Finite Group Databasebeta

← #6506472

Subgroups — #6506472

Order 512.

2351 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37477C211yesmaximal, characteristic, complemented
2#35149C211yesmaximal, characteristic
3#29118C212yesmaximal, complemented
4#29115C212yesmaximal, complemented
5#27284C214yesmaximal, complemented
6#27278C214yesmaximal, complemented
7#27276C214yesmaximal, complemented
8#27270C214yesmaximal, complemented
9#25868C212yesmaximal, complemented
10#25865C212yesmaximal, complemented
11#21769C212yesmaximal, complemented
12#20601C212yesmaximal, complemented
13#20529C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#535C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#491C2212yescomplemented
19C2×#489C2212yescomplemented
20C2×#488C2212yescomplemented
21C2×#486C2212yescomplemented
22C2×#417C2214yes—
23C2×#417C2214yes—
24C2×#407C2212yes—
25#2849—22no—
26#2841—22no—
27#2659—22no—
28#2646—22no—
29#2629C2212yes—
30#2628C2212yes—
31#2613—22no—
32#2442C2214yescomplemented
33#2440C2214yescomplemented
34#2437—22no—
35#2432C2214yes—
36#2431C2214yes—
37#2407—22no—
38#2407—22no—
39#2310C2211yescharacteristic
40#2309C2211yescharacteristic
41#2280C2211yescharacteristic
42C2×#380C2212yes—
43#2074C2214yescomplemented
44#2071C2214yescomplemented
45#2043C2214yescomplemented
46#2042C2214yescomplemented
47#2042C2214yescomplemented
48#2041C2214yescomplemented
49#2025C2214yescomplemented
50#2024C2214yescomplemented
51#1961C2212yescomplemented
52#1959C2212yescomplemented
53#1958C2212yescomplemented
54#1956C2212yescomplemented
55#1936C2214yescomplemented
56#1935C2214yescomplemented
57#1934C2214yescomplemented
58#1933C2214yescomplemented
59#1895C2214yescomplemented
60#1894C2214yescomplemented
61#1874C2212yescomplemented
62#1874C2212yescomplemented
63#1872C2212yescomplemented
64#1872C2212yescomplemented
65#1825C2214yes—
66#1788C2214yes—
67#1788C2214yes—
68#1788C2214yes—
69#1787C2214yes—
70#1786C2214yes—
71#1753C2214yescomplemented
72#1753C2214yescomplemented
73#1750C2212yes—
74#1750C2212yes—
75C22×#41—22no—
76C22×#41—28no—
77C22×#41—44no—
78C22×#41—44no—
79C22×#41C2314yes—
80C22×#41C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4—21no—
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C22×M16C2314yes—
87#536—44no—
88#535—44no—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#127C2312yes—
92C2×#125C2312yes—
93C2×#124C2312yes—
94C2×#124C2312yes—
95#525—28no—
96C2×#123—21no—
97C2×#123—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8C2312yes—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×#120—21no—
105C2×#120—22no—
106C2×C4×D4—22no—
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—22no—
113#491—44no—
114#489—44no—
115#488—44no—
116#486—44no—
117#417—48no—
118#417—48no—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131#407—28no—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×#97C2312yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2312yes—
139#400—22no—
140#400—24no—
141#399—22no—
142#398—24no—
143#398—24no—
144#398—24no—
145#396—24no—
146#395—22no—
147#395—24no—
148#394—24no—
149#394—24no—
150#393—22no—
151#392—24no—
152#392—24no—
153#391—21no—
154#391—21no—
155#391C2314yes—
156#390—22no—
157#390C2314yes—
158#389C2314yes—
159#389C2314yes—
160#386—24no—
161#386—24no—
162#385—24no—
163#385—24no—
164#385—24no—
165#385—24no—
166#384C2312yes—
167#382C2312yes—
168#381—22no—
169#381D412yes—
170#380—22no—
171#380—22no—
172#380—24no—
173#380D412yes—
174#380D412yes—
175#380C2314yes—
176#380C2314yes—
177#379—22no—
178#379D412yes—
179C4×C22⋊C4C2314yes—
180C2×#95—22no—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2314yes—
186#336C2314yes—
187C25—21no—
188C25C2×D411yescharacteristic
189C2×#41—28no—
190C2×#41—28no—
191C2×#41—28no—
192C2×#41—28no—
193C2×#41—24no—
194C2×#41—44no—
195C2×#41—44no—
196C2×#41—42no—
197C2×#41—42no—
198C2×#41—48no—
199C2×#41—48no—
200C2×#41—464no—
201C2×#41—48no—
202C22×Q8—24no—
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—42no—
207C22×Q8C2411yescharacteristic
208C22×Q8C2412yes—
209C22×D4—24no—
210C22×D4—21no—
211C22×D4—24no—
212C22×D4—22no—
213C22×D4—21no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—24no—
217C22×D4—42no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4C2411yescharacteristic
222C22×D4C2412yes—
223C23×C4—21no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—21no—
227C23×C4—24no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4C2×D412yes—
234C23×C4C2×D411yescharacteristic
235C23×C4#4118yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C2×M16—28no—
239C2×M16—44no—
240C2×M16—44no—
241C22×C8C2414yes—
242C22×C8C2414yes—
243C4⋊Q8—24no—
244C4⋊Q8—42no—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247#127—24no—
248#127—42no—
249#125—42no—
250#125—42no—
251#124—24no—
252#124—24no—
