Finite Group Databasebeta

← #6633292

Subgroups — #6633292

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37421C211yesmaximal, characteristic, complemented
2#35554C211yesmaximal, characteristic, complemented
3#32914C211yesmaximal, characteristic, complemented
4#32911C211yesmaximal, characteristic, complemented
5#31665C214yesmaximal, complemented
6#31661C214yesmaximal, complemented
7#30761C214yesmaximal, complemented
8#30753C214yesmaximal, complemented
9#29723C211yesmaximal, characteristic, complemented
10#29721C211yesmaximal, characteristic, complemented
11#29339C212yesmaximal, complemented
12#25725C212yesmaximal, complemented
13#23945C212yesmaximal, complemented
14#23945C212yesmaximal, complemented
15#21917C211yesmaximal, characteristic, complemented
16C2×#542C2212yes—
17C22×#127C2211yescharacteristic
18C2×#518C2211yescharacteristic
19C2×#496C2214yes—
20C2×#461C2212yescomplemented
21C2×#459C2212yescomplemented
22C2×#450C2212yes—
23C2×#450C2212yes—
24C2×#447C2212yes—
25C2×#447C2212yes—
26#2850C2212yes—
27#2712—22no—
28#2702—24no—
29#2693—22no—
30#2621C2211yescharacteristic
31#2619C2211yescharacteristic
32#2568C2212yescomplemented
33#2563C2212yescomplemented
34#2545C2212yescomplemented
35#2544C2212yescomplemented
36#2445—24no—
37#2445C2214yes—
38#2445C2214yes—
39#2405—22no—
40#2405—22no—
41#2309C2211yescharacteristic
42#2309C2211yescharacteristic
43#2309C2211yescharacteristic
44#2296C2211yescharacteristic
45#2296C2211yescharacteristic
46#2101C2214yescomplemented
47#2073C2212yescomplemented
48#2073C2212yescomplemented
49#2055C2214yescomplemented
50#2054C2214yescomplemented
51#2042C2214yescomplemented
52#2023C2214yescomplemented
53#2023C2214yescomplemented
54#2014C2214yes—
55#2013C2212yescomplemented
56#2012C2214yescomplemented
57#2012C2214yescomplemented
58#2011C2214yes—
59#2011C2212yescomplemented
60#1961C2214yes—
61#1936C2214yescomplemented
62#1935C2214yescomplemented
63#1917C2214yescomplemented
64#1917C2214yescomplemented
65#1880C2214yescomplemented
66#1880C2214yescomplemented
67#1879C2214yescomplemented
68#1879C2214yescomplemented
69#1879C2214yescomplemented
70#1877C2214yescomplemented
71#1875C2212yes—
72#1875C2212yes—
73#1873C2212yes—
74#1873C2212yes—
75#1818C2212yes—
76#1804C2212yescomplemented
77#1804C2212yescomplemented
78#1798C2211yescharacteristic
79#1798C2211yescharacteristic
80#1749C2212yes—
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×#127C2314yes—
95C2×#127C2311yescharacteristic
96C2×#126—22no—
97C2×#124—22no—
98C2×#124C2312yes—
99C2×#124C2312yes—
100C2×#123—24no—
101C2×#123—24no—
102C2×#123—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8C2312yes—
105C2×#121C2312yes—
106#518—44no—
107C2×C4×Q8C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C22⋊C4—22no—
117C22×C42C2311yescharacteristic
118#496—48no—
119#461—44no—
120#459—44no—
121#450—44no—
122#450—44no—
123#447—44no—
124#447—44no—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2314yes—
138#400—24no—
139#398—22no—
140#397—22no—
141#396—24no—
142#394—22no—
143#394—24no—
144#394—24no—
145#394—24no—
146#394—24no—
147#394—24no—
148#393—24no—
149#392—22no—
150#391—24no—
151#391C2312yes—
152#391C2312yes—
153#390—22no—
154#390—22no—
155#390—22no—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#390C2314yes—
161#390C2314yes—
162#389D414yes—
163#389C2314yes—
164#388—24no—
165#388—24no—
166#386—24no—
167#386—24no—
168#386—24no—
169#386—24no—
170#385—22no—
171#385D412yes—
172#385D412yes—
173#382C2311yescharacteristic
174#382C2312yes—
175#382C2311yescharacteristic
176#380—24no—
177#379—21no—
178#379—21no—
179#379—21no—
180#379—21no—
181#379—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2312yes—
190#336C2312yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
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—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—42no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—48no—
213C22×D4—88no—
214C22×D4C2412yes—
215C22×D4C2412yes—
216C22×D4C2412yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4C2×D412yes—
221C23×C4C2411yescharacteristic
222C2×SD16—28no—
223C2×SD16—44no—
224C2×SD16—44no—
225C2×SD16—44no—
226C2×SD16—44no—
227C2×SD16—44no—
228C2×D8—28no—
229C2×D8—44no—
230C2×D8—44no—
231C2×D8—44no—
232C2×D8—44no—
233C2×D8—44no—
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—42no—
237C4⋊Q8—42no—
238C4⋊Q8—42no—
239#127—28no—
240#127—44no—
