Finite Group Databasebeta

← #637041

Subgroups — #637041

Order 512.

3932 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#125C211yesmaximal, characteristic, complemented
2C2×#2487C211yesmaximal, characteristic
3C2×#2486C211yesmaximal, characteristic
4C2×C4×#123C211yesmaximal, characteristic
5#16683C214yesmaximal, complemented
6#16682C214yesmaximal, complemented
7#15218C214yesmaximal, complemented
8#15216C214yesmaximal, complemented
9#15073C212yesmaximal, complemented
10#15069C212yesmaximal
11#14685C211yesmaximal, characteristic, complemented
12#14263C212yesmaximal, complemented
13#14256C212yesmaximal
14#13924C211yesmaximal, characteristic, complemented
15#13904C211yesmaximal, characteristic, complemented
16C22×#125—28no—
17C22×#125—28no—
18C22×#125—28no—
19C22×#125C2214yes—
20C22×#125C2214yes—
21C22×#123C2211yescharacteristic
22C22×#123C2212yes—
23C22×C4×D4C2211yescharacteristic
24C23×C4⋊C4C2212yes—
25C23×C4⋊C4C2212yes—
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2211yescharacteristic
28C23×C42C2211yescharacteristic
29#2487—28no—
30#2486—28no—
31C2×#400C2212yes—
32C2×#399C2212yes—
33C2×#398C2212yes—
34C2×#397C2212yes—
35C4×#123—28no—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#388C2212yes—
39C2×#387C2212yes—
40C2×#386C2211yescharacteristic
41C2×#386C2212yes—
42C2×#385C2211yescharacteristic
43C2×#385C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#381C2211yescharacteristic
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56#1445C2214yes—
57#1445C2214yes—
58#1444C2218yes—
59#1444C2218yes—
60#1442C2218yes—
61#1442C2218yes—
62#1441C2218yes—
63#1441C2218yes—
64#1439C2218yes—
65#1439C2218yes—
66C4⋊#95C2214yes—
67C4⋊#95C2214yes—
68C4⋊#95C2214yes—
69#1434C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C23×D4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×#125—2128no—
81C2×#125—216no—
82C2×#125—216no—
83C2×#125—216no—
84C2×#125—216no—
85C2×#125—216no—
86C2×#125—216no—
87C2×#123—24no—
88C2×#123—24no—
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—24no—
92C2×#123—44no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4D418yes—
107C22×C4⋊C4D418yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2318yes—
110C22×C4⋊C4C2318yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C42—24no—
132C22×C42—24no—
133C22×C42D418yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138#400—28no—
139#399—28no—
140#398—28no—
141#397—28no—
142#389—28no—
143#389—28no—
144#388—28no—
145#387—28no—
146#386—28no—
147#386—24no—
148#385—28no—
149#385—24no—
150#381—28no—
151#381—24no—
152#381—28no—
153#381—44no—
154#381—44no—
155#380—24no—
156#380—24no—
157#380—28no—
158#380—28no—
159C4×C4⋊C4—28no—
160C4×C22⋊C4—24no—
161C4×C22⋊C4—28no—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2318yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2318yes—
175C2×#95C2318yes—
176C2×#95C2318yes—
177C2×#95C2318yes—
178C2×#95C2318yes—
179C25C2411yescharacteristic
180C25C2411yescharacteristic
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4#4114yes—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—28no—
209C23×C4—44no—
210C23×C4—48no—
211C23×C4C2×D418yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D418yes—
217C23×C4C2×D414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231#125—2128no—
232#125—2128no—
233#123—232no—
234#123—432no—
235C4×D4—416no—
236C2×C4⋊C4—264no—
237C2×C4⋊C4—264no—
238C2×C4⋊C4—232no—
239C2×C4⋊C4—232no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D418yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4#4114yes—
303C2×C42—232no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42C2×D418yes—
314C2×C42C2×D418yes—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—216no—
327#95—48no—
328#95—48no—
329#95—48no—
330#95—48no—
331#95—48no—
332C24—24no—
333C24—24no—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—24no—
340C24—21no—
341C24—24no—
342C24C22×D412yes—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C2×#4112yes—
346C24C2×#4112yes—
347C24C2×#4112yes—
348C24H(5,𝔽2)12yes—
349C242−1+412yes—
350C24C2511yescharacteristic
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C22×C4—24no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—216no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—216no—
369C22×C4—216no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—48no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—48no—
427C22×C4—416no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4#120116yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D418yes—
445C22×C4C22×D418yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4112yes—
454C22×C42−1+418yes—
455C4⋊C4—264no—
456C4⋊C4—264no—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—416no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—88no—
468C22⋊C4—216no—
469C22⋊C4—28no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—88no—
478C42—216no—
479C42—216no—
480C42—44no—
481C42—48no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—28no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23C2×#12014yes—
510C23#52514yes—
511C23#53714yes—
512C23#54614yes—
513C23#54714yes—
514C23#54814yes—
515C23#55014yes—
516C23C23×D411yescharacteristic
517C23C23×D411yescharacteristic
518C23C23×D411yescharacteristic
519C23C22×#4111yescharacteristic
520C23C22×#4111yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×2−1+414yes—
523C23C2×2−1+411yescharacteristic
524C23#58514yes—
525D4—416no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—232no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—416no—
548C2×C4—432no—
549C2×C4—416no—
550C2×C4—432no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—88no—
594C2×C4C2×#12014yes—
595C2×C4C2×#12014yes—
596C2×C4#53618yes—
597C2×C4#53618yes—
598C2×C4#53618yes—
599C22—28no—
600C22—24no—
601C22—28no—
602C22—24no—
603C22—28no—
604C22—28no—
605C22—28no—
606C22—28no—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—44no—
611C22—82no—
612C22—81no—
613C22—81no—
614C22—82no—
615C22—82no—
616C22#240718yes—
617C22C22×#12011yescharacteristic
618C22C2×#52511yescharacteristic
619C22C2×#53612yes—
620C22C2×#53612yes—
621C22C2×#53612yes—
622C22C2×#53711yescharacteristic
623C22#344912yes—
624C22C2×#54611yescharacteristic
625C22C2×#54711yescharacteristic
626C22C2×#54811yescharacteristic
627C22C2×#55011yescharacteristic
628C22#348512yes—
629C22#348712yes—
630C22#351514yes—
631C22#351514yes—
632C22#355914yes—
633C22#356314yes—
634C4—24no—
635C4—44no—
636C4—44no—
637C4—48no—
638C4—44no—
639C4—48no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—82no—
644C4—84no—
645C4—84no—
646C4#261418yes—
647C2—28no—
648C2—82no—
649C2C2×#261411yescharacteristic
650C2C2×#240711yescharacteristic
651C2#3513812yes—
652C2#3514911yescharacteristic
653C2#3754912yes—
654C2#3890712yes—
655C2#3891912yes—
656C2#3912212yes—
657C2#3913412yes—
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
1. C23×#125C2semidirect
5. #16683C2semidirect
6. #16682C2semidirect
7. #15218C2semidirect
8. #15216C2semidirect
9. #15073C2semidirect
11. #14685C2semidirect
12. #14263C2semidirect
14. #13924C2semidirect
15. #13904C2semidirect

Maximal subgroups

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

subgroupindex
C23×#1252
C2×#24872
C2×#24862
C2×C4×#1232
#166832
#166822
#152182
#152162
#150732
#150692
#146852
#142632
#142562
#139242
#139042