Finite Group Databasebeta

← #918266

Subgroups — #918266

Order 512.

3264 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C22×#393C211yesmaximal, characteristic
3#17201C212yesmaximal, complemented
4#17198C212yesmaximal, complemented
5#16831C212yesmaximal, complemented
6#16823C212yesmaximal, complemented
7#16447C214yesmaximal, complemented
8#15586C214yesmaximal, complemented
9#14934C212yesmaximal, complemented
10#14919C212yesmaximal, complemented
11#14918C212yesmaximal
12#14336C212yesmaximal, complemented
13#13992C212yesmaximal, complemented
14#13869C212yesmaximal
15#13869C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21#2561—28no—
22C2×#397C2211yescharacteristic
23C2×#397C2211yescharacteristic
24C2×#393—28no—
25C2×#393C2211yescharacteristic
26C2×#393C2211yescharacteristic
27C2×#393C2212yes—
28C2×#393C2212yes—
29C2×#393C2212yes—
30C2×#393C2212yes—
31C2×#388C2212yes—
32C2×#388C2212yes—
33C2×#385C2212yes—
34C2×#381C2212yes—
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#379C2212yes—
41C2×#379C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2212yes—
45#1446C2214yescomplemented
46#1446C2214yescomplemented
47#1446C2214yescomplemented
48#1446C2214yescomplemented
49#1446C2214yescomplemented
50#1445C2214yescomplemented
51#1443C2214yescomplemented
52#1443C2214yescomplemented
53#1443C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2214yescomplemented
56#1442C2214yescomplemented
57#1442C2214yescomplemented
58#1442C2214yescomplemented
59#1440C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2212yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1436C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8—24no—
79C23×Q8—28no—
80C23×Q8—42no—
81C23×Q8—44no—
82C23×Q8—42no—
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—42no—
93C2×C22⋊Q8—42no—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—42no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4—42no—
102C22×C22⋊C4—42no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4D418yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C42C2312yes—
123#397—24no—
124#397—24no—
125#393—24no—
126#393—24no—
127#393—28no—
128#393—28no—
129#393—28no—
130#393—28no—
131#388—28no—
132#388—28no—
133#385—44no—
134#381—28no—
135#381—28no—
136#381—44no—
137#381—44no—
138#381—44no—
139#381—44no—
140#379—28no—
141#379—42no—
142C4×C22⋊C4—24no—
143C2×#95—28no—
144C2×#95—24no—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C25—24no—
175C25—24no—
176C25—22no—
177C25—24no—
178C25—22no—
179C25—42no—
180C25—44no—
181C25C2×D414yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—28no—
189C22×Q8—28no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8—41no—
197C22×Q8—41no—
198C22×Q8—44no—
199C22×Q8—432no—
200C22×Q8—416no—
201C22×Q8—416no—
202C22×Q8—432no—
203C22×Q8—416no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4#4118yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2414yes—
242C22⋊Q8—416no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—416no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—432no—
289C2×C22⋊C4—416no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D414yes—
300C2×C42—24no—
301C2×C42—44no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—28no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—28no—
349C24—216no—
350C24—44no—
351C24—48no—
352C24—416no—
353C24—48no—
354C24—48no—
355C24—416no—
356C24—48no—
357C24—44no—
358C24—44no—
359C24—88no—
360C24—88no—
361C24—84no—
362C24—84no—
363C24C22×D412yes—
364C24C22×D412yes—
365C24C22×D412yes—
366C24C2×#4114yes—
367C24H(5,𝔽2)12yes—
368C24H(5,𝔽2)12yes—
369C24H(5,𝔽2)14yes—
370C242−1+414yes—
371C24C2511yescharacteristic
372C2×Q8—416no—
373C2×Q8—416no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—416no—
379C2×Q8—416no—
380C2×Q8—44no—
381C2×Q8—4128no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—84no—
385C2×Q8—84no—
386C2×Q8—84no—
387C22×C4—28no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—88no—
469C22×C4—84no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—84no—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4118yes—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—88no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C42—84no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—24no—
506C23—24no—
507C23—28no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—416no—
525C23—432no—
526C23—42no—
527C23—41no—
528C23—44no—
529C23—41no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—416no—
535C23—41no—
536C23—44no—
537C23—44no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—816no—
546C23—88no—
547C23—832no—
548C23—816no—
549C23—816no—
550C23#53414yes—
551C23#53614yes—
552C23#53818yes—
553C23#54618yes—
554C23#55114yes—
555C23H(3,𝔽4)14yes—
556C23C23×D411yescharacteristic
557C23C22×#4112yes—
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)12yes—
561C23C2×2−1+412yes—
562C23#58512yes—
563C23#58514yes—
564Q8—432no—
565Q8—432no—
566Q8—88no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—416no—
583C2×C4—432no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—88no—
636C2×C4—816no—
637C2×C4—816no—
638C22—28no—
639C22—24no—
640C22—216no—
641C22—28no—
642C22—28no—
643C22—28no—
644C22—24no—
645C22—24no—
646C22—28no—
647C22—44no—
648C22—48no—
649C22—42no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—44no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—832no—
659C22—84no—
660C22—82no—
661C22—82no—
662C22#268418yes—
663C22C2×#53411yescharacteristic
664C22C2×#53611yescharacteristic
665C22#344911yescharacteristic
666C22C2×#53812yes—
667C22C2×#54612yes—
668C22C2×#55111yescharacteristic
669C22#348512yes—
670C22#349514yes—
671C22#351014yes—
672C22C2×H(3,𝔽4)11yescharacteristic
673C22#353512yes—
674C22#353512yes—
675C22#353514yes—
676C22#354514yes—
677C22#354512yes—
678C22#354512yes—
679C4—42no—
680C4—42no—
681C4—48no—
682C4—42no—
683C4—42no—
684C4—82no—
685C4—84no—
686C4—82no—
687C4—84no—
688C4—84no—
689C4—82no—
690C4—82no—
691C2—28no—
692C2—44no—
693C2—82no—
694C2#3711911yescharacteristic
695C2#3744711yescharacteristic
696C2#3783512yes—
697C2#3946112yes—
698C2#4231814yes—
699C2#4478612yes—
700C2C2×#268411yescharacteristic
701C2#5292512yes—
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
3. #17201C2semidirect
4. #17198C2semidirect
5. #16831C2semidirect
6. #16823C2semidirect
7. #16447C2semidirect
8. #15586C2semidirect
9. #14934C2semidirect
10. #14919C2semidirect
12. #14336C2semidirect
13. #13992C2semidirect
45. #1446C22semidirect
46. #1446C22semidirect
47. #1446C22semidirect
48. #1446C22semidirect
49. #1446C22semidirect
50. #1445C22semidirect
51. #1443C22semidirect
52. #1443C22semidirect
53. #1443C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1442C22semidirect
67. #1436C22semidirect
68. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C22×#3932
#172012
#171982
#168312
#168232
#164472
#155862
#149342
#149192
#149182
#143362
#139922
#138692