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—
254#121—24no—
255#121—24no—
256C4×Q8—42no—
257C4×Q8—42no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—42no—
263C4×D4—42no—
264C4×D4—48no—
265#117—22no—
266#117—22no—
267#117—42no—
268#117—42no—
269#117—42no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4C2×D411yescharacteristic
303C2×C22⋊C4C2×D411yescharacteristic
304C2×C22⋊C4C2×D412yes—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—42no—
309C2×C42C2×D412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2412yes—
313C2×C42C2412yes—
314C4⋊C8—42no—
315C4⋊C8—42no—
316#103—44no—
317#103—44no—
318#102—44no—
319#102—44no—
320#102—44no—
321#102—44no—
322#102—44no—
323#102—44no—
324C22⋊C8—24no—
325C22⋊C8—24no—
326#97—44no—
327#95—24no—
328#95—22no—
329#95—24no—
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—84no—
345C24—84no—
346C24C22×D411yescharacteristic
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—44no—
352C2×Q8—84no—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—42no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—82no—
384C2×D4—84no—
385C2×D4—832no—
386C2×D4—84no—
387C2×D4H(5,𝔽2)14yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4C22×D414yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D411yescharacteristic
409C22×C4H(5,𝔽2)12yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C4C2511yescharacteristic
412SD16—88no—
413SD16—88no—
414D8—88no—
415D8—88no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—82no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C42—24no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—44no—
468C23—24no—
469C23—21no—
470C23—21no—
471C23—24no—
472C23—48no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—84no—
488C23—88no—
489C23—88no—
490C23—164no—
491C23—164no—
492C23—164no—
493C23—164no—
494C23#53512yes—
495C23C23×D411yescharacteristic
496Q8—44no—
497Q8—44no—
498Q8—164no—
499D4—24no—
500D4—24no—
501D4—48no—
502D4—42no—
503D4—42no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—44no—
508D4—84no—
509D4—816no—
510D4—816no—
511D4—164no—
512D4—164no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—48no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—164no—
541C2×C4—164no—
542C2×C4—164no—
543C2×C4#53512yes—
544C2×C4C2×H(5,𝔽2)11yescharacteristic
545C2×C4C2×H(5,𝔽2)11yescharacteristic
546C2×C4C2×H(5,𝔽2)12yes—
547C2×C4#58512yes—
548C8—44no—
549C8—44no—
550C22—22no—
551C22—24no—
552C22—42no—
553C22—42no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—84no—
558C22—84no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—84no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—82no—
568C22—82no—
569C22—168no—
570C22—168no—
571C22C2×#53511yescharacteristic
572C22#348411yescharacteristic
573C22#348711yescharacteristic
574C22#358514yes—
575C4—22no—
576C4—42no—
577C4—42no—
578C4—44no—
579C4—42no—
580C4—42no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—82no—
586C4#353412yes—
587C4#353412yes—
588C2—42no—
589C2—84no—
590C2—162no—
591C2—162no—
592C2#3671011yescharacteristic
593C2#6082512yes—
594C2#6164412yes—
595C2#6194512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #37421C2semidirect
2. #35554C2semidirect
3. #32914C2semidirect
4. #32911C2semidirect
5. #31665C2semidirect
6. #31661C2semidirect
7. #30761C2semidirect
8. #30753C2semidirect
9. #29723C2semidirect
10. #29721C2semidirect
11. #29339C2semidirect
12. #25725C2semidirect
13. #23945C2semidirect
14. #23945C2semidirect
15. #21917C2semidirect
20. C2×#461C22semidirect
21. C2×#459C22semidirect
32. #2568C22semidirect
33. #2563C22semidirect
34. #2545C22semidirect
35. #2544C22semidirect
46. #2101C22semidirect
47. #2073C22semidirect
48. #2073C22semidirect
49. #2055C22semidirect
50. #2054C22semidirect
51. #2042C22semidirect
52. #2023C22semidirect
53. #2023C22semidirect
55. #2013C22semidirect
56. #2012C22semidirect
57. #2012C22semidirect
59. #2011C22semidirect
61. #1936C22semidirect
62. #1935C22semidirect
63. #1917C22semidirect
64. #1917C22semidirect
65. #1880C22semidirect
66. #1880C22semidirect
67. #1879C22semidirect
68. #1879C22semidirect
69. #1879C22semidirect
70. #1877C22semidirect
76. #1804C22semidirect
77. #1804C22semidirect

Maximal subgroups

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

subgroupindex
#374212
#355542
#329142
#329112
#316652
#316612
#307612
#307532
#297232
#297212
#293392
#257252
#239452
#219172