Finite Group Databasebeta

← #4933813

Subgroups — #4933813

Order 512.

3658 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2435C211yesmaximal, characteristic
3#16559C212yesmaximal, complemented
4#16368C214yesmaximal, complemented
5#15820C212yesmaximal, complemented
6#15670C212yesmaximal, complemented
7#15467C214yesmaximal, complemented
8#15391C214yesmaximal, complemented
9#15050C212yesmaximal, complemented
10#14727C214yesmaximal, complemented
11#14473C212yesmaximal, complemented
12#13867C212yesmaximal
13#13832C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19#2804—28no—
20#2435—28no—
21C2×#400C2212yescomplemented
22C2×#397C2212yes—
23C2×#397C2212yes—
24C2×#394C2214yes—
25C2×#393C2212yes—
26C2×#393C2214yescomplemented
27C2×#393C2214yescomplemented
28C2×#391C2212yescomplemented
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#385C2214yescomplemented
36C2×#382C2212yescomplemented
37C2×#379C2214yes—
38C2×#379C2212yes—
39C2×#379C2214yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41#1446C2214yescomplemented
42#1446C2212yescomplemented
43#1445C2212yescomplemented
44#1445C2212yescomplemented
45#1443C2214yescomplemented
46#1442C2214yescomplemented
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yescomplemented
50#1440C2214yescomplemented
51#1440C2214yescomplemented
52#1440C2214yescomplemented
53#1440C2214yescomplemented
54#1440C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1438C2214yes—
58#1438C2214yes—
59#1437C2212yes—
60#1437C2214yescomplemented
61#1437C2214yescomplemented
62#1436C2212yescomplemented
63#1434C2212yescomplemented
64#1434C2214yescomplemented
65C22⋊#95C2214yes—
66C22⋊#95C2214yescomplemented
67C22⋊#95C2214yescomplemented
68C22⋊#95C2214yescomplemented
69C26C2312yes—
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C23×D4—24no—
73C23×D4—28no—
74C23×D4—42no—
75C23×D4—42no—
76C23×D4—44no—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×#124—24no—
84C2×#124—28no—
85C2×#124—24no—
86C2×#124—44no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—42no—
89C2×C22⋊Q8—42no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4—48no—
95C22×C22⋊C4—44no—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115#400—28no—
116#397—44no—
117#397—44no—
118#394—216no—
119#393—216no—
120#393—44no—
121#393—48no—
122#391—28no—
123#390—24no—
124#390D418yes—
125#386—28no—
126#386—28no—
127#386—44no—
128#386—44no—
129#385—216no—
130#382—44no—
131#379—216no—
132#379—48no—
133#379—44no—
134C4×C22⋊C4—24no—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C25—24no—
157C25—24no—
158C25—24no—
159C25—24no—
160C25—24no—
161C25—28no—
162C25—42no—
163C25—42no—
164C25—44no—
165C25—44no—
166C25—44no—
167C25C2412yes—
168C25C2412yes—
169C25C2411yescharacteristic
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—42no—
176C22×Q8—44no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×Q8—44no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—42no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—416no—
196C22×D4—432no—
197C22×D4—416no—
198C22×D4—432no—
199C22×D4—416no—
200C22×D4C2×D414yes—
201C22×D4#4118yes—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#124—416no—
228#124—416no—
229C22⋊Q8—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C2×D414yes—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—432no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C2×D4116yes—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42C2×D414yes—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—42no—
316#95—44no—
317#95—44no—
318C24—21no—
319C24—22no—
320C24—21no—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—22no—
331C24—24no—
332C24—216no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—48no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—416no—
348C24—416no—
349C24—416no—
350C24—416no—
351C24—48no—
352C24—48no—
353C24—48no—
354C24—48no—
355C24—88no—
356C24—88no—
357C24—88no—
358C24—84no—
359C24—88no—
360C24—84no—
361C24—88no—
362C24—88no—
363C24—88no—
