Finite Group Databasebeta

← #5486629

Subgroups — #5486629

Order 512.

2840 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17135C211yesmaximal, characteristic, complemented
2#17093C214yesmaximal, complemented
3#16866C214yesmaximal, complemented
4#16526C214yesmaximal, complemented
5#16106C214yesmaximal, complemented
6#15948C211yesmaximal, characteristic, complemented
7#15268C214yesmaximal, complemented
8#15261C211yesmaximal, characteristic, complemented
9#15258C211yesmaximal, characteristic
10#14754C214yesmaximal, complemented
11#14661C211yesmaximal, characteristic, complemented
12#14657C211yesmaximal, characteristic, complemented
13#14167C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#123C2211yescharacteristic
16C22×#120C2211yescharacteristic
17C22×#120C2212yes—
18C2×#397C2214yescomplemented
19C2×#396C2214yes—
20C2×#396C2214yescomplemented
21C2×#394C2214yes—
22C2×#394C2214yes—
23C2×#394C2212yescomplemented
24C2×#392C2212yes—
25C2×#392C2214yes—
26C2×#390C2214yes—
27C2×#388C2214yes—
28C2×#386C2212yes—
29C2×#386C2212yescomplemented
30C2×#383C2212yescomplemented
31C2×#379C2212yes—
32C2×#379C2212yes—
33C2×C4×C22⋊C4C2212yescomplemented
34#1445C2214yescomplemented
35#1444C2214yescomplemented
36#1443C2214yescomplemented
37#1442C2214yescomplemented
38#1440C2214yes—
39#1440C2212yes—
40#1440C2212yes—
41#1440C2214yes—
42#1440C2212yescomplemented
43#1439C2214yes—
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yescomplemented
47#1439C2214yescomplemented
48#1438C2211yescharacteristic
49#1438C2211yescharacteristic
50#1438C2214yes—
51#1438C2211yescharacteristic
52#1438C2214yes—
53#1438C2212yes—
54#1438C2211yescharacteristic
55#1438C2214yes—
56#1438C2212yes—
57#1438C2214yes—
58#1438C2212yescomplemented
59#1438C2214yescomplemented
60#1438C2212yescomplemented
61#1437C2214yes—
62#1437C2212yes—
63#1437C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2212yescomplemented
66#1436C2214yescomplemented
67C26C2311yescharacteristic
68C23×D4C2314yes—
69C23×D4C2312yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C23×D4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C2×#127—24no—
76C2×#127—44no—
77C2×#123—24no—
78C2×#123—44no—
79C2×#120—24no—
80C2×#120—44no—
81C2×#120—48no—
82C2×#120—44no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111#397—48no—
112#396—48no—
113#396—48no—
114#394—48no—
115#394—48no—
116#394—44no—
117#392—48no—
118#392—44no—
119#390—48no—
120#388—48no—
121#386—44no—
122#386—44no—
123#383—44no—
124#379—44no—
125#379—44no—
126C4×C22⋊C4—44no—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C25—22no—
145C25—24no—
146C25—42no—
147C25—44no—
148C25—44no—
149C25—42no—
150C25C2412yes—
151C25C2411yescharacteristic
152C25C2411yescharacteristic
153C25C2412yes—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—28no—
159C22×D4—22no—
160C22×D4—24no—
161C22×D4—28no—
162C22×D4—24no—
163C22×D4—42no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—42no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—42no—
183C22×D4#4114yes—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197#127—416no—
198#123—416no—
199#120—416no—
200#120—432no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—44no—
284#95—42no—
285#95—44no—
286#95—42no—
287#95—44no—
288#95—42no—
289#95—44no—
290#95—44no—
291C24—28no—
292C24—22no—
293C24—24no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—22no—
300C24—24no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—22no—
315C24—24no—
316C24—22no—
317C24—24no—
318C24—24no—
319C24—48no—
320C24—48no—
321C24—82no—
322C24—82no—
323C24—88no—
324C24—84no—
325C24—88no—
326C24—88no—
327C24—84no—
328C24—84no—
329C24—84no—
330C24—88no—
331C24—88no—
332C24C2×#4112yes—
333C24C2×#4112yes—
334C24H(5,𝔽2)14yes—
335C24C2511yescharacteristic
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—416no—
342C2×D4—416no—
343C2×D4—48no—
344C2×D4—44no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—88no—
353C2×D4—88no—
354C2×D4—88no—
355C2×D4—88no—
356C2×D4—88no—
357C2×D4—88no—
358C2×D4—84no—
359C2×D4—88no—
360C2×D4—88no—
361C2×D4—88no—
362C2×D4—88no—
363C2×D4—88no—
364C2×D4—88no—
365C2×D4—88no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—88no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4C2×#4112yes—
419C22×C4H(5,𝔽2)14yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C4H(5,𝔽2)18yes—
422C4⋊C4—48no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C4⋊C4—88no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—416no—
429C22⋊C4—416no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C42—28no—
449C42—88no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—28no—
458C23—28no—
459C23—24no—
460C23—28no—
461C23—28no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—44no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—88no—
521C23—816no—
522C23—832no—
523C23—816no—
524C23—816no—
525C23—832no—
526C23C22×#4111yescharacteristic
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)12yes—
530C23C2×H(5,𝔽2)14yes—
531C23#58514yes—
532D4—88no—
533D4—832no—
534D4—816no—
535D4—816no—
536D4—88no—
537D4—816no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—88no—
591C22—48no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—48no—
604C22—832no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—82no—
616C22—84no—
617C22—84no—
618C22—82no—
619C22—82no—
620C22—84no—
621C22—84no—
622C22#349011yescharacteristic
623C22#349011yescharacteristic
624C22#349011yescharacteristic
625C22#351414yes—
626C22#352912yes—
627C22#353212yes—
628C22#353312yes—
629C22#353312yes—
630C22#353414yes—
631C22#353712yes—
632C22#353714yes—
633C22#354312yes—
634C22#354614yes—
635C22#354614yes—
636C4—48no—
637C4—44no—
638C4—42no—
639C4—44no—
640C4—84no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4—84no—
645C2—82no—
646C2—84no—
647C2—82no—
648C2—162no—
649C2#4574112yes—
650C2#4638411yescharacteristic
651C2#4897112yes—
652C2#5194512yes—
653C2#5247914yes—
654C2#5505414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15258C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17135C2semidirect
2. #17093C2semidirect
3. #16866C2semidirect
4. #16526C2semidirect
5. #16106C2semidirect
6. #15948C2semidirect
7. #15268C2semidirect
8. #15261C2semidirect
10. #14754C2semidirect
11. #14661C2semidirect
12. #14657C2semidirect
13. #14167C2semidirect
18. C2×#397C22semidirect
20. C2×#396C22semidirect
23. C2×#394C22semidirect
29. C2×#386C22semidirect
30. C2×#383C22semidirect
33. C2×C4×C22⋊C4C22semidirect
34. #1445C22semidirect
35. #1444C22semidirect
36. #1443C22semidirect
37. #1442C22semidirect
42. #1440C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
58. #1438C22semidirect
59. #1438C22semidirect
60. #1438C22semidirect
63. #1437C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect

Maximal subgroups

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

subgroupindex
#171352
#170932
#168662
#165262
#161062
#159482
#152682
#152612
#152582
#147542
#146612
#146572
#141672