Finite Group Databasebeta

← #612485

Subgroups — #612485

Order 512.

7436 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2848C211yesmaximal, characteristic, complemented
3C2×#2449C213yesmaximal, complemented
4C2×C4×#120C211yesmaximal, characteristic
5#14786C213yesmaximal, complemented
6#14777C213yesmaximal, complemented
7#14729C213yesmaximal, complemented
8#14720C213yesmaximal, complemented
9#14694C211yesmaximal, characteristic
10#14694C211yesmaximal, characteristic
11#14693C211yesmaximal, characteristic, complemented
12#14693C211yesmaximal, characteristic, complemented
13#14043C213yesmaximal, complemented
14#14015C213yesmaximal, complemented
15#13882C213yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C25×C4C2211yescharacteristic
19C22×C22⋊Q8C2213yes—
20C22×C22⋊Q8C2211yescharacteristic, complemented
21C22×#120C2211yescharacteristic
22C22×C4×D4C2213yes—
23C23×C4⋊C4—224no—
24C23×C4⋊C4C2213yes—
25C23×C4⋊C4C2213yes—
26C23×C4⋊C4C2213yes—
27C23×C4⋊C4C2213yes—
28#2848—28no—
29#2449—224no—
30C2×#399C2213yescomplemented
31C2×#398C2213yescomplemented
32C2×#396C2213yescomplemented
33C2×#395C2213yescomplemented
34C4×#120—28no—
35C2×#389C2213yescomplemented
36C2×#387C2213yescomplemented
37C2×#386C2213yes—
38C2×#386C2213yes—
39C2×#386C2213yes—
40C2×#386C2213yes—
41C2×#386C2213yes—
42C2×#385C2213yes—
43C2×#385C2213yes—
44C2×#385C2213yes—
45C2×#385C2213yes—
46C2×#385C2213yes—
47C2×#385C2213yes—
48C2×#385C2213yes—
49C2×#385C2213yescomplemented
50C2×#380C2213yes—
51C2×#379C2211yescharacteristic
52C2×#379C2211yescharacteristic
53C2×#379C2211yescharacteristic
54C2×C4×C22⋊C4C2213yes—
55#1445C2216yescomplemented
56#1444C2216yescomplemented
57#1444C2216yescomplemented
58#1441C2216yescomplemented
59#1441C2216yescomplemented
60#1439C2216yescomplemented
61#1436C2216yescomplemented
62C22⋊#95C2213yes—
63C22⋊#95C2213yes—
64C22⋊#95C2213yes—
65C22⋊#95C2213yes—
66C22⋊#95C2213yes—
67C22⋊#95C2213yes—
68C22⋊#95C2213yes—
69C22⋊#95C2213yes—
70C26C2311yescharacteristic
71C23×Q8C2311yescharacteristic
72C23×D4C2313yes—
73C24×C4—24no—
74C24×C4—212no—
75C24×C4—212no—
76C24×C4—212no—
77C24×C4—212no—
78C24×C4—24no—
79C24×C4—24no—
80C24×C4—46no—
81C24×C4—46no—
82C24×C4C2313yes—
83C24×C4C2313yes—
84C24×C4C2313yes—
85C24×C4C2313yes—
86C24×C4C2311yescharacteristic
87C24×C4C2313yes—
88C24×C4C2313yes—
89C2×C22⋊Q8—212no—
90C2×C22⋊Q8—212no—
91C2×C22⋊Q8—212no—
92C2×C22⋊Q8—46no—
93C2×#120—24no—
94C2×#120—24no—
95C2×#120Q818yescomplemented
96C2×C4×D4—212no—
97C2×C4×D4—212no—
98C2×C4×D4—46no—
99C22×C4⋊C4—2128no—
100C22×C4⋊C4—212no—
101C22×C4⋊C4—212no—
