Finite Group Databasebeta

← #640431

Subgroups — #640431

Order 512.

3957 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2849C211yesmaximal, characteristic
2C22×#381C211yesmaximal, characteristic, complemented
3#16697C214yesmaximal, complemented
4#16690C214yesmaximal, complemented
5#15026C211yesmaximal, characteristic, complemented
6#15019C214yesmaximal, complemented
7#14955C211yesmaximal, characteristic, complemented
8#14946C211yesmaximal, characteristic, complemented
9#14945C211yesmaximal, characteristic, complemented
10#14927C214yesmaximal, complemented
11#14265C212yesmaximal, complemented
12#14259C212yesmaximal, complemented
13#14134C212yesmaximal
14#13877C212yesmaximal, complemented
15C2×C22⋊#95C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2212yes—
18C22×#120C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21#2849—28no—
22C2×#386C2212yes—
23C2×#386C2211yescharacteristic
24C2×#386C2211yescharacteristic
25C2×#386C2212yes—
26C2×#386C2214yes—
27C2×#386C2212yes—
28C2×#385C2212yes—
29C2×#385C2211yescharacteristic
30C2×#385C2211yescharacteristic
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#381—28no—
34C2×#381C2212yes—
35C2×#381C2214yes—
36C2×#381C2214yes—
37C2×#380C2214yes—
38C2×#379C2211yescharacteristic
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42C22×#95C2214yes—
43#1445C2214yescomplemented
44#1445C2214yescomplemented
45#1443C2214yescomplemented
46#1443C2214yescomplemented
47#1443C2214yescomplemented
48#1443C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1440C2214yes—
56#1440C2214yescomplemented
57#1440C2214yescomplemented
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1438C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2212yes—
73C26C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4—28no—
78C24×C4—48no—
79C24×C4D418yes—
80C24×C4C2312yes—
81C24×C4C2314yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#123—28no—
88C2×#123—48no—
89C2×#120—24no—
90C2×#120—28no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4—48no—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116#386—24no—
117#386—24no—
118#386—28no—
119#386—28no—
120#386—48no—
121#386—44no—
122#385—24no—
123#385—24no—
124#385—28no—
125#385—28no—
126#385—44no—
127#381—28no—
128#381—216no—
129#381—216no—
130#380—48no—
131#379—24no—
132C2×#95—24no—
133C2×#95—24no—
134C2×#95—28no—
135C2×#95—48no—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C25—24no—
161C25—22no—
162C25—28no—
163C25—22no—
164C25—44no—
165C25C2×D414yes—
166C25#4118yes—
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—28no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—44no—
182C23×C4—232no—
183C23×C4—28no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—28no—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—432no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—464no—
212C23×C4C2×D418yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D418yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227#123—432no—
228#120—416no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—432no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—432no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4#4118yes—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—28no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305C24—22no—
306C24—24no—
307C24—21no—
308C24—21no—
309C24—24no—
310C24—24no—
311C24—28no—
312C24—216no—
313C24—28no—
314C24—232no—
315C24—216no—
316C24—216no—
317C24—22no—
318C24—21no—
319C24—24no—
320C24—21no—
321C24—24no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—21no—
327C24—21no—
328C24—24no—
329C24—48no—
330C24—416no—
331C24—416no—
332C24—416no—
333C24—416no—
334C24—416no—
335C24#123116yes—
336C24C22×D412yes—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C2×#4114yes—
340C24C2×#4114yes—
341C24C2×#4114yes—
342C24C2×#4114yes—
343C24H(5,𝔽2)18yes—
344C24H(5,𝔽2)12yes—
345C242−1+412yes—
346C24C2511yescharacteristic
347C2×D4—48no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—48no—
353C2×D4—44no—
354C2×D4—416no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—88no—
358C22×C4—216no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—4128no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—432no—
429C22×C4—432no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—416no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C2×#4118yes—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C22⋊C4—216no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—416no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—416no—
462C22⋊C4—48no—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C23—22no—
468C23—24no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—24no—
474C23—24no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—232no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—24no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—232no—
495C23—416no—
496C23—432no—
497C23—464no—
498C23—464no—
499C23—432no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23C2×#12314yes—
518C23C2×#12314yes—
519C23C2×#12314yes—
520C23#53214yes—
521C23#53418yes—
522C23#53714yes—
523C23#538116yes—
524C23#55118yes—
525C23#55218yes—
526C23C23×D411yescharacteristic
527C23C22×#4112yes—
528C23C22×#4112yes—
529C23C2×H(5,𝔽2)14yes—
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×2−1+411yescharacteristic
532C23#58514yes—
533D4—88no—
534D4—88no—
535D4—88no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—416no—
546C2×C4—416no—
547C2×C4—432no—
548C2×C4—432no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—832no—
598C2×C4—816no—
599C2×C4—816no—
600C22—24no—
601C22—24no—
602C22—28no—
603C22—28no—
604C22—216no—
605C22—216no—
606C22—216no—
607C22—216no—
608C22—24no—
609C22—28no—
610C22—28no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—464no—
616C22—44no—
617C22—48no—
618C22—48no—
619C22—44no—
620C22—81no—
621C22—82no—
622C22—84no—
623C22—84no—
624C22—81no—
625C22—82no—
626C22—81no—
627C22#249818yes—
628C22#262218yes—
629C22C22×#12311yescharacteristic
630C22C2×#53211yescharacteristic
631C22C2×#53412yes—
632C22C2×#53711yescharacteristic
633C22#344912yes—
634C22C2×#53814yes—
635C22C2×#55112yes—
636C22C2×#55212yes—
637C22#349114yes—
638C22#351314yes—
639C22#351314yes—
640C22#353414yes—
641C22#353514yes—
642C4—48no—
643C4—44no—
644C4—44no—
645C4—81no—
646C4—82no—
647C4—84no—
648C4—84no—
649C4—81no—
650C4—82no—
651C4—84no—
652C4—81no—
653C2—28no—
654C2—28no—
655C2—44no—
656C2—161no—
657C2C2×#262211yescharacteristic
658C2C2×#249811yescharacteristic
659C2#3506412yes—
660C2#3509711yescharacteristic
661C2#3745212yes—
662C2#3805314yes—
663C2#3806514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#381C2semidirect
3. #16697C2semidirect
4. #16690C2semidirect
5. #15026C2semidirect
6. #15019C2semidirect
7. #14955C2semidirect
8. #14946C2semidirect
9. #14945C2semidirect
10. #14927C2semidirect
11. #14265C2semidirect
12. #14259C2semidirect
14. #13877C2semidirect
15. C2×C22⋊#95C2semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1443C22semidirect
46. #1443C22semidirect
47. #1443C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
56. #1440C22semidirect
57. #1440C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28492
C22×#3812
#166972
#166902
#150262
#150192
#149552
#149462
#149452
#149272
#142652
#142592
#141342
#138772
C2×C22⋊#952