Finite Group Databasebeta

← #851467

Subgroups — #851467

Order 512.

3118 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16719C214yesmaximal, complemented
2#16118C214yesmaximal, complemented
3#15282C214yesmaximal
4#15216C211yesmaximal, characteristic, complemented
5#15216C212yesmaximal, complemented
6#15195C211yesmaximal, characteristic, complemented
7#15132C212yesmaximal, complemented
8#14680C212yesmaximal
9#14577C214yesmaximal, complemented
10#13993C212yesmaximal, complemented
11#13974C212yesmaximal, complemented
12#13876C212yesmaximal, complemented
13C4×#379C211yesmaximal, characteristic
14C25×C4C2211yescharacteristic
15C22×#125C2211yescharacteristic, complemented
16C2×#397C2214yes—
17C2×#389C2212yescomplemented
18C2×#388C2214yes—
19C2×#388C2214yes—
20C2×#384C2211yescharacteristic, complemented
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#379C2212yes—
28C2×#379C2214yes—
29C2×C4×C4⋊C4C2212yescomplemented
30C2×C4×C22⋊C4C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1443C2214yescomplemented
35#1443C2214yescomplemented
36#1443C2214yescomplemented
37#1442C2214yescomplemented
38#1442C2214yescomplemented
39#1442C2214yescomplemented
40#1442C2214yescomplemented
41#1442C2214yescomplemented
42#1442C2214yescomplemented
43#1440C2214yes—
44#1440C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1437C2214yes—
52#1436C2214yescomplemented
53#1436C2214yescomplemented
54#1436C2214yescomplemented
55C4⋊#95C2211yescharacteristic, complemented
56C4⋊#95C2212yescomplemented
57C4⋊#95C2212yescomplemented
58C4⋊#95C2212yescomplemented
59C4⋊#95C2212yescomplemented
60C4⋊#95C2211yescharacteristic, complemented
61C22⋊#95C2212yes—
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C26C2311yescharacteristic
68C24×C4—28no—
69C24×C4—24no—
70C24×C4—42no—
71C24×C4—44no—
72C24×C4—42no—
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C2×#125—44no—
78C2×#125—42no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2314yes—
102#397—216no—
103#389—44no—
104#388—48no—
105#388—48no—
106#384—42no—
107#380—28no—
108#380—28no—
109#380—44no—
110#380—44no—
111#380—44no—
112#380—44no—
113#379—28no—
114#379—48no—
115C4×C4⋊C4—44no—
116C4×C22⋊C4—28no—
117C4×C22⋊C4—42no—
118C4×C22⋊C4—44no—
119C4×C22⋊C4—42no—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2311yescharacteristic
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2311yescharacteristic
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C25—22no—
146C25—24no—
147C25—24no—
148C25—24no—
149C25—24no—
150C25—42no—
151C25—41no—
152C25—41no—
153C25—42no—
154C25C2412yes—
155C25C2411yescharacteristic
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—42no—
172C23×C4—41no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—41no—
178C23×C4—416no—
179C23×C4—432no—
180C23×C4—432no—
181C23×C4—416no—
182C23×C4—416no—
183C23×C4#4118yes—
184C23×C4#4114yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196#125—416no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4#4118yes—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—44no—
264#95—24no—
265#95—24no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—41no—
275#95—42no—
276#95—42no—
277#95—41no—
278#95—42no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—21no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—28no—
309C24—216no—
310C24—44no—
311C24—48no—
312C24—48no—
313C24—48no—
314C24—48no—
315C24—48no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24—44no—
320C24—44no—
321C24—48no—
322C24—48no—
323C24—48no—
324C24—48no—
325C24—44no—
326C24—44no—
327C24—44no—
328C24—48no—
329C24C2×#4112yes—
330C24C2×#4114yes—
331C24C2×#4114yes—
332C24H(5,𝔽2)14yes—
333C242−1+414yes—
334C24C2511yescharacteristic
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—216no—
358C22×C4—28no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—216no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
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—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—4128no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4C2×#4114yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)14yes—
432C22×C4H(5,𝔽2)12yes—
433C4⋊C4—416no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—88no—
439C4⋊C4—88no—
440C22⋊C4—216no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—416no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C42—48no—
453C42—44no—
454C42—44no—
455C42—88no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—416no—
485C23—432no—
486C23—432no—
487C23—432no—
488C23—432no—
489C23—416no—
490C23—432no—
491C23—416no—
492C23—432no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23#54114yes—
520C23#55318yes—
521C23C22×#4111yescharacteristic
522C23C22×#4112yes—
523C23C22×#4112yes—
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×H(5,𝔽2)12yes—
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×H(5,𝔽2)12yes—
528C23C2×2−1+412yes—
529C23#58512yes—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—432no—
542C2×C4—416no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—432no—
559C2×C4—416no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4#53818yes—
603C2×C4#53814yes—
604C2×C4#55114yes—
605C2×C4#55118yes—
606C2×C4H(3,𝔽4)14yes—
607C22—24no—
608C22—28no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—464no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22#344411yescharacteristic
633C22C2×#53811yescharacteristic
634C22C2×#53812yes—
635C22C2×#54111yescharacteristic
636C22C2×#55111yescharacteristic
637C22C2×#55112yes—
638C22C2×#55312yes—
639C22#347512yes—
640C22#349212yes—
641C22#349412yes—
642C22#349512yes—
643C22#350612yes—
644C22#351912yes—
645C22#352012yes—
646C22#352012yes—
647C22C2×H(3,𝔽4)11yescharacteristic
648C22#353612yes—
649C22#353711yescharacteristic
650C22#353911yescharacteristic
651C22#354112yes—
652C22#354212yes—
653C4—44no—
654C4—42no—
655C4—44no—
656C4—44no—
657C4—82no—
658C4—84no—
659C4—84no—
660C4—84no—
661C4—84no—
662C4—82no—
663C4#273618yes—
664C2—44no—
665C2—48no—
666C2#3508611yescharacteristic
667C2#3620812yes—
668C2#3728311yescharacteristic
669C2#3785012yes—
670C2#3914412yes—
671C2#3993212yes—
672C2#4560112yes—
673C2C2×#273611yescharacteristic
674C2#5920612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16719C2semidirect
2. #16118C2semidirect
4. #15216C2semidirect
5. #15216C2semidirect
6. #15195C2semidirect
7. #15132C2semidirect
9. #14577C2semidirect
10. #13993C2semidirect
11. #13974C2semidirect
12. #13876C2semidirect
15. C22×#125C22semidirect
17. C2×#389C22semidirect
20. C2×#384C22semidirect
29. C2×C4×C4⋊C4C22semidirect
34. #1443C22semidirect
35. #1443C22semidirect
36. #1443C22semidirect
37. #1442C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1442C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
52. #1436C22semidirect
53. #1436C22semidirect
54. #1436C22semidirect
55. C4⋊#95C22semidirect
56. C4⋊#95C22semidirect
57. C4⋊#95C22semidirect
58. C4⋊#95C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
#167192
#161182
#152822
#152162
#151952
#151322
#146802
#145772
#139932
#139742
#138762
C4×#3792