Finite Group Databasebeta

← #4386404

Subgroups — #4386404

Order 512.

3670 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2435C211yesmaximal, characteristic
2C2×#2308C211yesmaximal, characteristic
3#16391C214yesmaximal, complemented
4#16032C214yesmaximal, complemented
5#15702C214yesmaximal, complemented
6#15300C214yesmaximal, complemented
7#15287C214yesmaximal, complemented
8#14918C214yesmaximal, complemented
9#14660C211yesmaximal, characteristic, complemented
10#14297C211yesmaximal, characteristic, complemented
11#14292C211yesmaximal, characteristic, complemented
12#13953C211yesmaximal, characteristic, complemented
13#13902C211yesmaximal, characteristic
14C24×D4C2211yescharacteristic
15C22×#124C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18#2435—28no—
19C2×#397C2214yescomplemented
20C2×#394C2214yes—
21#2308—28no—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#385C2214yescomplemented
28C2×#381C2211yescharacteristic
29C2×#381C2211yescharacteristic
30C2×#381C2214yescomplemented
31C2×#381C2214yescomplemented
32C2×#380C2211yescharacteristic
33C2×#379C2212yes—
34C2×#379C2212yes—
35C2×#379C2212yes—
36C2×#379C2212yes—
37C2×#379C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40#1446C2212yescomplemented
41#1446C2212yescomplemented
42#1445C2212yescomplemented
43#1445C2212yescomplemented
44#1441C2214yescomplemented
45#1441C2214yescomplemented
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yescomplemented
49#1440C2214yescomplemented
50#1440C2214yescomplemented
51#1440C2214yescomplemented
52#1440C2214yescomplemented
53#1440C2214yescomplemented
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1438C2214yes—
62#1436C2212yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2212yescomplemented
66C4⋊#95C2212yescomplemented
67#1434C2212yescomplemented
68C22⋊#95C2214yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×Q8C2311yescharacteristic
72C23×D4—24no—
73C23×D4—24no—
74C23×D4—44no—
75C23×D4—44no—
76C23×D4D418yes—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C24×C4C2314yes—
80C24×C4C2311yescharacteristic
81C2×#124—28no—
82C2×#124—44no—
83C2×#124—44no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#397—48no—
116#394—216no—
117#390—28no—
118#390—44no—
119#390—44no—
120#386—216no—
121#386—216no—
122#385—48no—
123#381—24no—
124#381—24no—
125#381—48no—
126#381—48no—
127#380—24no—
128#379—28no—
129#379—28no—
130#379—28no—
131#379—44no—
132#379—44no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—24no—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C25—22no—
157C25—22no—
158C25—24no—
159C25—24no—
160C25—22no—
161C25—22no—
162C25—42no—
163C25—42no—
164C25—44no—
165C25—42no—
166C25—44no—
167C25—42no—
168C25—42no—
169C25—42no—
170C25C2×D414yes—
171C25C2×D414yes—
172C25C2412yes—
173C25C2411yescharacteristic
174C25C2412yes—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—232no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—432no—
194C22×D4—416no—
195C22×D4—416no—
196C22×D4—432no—
197C22×D4C2×D418yes—
198C22×D4C2×D418yes—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C2×D414yes—
209C23×C4#4114yes—
210C23×C4#4114yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221#124—432no—
222C22⋊Q8—416no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—432no—
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—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317C24—28no—
318C24—216no—
319C24—28no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—22no—
333C24—28no—
334C24—216no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—28no—
346C24—48no—
347C24—48no—
348C24—48no—
349C24—48no—
350C24—48no—
351C24—44no—
352C24—48no—
353C24—44no—
354C24—84no—
355C24—84no—
356C24—82no—
357C24—82no—
358C24—84no—
359C24—84no—
360C24—82no—
361C24—84no—
362C24—84no—
363C24—82no—
364C24—88no—
365C24—88no—
366C24—88no—
367C24—88no—
368C24—88no—
369C24—88no—
370C24—88no—
371C24—88no—
372C24C22×D414yes—
373C24C22×D412yes—
374C24C22×D414yes—
375C24C2×#4112yes—
376C24C2×#4112yes—
377C24H(5,𝔽2)14yes—
378C24H(5,𝔽2)14yes—
379C24C2511yescharacteristic
380C2×Q8—48no—
381C2×Q8—44no—
382C2×Q8—82no—
383C2×Q8—84no—
384C2×Q8—82no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—216no—
388C2×D4—216no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—4128no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
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—48no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4#123116yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4H(5,𝔽2)14yes—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—84no—
473C4⋊C4—88no—
474C4⋊C4—88no—
475C22⋊C4—48no—
476C22⋊C4—416no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—48no—
483C22⋊C4—416no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—84no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C42—48no—
493C42—82no—
494C42—82no—
495C23—232no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—22no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—24no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—416no—
525C23—44no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—88no—
558C23—88no—
559C23—832no—
560C23—88no—
561C23—88no—
562C23—816no—
563C23—816no—
564C23—816no—
565C23—816no—
566C23—832no—
567C23—816no—
568C23—816no—
569C23—816no—
570C23—816no—
571C23C2×#12314yes—
572C23#53418yes—
573C23C23×D411yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4111yescharacteristic
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)12yes—
578C23C2×H(5,𝔽2)12yes—
579C23#58514yes—
580C23#58512yes—
581Q8—88no—
582D4—432no—
583D4—432no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—416no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
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—44no—
641C2×C4—832no—
642C2×C4C2×#12314yes—
643C2×C4C2×#12314yes—
644C22—24no—
645C22—216no—
646C22—24no—
647C22—28no—
648C22—24no—
649C22—44no—
650C22—48no—
651C22—48no—
652C22—48no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—48no—
662C22—44no—
663C22—44no—
664C22—48no—
665C22—48no—
666C22—44no—
667C22—44no—
668C22—832no—
669C22—832no—
670C22—84no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—84no—
676C22C22×#12311yescharacteristic
677C22C2×#53412yes—
678C22#348512yes—
679C22#349011yescharacteristic
680C22#349012yes—
681C22#349011yescharacteristic
682C22#349412yes—
683C22#349612yes—
684C22#350811yescharacteristic
685C22#351012yes—
686C22#351012yes—
687C22#351011yescharacteristic
688C22#353314yes—
689C22#353414yes—
690C22#354214yes—
691C22#354514yes—
692C4—24no—
693C4—24no—
694C4—44no—
695C4—84no—
696C4—84no—
697C4—84no—
698C4—82no—
699C4—82no—
700C4—84no—
701C4—82no—
702C2—48no—
703C2—44no—
704C2—84no—
705C2#3499211yescharacteristic
706C2#3554111yescharacteristic
707C2#3562611yescharacteristic
708C2#4050814yes—
709C2#4703312yes—
710C2#4703512yes—
711C2#5066012yes—
712C2#5104112yes—
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
3. #16391C2semidirect
4. #16032C2semidirect
5. #15702C2semidirect
6. #15300C2semidirect
7. #15287C2semidirect
8. #14918C2semidirect
9. #14660C2semidirect
10. #14297C2semidirect
11. #14292C2semidirect
12. #13953C2semidirect
19. C2×#397C22semidirect
27. C2×#385C22semidirect
30. C2×#381C22semidirect
31. C2×#381C22semidirect
40. #1446C22semidirect
41. #1446C22semidirect
42. #1445C22semidirect
43. #1445C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
48. #1440C22semidirect
49. #1440C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24352
C2×#23082
#163912
#160322
#157022
#153002
#152872
#149182
#146602
#142972
#142922
#139532
#139022