102C22×C4⋊C4—212no—
103C22×C4⋊C4—224no—
104C22×C4⋊C4—224no—
105C22×C4⋊C4—224no—
106C22×C4⋊C4—224no—
107C22×C4⋊C4D4124yescomplemented
108C22×C4⋊C4C2313yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2316yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2313yes—
113C22×C4⋊C4C2313yes—
114C22×C4⋊C4C2313yes—
115C22×C4⋊C4C2313yes—
116C22×C4⋊C4C2313yes—
117C22×C4⋊C4C2313yes—
118C22×C4⋊C4C2313yes—
119C22×C4⋊C4C2313yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2313yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2313yes—
125C22×C22⋊C4C2313yes—
126C22×C22⋊C4C2313yes—
127C22×C22⋊C4C2313yes—
128C22×C22⋊C4C2313yes—
129C22×C22⋊C4C2313yes—
130C22×C22⋊C4C2313yes—
131C22×C22⋊C4C2313yes—
132C22×C22⋊C4C2313yes—
133C22×C42C2313yes—
134#399—212no—
135#398—212no—
136#396—212no—
137#395—212no—
138#389—212no—
139#387—212no—
140#386—212no—
141#386—212no—
142#386—212no—
143#386—212no—
144#386—46no—
145#385—212no—
146#385—212no—
147#385—212no—
148#385—212no—
149#385—212no—
150#385—212no—
151#385—212no—
152#385—46no—
153#380—212no—
154#379—24no—
155#379—24no—
156#379D418yes—
157C4×C22⋊C4—212no—
158C2×#95C2313yes—
159C2×#95C2313yes—
160C2×#95C2313yes—
161C2×#95C2313yes—
162C2×#95C2313yes—
163C2×#95C2313yes—
164C2×#95C2313yes—
165C2×#95C2313yes—
166C2×#95C2316yes—
167C2×#95C2316yes—
168C2×#95C2316yes—
169C2×#95C2316yes—
170C2×#95C2316yes—
171C2×#95C2316yes—
172C25—26no—
173C25—26no—
174C25—26no—
175C25—26no—
176C25C2×D414yes—
177C25C2×Q814yes—
178C25#4114yes—
179C25C2411yescharacteristic
180C25C2413yes—
181C22×Q8—22no—
182C22×Q8—26no—
183C22×Q8—43no—
184C22×D4—26no—
185C22×D4—26no—
186C22×D4—26no—
187C22×D4—26no—
188C22×D4—26no—
189C22×D4—26no—
190C22×D4C2×Q8112yes—
191C23×C4—232no—
192C23×C4—296no—
193C23×C4—296no—
194C23×C4—26no—
195C23×C4—26no—
196C23×C4—26no—
197C23×C4—26no—
198C23×C4—212no—
199C23×C4—212no—
200C23×C4—212no—
201C23×C4—212no—
202C23×C4—232no—
203C23×C4—232no—
204C23×C4—26no—
205C23×C4—26no—
206C23×C4—26no—
207C23×C4—26no—
208C23×C4—26no—
209C23×C4—26no—
210C23×C4—212no—
211C23×C4—212no—
212C23×C4—212no—
213C23×C4—212no—
214C23×C4—26no—
215C23×C4—26no—
216C23×C4—26no—
217C23×C4—26no—
218C23×C4—22no—
219C23×C4—26no—
220C23×C4—46no—
221C23×C4—46no—
222C23×C4—46no—
223C23×C4—46no—
224C23×C4—43no—
225C23×C4—448no—
226C23×C4—448no—
227C23×C4—448no—
228C23×C4—448no—
229C23×C4C2×D4112yes—
230C23×C4C2×D4112yes—
231C23×C4C2×D4112yes—
232C23×C4#41112yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2413yes—
236C23×C4C2413yes—
237C23×C4C2413yes—
238C23×C4C2411yescharacteristic
239C23×C4C2413yes—