253#124—42no—
254#124—42no—
255#123—48no—
256#123—44no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—48no—
260C22⋊Q8—88no—
261#121—24no—
262#121—24no—
263#121—44no—
264#121—44no—
265#120—48no—
266#120—44no—
267C4×D4—48no—
268#117—24no—
269#117—42no—
270#117—42no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—84no—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×D414yes—
303C2×C42—21no—
304C2×C42C2412yes—
305C2×C42C2412yes—
306#103—28no—
307#103—28no—
308#103—44no—
309#103—44no—
310#103—44no—
311#103—44no—
312#102—28no—
313#102—28no—
314#102—44no—
315#102—44no—
316#102—44no—
317#102—44no—
318C22⋊C8—28no—
319C22⋊C8—24no—
320C22⋊C8—24no—
321#97—42no—
322#97—42no—
323C4×C8—42no—
324C4×C8—42no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—24no—
333#95C2×D414yes—
334#95C2×D414yes—
335C24—24no—
336C24—22no—
337C24—21no—
338C24—42no—
339C24—41no—
340C24—41no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—82no—
345C24—82no—
346C24#12414yes—
347C24C22×D411yescharacteristic
348C24C22×D412yes—
349#41—232no—
350#41—416no—
351#41—416no—
352#41—416no—
353#41—464no—
354#41—88no—
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—416no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—84no—
366C2×Q8—82no—
367C2×Q8H(5,𝔽2)14yes—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—22no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—416no—
376C2×D4—48no—
377C2×D4—42no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—48no—
381C2×D4—432no—
382C2×D4—48no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4H(5,𝔽2)14yes—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—416no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—432no—
414C22×C4—42no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4#12418yes—
424C22×C4#12414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D414yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C4C2511yescharacteristic
434M16—88no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—42no—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—82no—
459C22⋊C4—44no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—42no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—44no—
490C23—48no—
491C23—48no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—88no—
502C23—84no—
503C23—84no—
504C23—82no—
505C23—82no—
506C23—84no—
507C23—88no—
508C23C2×#12412yes—
509C23C2×#12414yes—
510C23C23×D411yescharacteristic
511Q8—28no—
512Q8—48no—
513Q8—44no—
514Q8—44no—
515Q8—164no—
516D4—28no—
517D4—48no—
518D4—432no—
519D4—44no—
520D4—44no—
521D4—84no—
522D4—88no—
523D4—164no—
524D4—164no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—432no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—84no—
559C2×C4—164no—
560C2×C4—164no—
561C2×C4C2×#12412yes—
562C2×C4C2×#12414yes—
563C2×C4C22×#4112yes—
564C2×C4C2×H(5,𝔽2)12yes—
565C2×C4C2×H(5,𝔽2)12yes—
566C8—44no—
567C8—44no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—42no—
574C22—41no—
575C22—42no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—88no—
584C22—816no—
585C22—84no—
586C22—84no—
587C22—81no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22—82no—
592C22C22×#12411yescharacteristic
593C22#348411yescharacteristic
594C22#348411yescharacteristic
595C22#358512yes—
596C22#358612yes—
597C4—24no—
598C4—22no—
599C4—44no—
600C4—42no—
601C4—42no—
602C4—44no—
603C4—81no—
604C4—82no—
605C4—84no—
606C4—84no—
607C4#349014yes—
608C2—44no—
609C2—42no—
610C2—84no—
611C2—161no—
612C2#3536711yescharacteristic
613C2#6062112yes—
614C2#6164811yescharacteristic
615C2#6164911yescharacteristic
616C2#6165012yes—
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. #37477C2semidirect
3. #29118C2semidirect
4. #29115C2semidirect
5. #27284C2semidirect
6. #27278C2semidirect
7. #27276C2semidirect
8. #27270C2semidirect
9. #25868C2semidirect
10. #25865C2semidirect
11. #21769C2semidirect
12. #20601C2semidirect
13. #20529C2semidirect
18. C2×#491C22semidirect
19. C2×#489C22semidirect
20. C2×#488C22semidirect
21. C2×#486C22semidirect
32. #2442C22semidirect
33. #2440C22semidirect
43. #2074C22semidirect
44. #2071C22semidirect
45. #2043C22semidirect
46. #2042C22semidirect
47. #2042C22semidirect
48. #2041C22semidirect
49. #2025C22semidirect
50. #2024C22semidirect
51. #1961C22semidirect
52. #1959C22semidirect
53. #1958C22semidirect
54. #1956C22semidirect
55. #1936C22semidirect
56. #1935C22semidirect
57. #1934C22semidirect
58. #1933C22semidirect
59. #1895C22semidirect
60. #1894C22semidirect
61. #1874C22semidirect
62. #1874C22semidirect
63. #1872C22semidirect
64. #1872C22semidirect
71. #1753C22semidirect
72. #1753C22semidirect

Maximal subgroups

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

subgroupindex
#374772
#351492
#291182
#291152
#272842
#272782
#272762
#272702
#258682
#258652
#217692
#206012
#205292