Finite Group Databasebeta

← #4341776

Subgroups — #4341776

Order 512.

2658 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2717C212yesmaximal
2C2×#2486C211yesmaximal, characteristic
3C2×#2326C211yesmaximal, characteristic
4#16815C212yesmaximal, complemented
5#16629C212yesmaximal, complemented
6#16563C212yesmaximal, complemented
7#16399C212yesmaximal, complemented
8#15560C214yesmaximal, complemented
9#15486C214yesmaximal, complemented
10#15122C212yesmaximal, complemented
11#14880C212yesmaximal, complemented
12#14786C212yesmaximal, complemented
13#14766C212yesmaximal, complemented
14#14505C212yesmaximal
15#13960C211yesmaximal, characteristic
16C22×#124C2211yescharacteristic
17C22×#123C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2717—216no—
22#2486—28no—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#397C2214yes—
27C2×#397C2212yes—
28C2×#397C2214yes—
29C2×#395C2214yes—
30#2326—28no—
31C2×#391C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#389C2212yes—
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#387C2214yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#384C2211yescharacteristic
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×#380C2214yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#376C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1445C2212yes—
50#1445C2212yes—
51#1444C2212yes—
52#1444C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2212yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1434C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2214yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#124—24no—
85C2×#124—24no—
86C2×#124D418yes—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×C4×Q8—24no—
91C2×C4×Q8—42no—
92C2×C4×Q8—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⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#398—28no—
127#398—44no—
128#398—44no—
129#397—216no—
130#397—216no—
131#397—28no—
132#395—216no—
133#391—24no—
134#390—24no—
135#389—28no—
136#388—28no—
137#388—28no—
138#387—216no—
139#386—28no—
140#385—28no—
141#385—28no—
142#385—216no—
143#384—24no—
144#381—28no—
145#380—28no—
146#380—24no—
147#380—216no—
148#380—42no—
149#376—24no—
150C2×#95—42no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—41no—
182C22×Q8—41no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8C2×D414yes—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4C2×D414yes—
191C22×D4C2×D414yes—
192C22×D4#4114yes—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—44no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226#124—232no—
227#123—432no—
228C4×Q8—416no—
229C4×D4—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D414yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4C2×D418yes—
291C2×C22⋊C4C2×D418yes—
292C2×C22⋊C4C2×D414yes—
293C2×C22⋊C4C2×D414yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—41no—
303C2×C42—41no—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—21no—
340C24—24no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24C22×D412yes—
345C24C22×D412yes—
346C24C22×D412yes—
347C24C22×D412yes—
348C24C2×#4112yes—
349C24C2×#4112yes—
350C24C2511yescharacteristic
351C2×Q8—28no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4#121116yes—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
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—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—88no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+418yes—
458C4⋊C4—44no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C4⋊C4—84no—
471C22⋊C4—216no—
472C22⋊C4—216no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—416no—
476C22⋊C4—48no—
477C22⋊C4—416no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C42—28no—
482C42—28no—
483C42—44no—
484C42—48no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—28no—
504C23—28no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23C2×#12114yes—
518C23C2×#12114yes—
519C23C23×D411yescharacteristic
520C23C23×D411yescharacteristic
521C23C22×#4111yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×2−1+414yes—
525C23#58511yescharacteristic
526C23#58511yescharacteristic
527C23#58514yes—
528Q8—416no—
529D4—416no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—416no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—88no—
603C2×C4—816no—
604C2×C4—816no—
605C2×C4C2×#12114yes—
606C2×C4#53414yes—
607C22—28no—
608C22—28no—
609C22—24no—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22C22×#12111yescharacteristic
626C22C2×#53411yescharacteristic
627C22#344911yescharacteristic
628C22#348512yes—
629C22#348914yes—
630C22#349611yescharacteristic
631C22#349611yescharacteristic
632C22#349912yes—
633C22#350511yescharacteristic
634C22#350511yescharacteristic
635C22#351514yes—
636C22#353312yes—
637C22#353312yes—
638C22#353812yes—
639C22#355012yes—
640C22#356414yes—
641C22#356514yes—
642C4—24no—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—48no—
648C4—44no—
649C4—44no—
650C4—82no—
651C4—82no—
652C4—84no—
653C4—84no—
654C4—84no—
655C2—44no—
656C2—44no—
657C2#3500011yescharacteristic
658C2#3602012yes—
659C2#3695511yescharacteristic
660C2#3738611yescharacteristic
661C2#4231012yes—
662C2#4696812yes—
663C2#4778612yes—
664C2#5158912yes—
665C2#5170812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #16815C2semidirect
5. #16629C2semidirect
6. #16563C2semidirect
7. #16399C2semidirect
8. #15560C2semidirect
9. #15486C2semidirect
10. #15122C2semidirect
11. #14880C2semidirect
12. #14786C2semidirect
13. #14766C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27172
C2×#24862
C2×#23262
#168152
#166292
#165632
#163992
#155602
#154862
#151222
#148802
#147862
#147662
#145052
#139602