Finite Group Databasebeta

← #6633286

Subgroups — #6633286

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37299C211yesmaximal, characteristic, complemented
2#35554C211yesmaximal, characteristic, complemented
3#32917C212yesmaximal, complemented
4#31638C214yesmaximal, complemented
5#31634C214yesmaximal, complemented
6#30743C214yesmaximal, complemented
7#30735C214yesmaximal, complemented
8#29727C212yesmaximal, complemented
9#29332C211yesmaximal, characteristic, complemented
10#29331C211yesmaximal, characteristic, complemented
11#25721C211yesmaximal, characteristic
12#25717C211yesmaximal, characteristic, complemented
13#23945C212yesmaximal, complemented
14#23945C212yesmaximal, complemented
15#21913C211yesmaximal, characteristic, complemented
16C2×#542C2212yes—
17C22×#127C2211yescharacteristic
18C2×#518C2211yescharacteristic
19C2×#493C2212yes—
20C2×#492C2212yes—
21C2×#463C2214yescomplemented
22C2×#450C2212yes—
23C2×#450C2212yes—
24C2×#447C2212yes—
25C2×#447C2212yes—
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2727—22no—
29#2683—22no—
30#2672—24no—
31#2620C2212yes—
32#2567C2212yescomplemented
33#2545C2212yescomplemented
34#2544C2212yescomplemented
35#2542C2212yescomplemented
36#2445—22no—
37#2445—22no—
38#2445C2214yes—
39#2445C2214yes—
40#2405—24no—
41#2309C2211yescharacteristic
42#2309C2211yescharacteristic
43#2309C2211yescharacteristic
44#2296C2211yescharacteristic
45#2296C2211yescharacteristic
46#2100C2212yescomplemented
47#2099C2212yescomplemented
48#2071C2214yescomplemented
49#2055C2214yescomplemented
50#2054C2214yescomplemented
51#2043C2212yescomplemented
52#2041C2212yescomplemented
53#2023C2214yescomplemented
54#2023C2214yescomplemented
55#2014C2214yes—
56#2012C2214yescomplemented
57#2012C2214yescomplemented
58#2012C2214yescomplemented
59#2011C2214yes—
60#1957C2212yes—
61#1957C2212yes—
62#1934C2214yescomplemented
63#1933C2214yescomplemented
64#1916C2214yescomplemented
65#1916C2214yescomplemented
66#1880C2214yescomplemented
67#1880C2214yescomplemented
68#1879C2214yescomplemented
69#1879C2214yescomplemented
70#1879C2214yescomplemented
71#1877C2214yescomplemented
72#1875C2212yes—
73#1875C2212yes—
74#1873C2212yes—
75#1873C2212yes—
76#1818C2212yes—
77#1802C2214yescomplemented
78#1798C2212yes—
79C4×C22⋊C8C2211yescharacteristic
80C4×C22⋊C8C2211yescharacteristic
81C23×D4—22no—
82C23×D4C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C22×D8C2314yes—
87C22×D8C2314yes—
88#542—44no—
89#542—44no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2312yes—
92C2×#127—22no—
93C2×#127—44no—
94C2×#127C2312yes—
95C2×#127C2312yes—
96C2×#127C2311yescharacteristic
97C2×#126—22no—
98C2×#124—22no—
99C2×#124C2312yes—
100C2×#124C2312yes—
101C2×#123—24no—
102C2×#123—24no—
103C2×#123—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8C2312yes—
107C2×#121—22no—
108C2×#121C2312yes—
109#518—44no—
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4C2314yes—
113C2×C4×D4C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C22×C22⋊C4—22no—
119C22×C42C2311yescharacteristic
120#493—44no—
121#492—44no—
122#463—48no—
123#450—44no—
124#450—44no—
125#447—44no—
126#447—44no—
127C2×C4⋊C8C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140#399—24no—
141#396—24no—
142#396—24no—
143#395—24no—
144#394—24no—
145#394—24no—
146#394—24no—
147#394—24no—
148#394—24no—
149#394—24no—
150#391C2312yes—
151#391C2312yes—
152#390—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—22no—
157#390—22no—
158#390C2314yes—
159#390C2314yes—
160#390C2314yes—
161#390C2312yes—
162#390C2312yes—
163#389—24no—
164#389—22no—
165#389D412yes—
166#389D412yes—
167#389C2312yes—
168#389C2312yes—
169#388—24no—
170#386—24no—
171#386—24no—
172#386—24no—
173#385—22no—
174#385D414yes—
175#384C2311yescharacteristic
176#384C2312yes—
177#384C2311yescharacteristic
178#380—22no—
179#379—22no—
180#379—22no—
181#379—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2312yes—
193#336C2312yes—
194C25—21no—
195C25—21no—
196C22×Q8—22no—
197C22×Q8C2412yes—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—42no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—48no—
211C22×D4—84no—
212C22×D4—84no—
213C22×D4C2412yes—
214C22×D4C2412yes—
215C22×D4C2412yes—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4C2×D412yes—
219C23×C4C2411yescharacteristic
220C2×SD16—28no—
221C2×SD16—44no—
222C2×SD16—44no—
223C2×SD16—44no—
224C2×SD16—44no—
225C2×SD16—44no—
226C2×D8—28no—
227C2×D8—44no—
228C2×D8—44no—
229C2×D8—44no—
230C2×D8—44no—
231C2×D8—44no—
232C22×C8C2414yes—
233C22×C8C2414yes—
234C4⋊Q8—42no—
235C4⋊Q8—42no—
236C4⋊Q8—42no—
237#127—24no—
238#127—24no—
239#127—42no—
240#127—42no—
241#127—42no—
242#127—48no—
243#126—48no—
244#124—42no—