240C23×C4C2413yes—
241C23×C4C2413yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2413yes—
246C22⋊Q8—448no—
247C22⋊Q8—416no—
248#120—232no—
249C4×D4—448no—
250C2×C4⋊C4—296no—
251C2×C4⋊C4—296no—
252C2×C4⋊C4—296no—
253C2×C4⋊C4—212no—
254C2×C4⋊C4—212no—
255C2×C4⋊C4—212no—
256C2×C4⋊C4—212no—
257C2×C4⋊C4—212no—
258C2×C4⋊C4—212no—
259C2×C4⋊C4—296no—
260C2×C4⋊C4—26no—
261C2×C4⋊C4—26no—
262C2×C4⋊C4—26no—
263C2×C4⋊C4—26no—
264C2×C4⋊C4—26no—
265C2×C4⋊C4—26no—
266C2×C4⋊C4—26no—
267C2×C4⋊C4—26no—
268C2×C4⋊C4—26no—
269C2×C4⋊C4—26no—
270C2×C4⋊C4—26no—
271C2×C4⋊C4—212no—
272C2×C4⋊C4—26no—
273C2×C4⋊C4—43no—
274C2×C4⋊C4—43no—
275C2×C4⋊C4—43no—
276C2×C4⋊C4—43no—
277C2×C4⋊C4—46no—
278C2×C4⋊C4C2×D4112yes—
279C2×C4⋊C4C2×D4112yes—
280C2×C4⋊C4C2×D4112yes—
281C2×C4⋊C4C2×D4112yes—
282C2×C22⋊C4—26no—
283C2×C22⋊C4—26no—
284C2×C22⋊C4—26no—
285C2×C22⋊C4—26no—
286C2×C22⋊C4—26no—
287C2×C22⋊C4—26no—
288C2×C22⋊C4—26no—
289C2×C22⋊C4—26no—
290C2×C22⋊C4—26no—
291C2×C22⋊C4—26no—
292C2×C22⋊C4—26no—
293C2×C22⋊C4—26no—
294C2×C22⋊C4—26no—
295C2×C22⋊C4—26no—
296C2×C22⋊C4—26no—
297C2×C22⋊C4—26no—
298C2×C22⋊C4—26no—
299C2×C22⋊C4—26no—
300C2×C22⋊C4—26no—
301C2×C22⋊C4—26no—
302C2×C22⋊C4—26no—
303C2×C22⋊C4—26no—
304C2×C22⋊C4—26no—
305C2×C22⋊C4—212no—
306C2×C22⋊C4—26no—
307C2×C22⋊C4—43no—
308C2×C22⋊C4—46no—
309C2×C22⋊C4C2×D4112yes—
310C2×C22⋊C4C2×D4112yes—
311C2×C22⋊C4C2×Q8112yes—
312C2×C42—26no—
313C2×C42—26no—
314C2×C42—43no—
315#95—212no—
316#95—212no—
317#95—26no—
318#95—26no—
319#95—26no—
320#95—26no—
321#95—26no—
322#95—26no—
323#95—26no—
324#95—26no—
325#95—212no—
326#95—212no—
327#95—46no—
328#95—46no—
329C24—28no—
330C24—224no—
331C24—224no—
332C24—224no—
333C24—248no—
334C24—224no—
335C24—224no—
336C24—224no—
337C24—224no—
338C24—224no—
339C24—224no—
340C24—26no—
341C24—26no—
342C24—26no—
343C24—26no—
344C24—23no—
345C24—21no—
346C24—23no—
347C24—21no—
348C24—23no—
349C24—23no—
350C24C22⋊Q8116yescomplemented
351C24C22×D416yes—
352C24C22×D416yes—
353C24C22×Q812yes—
354C24C22×Q816yes—
355C24C2×#4116yes—
356C24C2×#4116yes—
357C24H(5,𝔽2)16yes—
358C242−1+416yes—
359C24C2511yescharacteristic
360C2×Q8—412no—
361C2×Q8—44no—
362C2×Q8—86no—
363C2×D4—224no—
364C2×D4—224no—
365C2×D4—224no—
366C2×D4—412no—
367C2×D4—412no—
368C2×D4—412no—
369C2×D4—412no—
370C22×C4—224no—
371C22×C4—2256no—
372C22×C4—248no—
373C22×C4—248no—
374C22×C4—248no—
375C22×C4—224no—
376C22×C4—224no—
377C22×C4—224no—
378C22×C4—21no—
379C22×C4—26no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—23no—
383C22×C4—23no—
384C22×C4—23no—
385C22×C4—23no—
386C22×C4—23no—
387C22×C4—224no—
