C22×D4C2412yes—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—24no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2×D412yes—
223C23×C4C2×D411yescharacteristic
224C23×C4#4118yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C22×C8—28no—
228C22×C8—44no—
229C22×C8—44no—
230C22×C8C2414yes—
231C22×C8C2414yes—
232C4⋊Q8—24no—
233C4⋊Q8—42no—
234#127—24no—
235#127—42no—
236#125—42no—
237#125—42no—
238#125—42no—
239#125—42no—
240#124—24no—
241#124—24no—
242#124—42no—
243#124—42no—
244#123—48no—
245#123—44no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—48no—
249#121—24no—
250#121—24no—
251#121—48no—
252#120—48no—
253#120—44no—
254C4×D4—48no—
255#117—24no—
256#117—42no—
257#117—42no—
258#117—44no—
259#117—44no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2414yes—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4C2×D412yes—
285C2×C22⋊C4C2×D412yes—
286C2×C22⋊C4C2×D414yes—
287C2×C22⋊C4C2×D414yes—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42C2412yes—
292C2×C42C2412yes—
293#103—28no—
294#103—28no—
295#103—44no—
296#103—44no—
297#103—44no—
298#103—44no—
299#102—28no—
300#102—28no—
301#102—44no—
302#102—44no—
303#102—44no—
304#102—44no—
305C22⋊C8—28no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308#97—42no—
309#97—42no—
310C4×C8—42no—
311C4×C8—42no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95C2×D414yes—
319#95C2×D414yes—
320C24—24no—
321C24—22no—
322C24—21no—
323C24—41no—
324C24—41no—
325C24—44no—
326C24—42no—
327C24—42no—
328C24—84no—
329C24#12414yes—
330C24C22×D411yescharacteristic
331C24C22×D412yes—
332#41—232no—
333#41—416no—
334#41—416no—
335#41—416no—
336#41—464no—
337#41—88no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—416no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8H(5,𝔽2)14yes—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—416no—
357C2×D4—48no—
358C2×D4—42no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—432no—
362C2×D4—48no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—44no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—84no—
371C2×D4H(5,𝔽2)14yes—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—416no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—432no—
393C22×C4—42no—
394C22×C4—48no—
395C22×C4—42no—
396C22×C4#12418yes—
397C22×C4#12414yes—
398C22×C4C22×D414yes—
399C22×C4C22×D414yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C2×#4114yes—
403C22×C4C2×#4114yes—
404C22×C4C2×#4114yes—
405C22×C4C2511yescharacteristic
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—88no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—48no—
422C4⋊C4—44no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—42no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—42no—
449C42—82no—
450C42—82no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—44no—
460C23—48no—
461C23—48no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—88no—
471C23—88no—
472C23—84no—
473C23—84no—
474C23—88no—
475C23C2×#12412yes—
476C23C2×#12414yes—
477C23C23×D411yescharacteristic
478Q8—28no—
479Q8—48no—
480Q8—44no—
481Q8—44no—
482D4—28no—
483D4—48no—
484D4—432no—
485D4—44no—
486D4—44no—
487D4—84no—
488D4—88no—
489D4—168no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—432no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—84no—
522C2×C4C2×#12412yes—
523C2×C4C2×#12414yes—
524C2×C4C22×#4112yes—
525C2×C4C2×H(5,𝔽2)12yes—
526C2×C4C2×H(5,𝔽2)12yes—
527C8—44no—
528C8—44no—
529C22—22no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—44no—
535C22—42no—
536C22—44no—
537C22—42no—
538C22—44no—
539C22—48no—
540C22—88no—
541C22—816no—
542C22—84no—
543C22—82no—
544C22—82no—
545C22—81no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22C22×#12411yescharacteristic
550C22#348411yescharacteristic
551C22#348411yescharacteristic
552C22#358414yes—
553C4—24no—
554C4—22no—
555C4—44no—
556C4—44no—
557C4—44no—
558C4—81no—
559C4—84no—
560C4—82no—
561C4—84no—
562C4#349014yes—
563C2—44no—
564C2—42no—
565C2—84no—
566C2—161no—
567C2#3536711yescharacteristic
568C2#6061212yes—
569C2#6163512yes—
570C2#6163712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35149C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37481C2semidirect
3. #29117C2semidirect
4. #29115C2semidirect
5. #27267C2semidirect
6. #27263C2semidirect
7. #27259C2semidirect
8. #27255C2semidirect
9. #25868C2semidirect
10. #25867C2semidirect
11. #21760C2semidirect
12. #20597C2semidirect
13. #20528C2semidirect
18. C2×#491C22semidirect
19. C2×#490C22semidirect
20. C2×#488C22semidirect
21. C2×#488C22semidirect
32. #2441C22semidirect
33. #2441C22semidirect
41. #2072C22semidirect
42. #2071C22semidirect
43. #2043C22semidirect
44. #2042C22semidirect
45. #2042C22semidirect
46. #2041C22semidirect
47. #2026C22semidirect
48. #2025C22semidirect
49. #1959C22semidirect
50. #1959C22semidirect
51. #1957C22semidirect
52. #1956C22semidirect
53. #1936C22semidirect
54. #1935C22semidirect
55. #1934C22semidirect
56. #1933C22semidirect
57. #1893C22semidirect
58. #1892C22semidirect
59. #1875C22semidirect
60. #1875C22semidirect
61. #1873C22semidirect
62. #1873C22semidirect
69. #1752C22semidirect
70. C8×C22⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#374812
#351492
#291172
#291152
#272672
#272632
#272592
#272552
#258682
#258672
#217602
#205972
#205282