245#124—42no—
246#124—44no—
247#124—48no—
248#123—48no—
249#123—88no—
250#123—88no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—48no—
254C22⋊Q8—88no—
255#121—24no—
256#121—24no—
257#121—84no—
258C4×Q8—42no—
259C4×Q8—42no—
260C4×D4—28no—
261C4×D4—44no—
262C4×D4—42no—
263C4×D4—42no—
264C4×D4—48no—
265#117—24no—
266#117—42no—
267#117—42no—
268#117—42no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4C2×D411yescharacteristic
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—48no—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—42no—
311C2×C42C2×D412yes—
312C2×C42C2412yes—
313C2×C42C2412yes—
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316C4⋊C8—44no—
317#103—44no—
318#103—44no—
319#102—44no—
320#102—44no—
321#102—44no—
322#102—44no—
323#102—44no—
324#102—44no—
325C22⋊C8—24no—
326C22⋊C8—24no—
327C4×C8—42no—
328C4×C8—42no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95C2×D414yes—
334#95C2×D414yes—
335C24—24no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—82no—
345C24—82no—
346C24—82no—
347C24—82no—
348C24C22×D411yescharacteristic
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—44no—
354C2×Q8—84no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—416no—
368C2×D4—42no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—88no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—832no—
385C2×D4—84no—
386C2×D4H(5,𝔽2)14yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4C22×D414yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C4C2511yescharacteristic
410SD16—88no—
411SD16—88no—
412D8—88no—
413D8—88no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—28no—
423C4⋊C4—28no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—82no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C22⋊C4—24no—
442C22⋊C4—42no—
443C22⋊C4—48no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—48no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C42—22no—
461C42—22no—
462C42—24no—
463C42—22no—
464C42—22no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—44no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—48no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—84no—
490C23—88no—
491C23—88no—
492C23—162no—
493C23—162no—
494C23—162no—
495C23—162no—
496C23—164no—
497C23—164no—
498C23#53512yes—
499C23C23×D411yescharacteristic
500Q8—44no—
501Q8—44no—
502Q8—164no—
503D4—28no—
504D4—48no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—44no—
509D4—44no—
510D4—816no—
511D4—816no—
512D4—84no—
513D4—164no—
514D4—164no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—48no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—168no—
544C2×C4—164no—
545C2×C4#53512yes—
546C2×C4#57614yes—
547C2×C4#57714yes—
548C2×C4C2×H(5,𝔽2)12yes—
549C2×C4C2×H(5,𝔽2)12yes—
550C2×C4#58512yes—
551C8—44no—
552C8—44no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—48no—
560C22—82no—
561C22—82no—
562C22—84no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—84no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—168no—
574C22—168no—
575C22C2×#53511yescharacteristic
576C22#348411yescharacteristic
577C22#348711yescharacteristic
578C22C2×#57612yes—
579C22C2×#57712yes—
580C4—22no—
581C4—24no—
582C4—24no—
583C4—42no—
584C4—42no—
585C4—44no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4#353414yes—
592C2—42no—
593C2—84no—
594C2—162no—
595C2—162no—
596C2#3671011yescharacteristic
597C2#6081711yescharacteristic
598C2#6082011yescharacteristic
599C2#6162912yes—
600C2#6193712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#25721C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37299C2semidirect
2. #35554C2semidirect
3. #32917C2semidirect
4. #31638C2semidirect
5. #31634C2semidirect
6. #30743C2semidirect
7. #30735C2semidirect
8. #29727C2semidirect
9. #29332C2semidirect
10. #29331C2semidirect
12. #25717C2semidirect
13. #23945C2semidirect
14. #23945C2semidirect
15. #21913C2semidirect
21. C2×#463C22semidirect
32. #2567C22semidirect
33. #2545C22semidirect
34. #2544C22semidirect
35. #2542C22semidirect
46. #2100C22semidirect
47. #2099C22semidirect
48. #2071C22semidirect
49. #2055C22semidirect
50. #2054C22semidirect
51. #2043C22semidirect
52. #2041C22semidirect
53. #2023C22semidirect
54. #2023C22semidirect
56. #2012C22semidirect
57. #2012C22semidirect
58. #2012C22semidirect
62. #1934C22semidirect
63. #1933C22semidirect
64. #1916C22semidirect
65. #1916C22semidirect
66. #1880C22semidirect
67. #1880C22semidirect
68. #1879C22semidirect
69. #1879C22semidirect
70. #1879C22semidirect
71. #1877C22semidirect
77. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#372992
#355542
#329172
#316382
#316342
#307432
#307352
#297272
#293322
#293312
#257212
#257172
#239452
#219132