364C24—88no—
365C24C22×D414yes—
366C24C2×#4114yes—
367C24C2×#4114yes—
368C24H(5,𝔽2)14yes—
369C24H(5,𝔽2)14yes—
370C24H(5,𝔽2)12yes—
371C24C2511yescharacteristic
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—84no—
377C2×Q8—84no—
378C2×Q8—84no—
379C2×Q8—84no—
380C2×Q8—88no—
381C2×Q8—88no—
382C2×D4—28no—
383C2×D4—216no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—44no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—4128no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4#124116yes—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—28no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
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—44no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—84no—
461C22×C4—88no—
462C22×C4—84no—
463C22×C4—88no—
464C22×C4—84no—
465C22×C4—88no—
466C22×C4—84no—
467C22×C4—88no—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C2×#4114yes—
472C22×C4H(5,𝔽2)12yes—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—88no—
485C22⋊C4—84no—
486C22⋊C4—88no—
487C22⋊C4—84no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C42—44no—
494C42—48no—
495C42—44no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—24no—
513C23—28no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—432no—
522C23—432no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—816no—
559C23—832no—
560C23—816no—
561C23—816no—
562C23—832no—
563C23—832no—
564C23—816no—
565C23—816no—
566C23—832no—
567C23C2×#12418yes—
568C23H(3,𝔽4)14yes—
569C23C23×D411yescharacteristic
570C23C22×#4112yes—
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×H(5,𝔽2)12yes—
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×H(5,𝔽2)12yes—
575C23#58514yes—
576C23#58512yes—
577Q8—88no—
578Q8—816no—
579D4—416no—
580D4—416no—
581D4—432no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—416no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
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—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—816no—
641C2×C4—816no—
642C2×C4C2×#12414yes—
643C22—28no—
644C22—216no—
645C22—28no—
646C22—24no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—48no—
657C22—48no—
658C22—48no—
659C22—48no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—48no—
666C22—48no—
667C22—864no—
668C22—82no—
669C22—84no—
670C22—82no—
671C22—82no—
672C22—84no—
673C22—82no—
674C22C22×#12411yescharacteristic
675C22#348411yescharacteristic
676C22#348411yescharacteristic
677C22#348512yes—
678C22#348812yes—
679C22#349012yes—
680C22#349012yes—
681C22#351314yes—
682C22#352114yes—
683C22C2×H(3,𝔽4)11yescharacteristic
684C22#353312yes—
685C22#353414yes—
686C22#354012yes—
687C22#354511yescharacteristic
688C22#354514yes—
689C22#355812yes—
690C4—24no—
691C4—42no—
692C4—44no—
693C4—44no—
694C4—84no—
695C4—82no—
696C4—82no—
697C4—84no—
698C4—84no—
699C4—84no—
700C2—48no—
701C2—44no—
702C2—84no—
703C2#3536711yescharacteristic
704C2#3584712yes—
705C2#4556812yes—
706C2#4597711yescharacteristic
707C2#4849114yes—
708C2#4899712yes—
709C2#4955012yes—
710C2#5205411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2804C2semidirect
3. #16559C2semidirect
4. #16368C2semidirect
5. #15820C2semidirect
6. #15670C2semidirect
7. #15467C2semidirect
8. #15391C2semidirect
9. #15050C2semidirect
10. #14727C2semidirect
11. #14473C2semidirect
13. #13832C2semidirect
21. C2×#400C22semidirect
26. C2×#393C22semidirect
27. C2×#393C22semidirect
28. C2×#391C22semidirect
35. C2×#385C22semidirect
36. C2×#382C22semidirect
41. #1446C22semidirect
42. #1446C22semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1443C22semidirect
46. #1442C22semidirect
49. #1440C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
54. #1440C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
60. #1437C22semidirect
61. #1437C22semidirect
62. #1436C22semidirect
63. #1434C22semidirect
64. #1434C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#24352
#165592
#163682
#158202
#156702
#154672
#153912
#150502
#147272
#144732
#138672
#138322