Finite Group Databasebeta

← #4353730

Subgroups — #4353730

Order 512.

2640 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2544C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic, complemented
3#16884C212yesmaximal, complemented
4#16558C212yesmaximal, complemented
5#16369C214yesmaximal, complemented
6#15884C212yesmaximal, complemented
7#15820C212yesmaximal, complemented
8#15602C214yesmaximal, complemented
9#15090C212yesmaximal, complemented
10#14842C212yesmaximal, complemented
11#14776C212yesmaximal, complemented
12#14735C212yesmaximal, complemented
13#14473C212yesmaximal, complemented
14#14419C212yesmaximal
15#13784C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2544—28no—
22#2487—28no—
23C2×#400C2212yes—
24C2×#400C2214yes—
25C2×#399C2212yes—
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#394C2212yes—
29C2×#394C2212yes—
30C2×#391C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#385C2212yes—
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1446C2214yes—
46#1445C2212yes—
47#1445C2212yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2212yes—
52#1442C2212yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1438C2214yes—
64#1438C2214yes—
65#1438C2214yes—
66#1437C2214yes—
67#1437C2212yes—
68#1436C2212yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C2×#127—24no—
87C2×#127—28no—
88C2×#124—24no—
89C2×#124—28no—
90C2×#124—24no—
91C2×#124—42no—
92C2×#124—42no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C2×C4×D4D418yes—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—42no—
98C22×C4⋊C4—42no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#400—28no—
126#400—48no—
127#399—28no—
128#397—44no—
129#397—44no—
130#394—28no—
131#394—28no—
132#391—24no—
133#390—24no—
134#389—28no—
135#386—28no—
136#386—28no—
137#386—28no—
138#386—28no—
139#386—48no—
140#386—48no—
141#385—28no—
142#380—28no—
143#380—24no—
144C4×C4⋊C4—24no—
145C4×C22⋊C4—42no—
146C2×#95—42no—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×Q8—22no—
175C22×Q8—24no—
176C22×Q8—41no—
177C22×Q8—41no—
178C22×Q8—42no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—22no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4—42no—
190C22×D4—41no—
191C22×D4—41no—
192C22×D4—42no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4C2×D414yes—
198C22×D4#4118yes—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229#127—416no—
230#124—416no—
231#124—416no—
232C4×D4—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D414yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
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×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4C2×D414yes—
294C2×C22⋊C4C2×D414yes—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—42no—
302C2×C42—44no—
303C2×C42C2×D414yes—
304#95—24no—
305#95—28no—
306#95—24no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—42no—
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—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24C22×D412yes—
352C24C22×D412yes—
353C24C2×#4114yes—
354C24C2×#4114yes—
355C24C2511yescharacteristic
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—84no—
360C2×Q8—84no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—44no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—84no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4#124116yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
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—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
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—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—84no—
450C22×C4—88no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—88no—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C2×#4114yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—88no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C42—28no—
487C42—44no—
488C42—44no—
489C42—84no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23C2×#12414yes—
544C23C2×#12414yes—
545C23C23×D411yescharacteristic
546C23C22×#4112yes—
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23#58512yes—
551C23#58514yes—
552C23#58512yes—
553Q8—88no—
554D4—416no—
555D4—88no—
556D4—816no—
557D4—816no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
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—88no—
623C2×C4—816no—
624C2×C4—816no—
625C2×C4C2×#12414yes—
626C2×C4#53414yes—
627C22—28no—
628C22—28no—
629C22—24no—
630C22—24no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—48no—
636C22—48no—
637C22—42no—
638C22—42no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—82no—
644C22—82no—
645C22—84no—
646C22—82no—
647C22—84no—
648C22—84no—
649C22—82no—
650C22—82no—
651C22—84no—
652C22—82no—
653C22—82no—
654C22C22×#12411yescharacteristic
655C22C2×#53411yescharacteristic
656C22#344911yescharacteristic
657C22#348512yes—
658C22#348812yes—
659C22#350512yes—
660C22#350512yes—
661C22#351314yes—
662C22#352114yes—
663C22#353312yes—
664C22#353312yes—
665C22#354612yes—
666C22#354911yescharacteristic
667C22#355111yescharacteristic
668C22#356214yes—
669C22#356614yes—
670C4—24no—
671C4—24no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—82no—
677C4—84no—
678C4—84no—
679C4—84no—
680C4—82no—
681C4—84no—
682C2—44no—
683C2—44no—
684C2—162no—
685C2#3501911yescharacteristic
686C2#3584712yes—
687C2#3695511yescharacteristic
688C2#3727311yescharacteristic
689C2#4142412yes—
690C2#4673314yes—
691C2#4910312yes—
692C2#4972812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2487C2semidirect
3. #16884C2semidirect
4. #16558C2semidirect
5. #16369C2semidirect
6. #15884C2semidirect
7. #15820C2semidirect
8. #15602C2semidirect
9. #15090C2semidirect
10. #14842C2semidirect
11. #14776C2semidirect
12. #14735C2semidirect
13. #14473C2semidirect
15. #13784C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25442
C2×#24872
#168842
#165582
#163692
#158842
#158202
#156022
#150902
#148422
#147762
#147352
#144732
#144192
#137842