388C22×C4—224no—
389C22×C4—248no—
390C22×C4—224no—
391C22×C4—224no—
392C22×C4—224no—
393C22×C4—224no—
394C22×C4—23no—
395C22×C4—23no—
396C22×C4—23no—
397C22×C4—23no—
398C22×C4—23no—
399C22×C4—23no—
400C22×C4—26no—
401C22×C4—26no—
402C22×C4—23no—
403C22×C4—26no—
404C22×C4—23no—
405C22×C4—23no—
406C22×C4—26no—
407C22×C4—26no—
408C22×C4—23no—
409C22×C4—23no—
410C22×C4—23no—
411C22×C4—23no—
412C22×C4—26no—
413C22×C4—23no—
414C22×C4—224no—
415C22×C4—224no—
416C22×C4—23no—
417C22×C4—23no—
418C22×C4—23no—
419C22×C4—23no—
420C22×C4—23no—
421C22×C4—21no—
422C22×C4—23no—
423C22×C4—26no—
424C22×C4—21no—
425C22×C4—23no—
426C22×C4—23no—
427C22×C4—26no—
428C22×C4—23no—
429C22×C4—23no—
430C22×C4—26no—
431C22×C4—26no—
432C22×C4—412no—
433C22×C4—412no—
434C22×C4—412no—
435C22×C4—424no—
436C22×C4—424no—
437C22×C4—424no—
438C22×C4—424no—
439C22×C4—424no—
440C22×C4—424no—
441C22×C4—424no—
442C22×C4—424no—
443C22×C4—4128no—
444C22×C4—4128no—
445C22×C4—412no—
446C22×C4—412no—
447C22×C4—412no—
448C22×C4—412no—
449C22×C4—44no—
450C22×C4—86no—
451C22×C4C22×D416yes—
452C22×C4C22×D416yes—
453C22×C4C22×D416yes—
454C22×C4C22×D412yes—
455C22×C4C22×D416yes—
456C22×C4C22×D416yes—
457C22×C4C22×D416yes—
458C22×C4C22×D416yes—
459C22×C4C22×Q816yes—
460C22×C4C2×#41112yes—
461C4⋊C4—248no—
462C4⋊C4—224no—
463C4⋊C4—224no—
464C4⋊C4—224no—
465C4⋊C4—412no—
466C4⋊C4—412no—
467C4⋊C4—412no—
468C4⋊C4—412no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—412no—
472C4⋊C4—86no—
473C4⋊C4#120116yescomplemented
474C22⋊C4—224no—
475C22⋊C4—224no—
476C22⋊C4—224no—
477C22⋊C4—412no—
478C22⋊C4—412no—
479C22⋊C4—412no—
480C22⋊C4—412no—
481C22⋊C4—412no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—412no—
485C22⋊C4—86no—
486C42—412no—
487C23—232no—
488C23—232no—
489C23—296no—
490C23—296no—
491C23—232no—
492C23—296no—
493C23—26no—
494C23—26no—
495C23—212no—
496C23—212no—
497C23—212no—
498C23—212no—
499C23—212no—
500C23—212no—
501C23—212no—
502C23—296no—
503C23—26no—
504C23—212no—
505C23—26no—
506C23—212no—
507C23—212no—
508C23—212no—
509C23—212no—
510C23—22no—
511C23—26no—
512C23—26no—
513C23—41no—
514C23—43no—
515C23—43no—
516C23—43no—
517C23—43no—
518C23—43no—
519C23—43no—
520C23—43no—
521C23—46no—
522C23C2×C22⋊Q8112yes—
523C23#525112yes—
524C23#539112yes—
525C23D42112yes—
526C23#547112yes—
527C23D4×Q8112yes—
528C23#554112yes—
529C23C23×D413yes—
530C23C23×D411yescharacteristic
531C23C23×Q811yescharacteristic
532C23C22×#4111yescharacteristic
533C23C22×#4113yes—
534C23C2×H(5,𝔽2)13yes—
535C23C2×2−1+413yes—
536Q8—88no—
537D4—448no—
538C2×C4—296no—
539C2×C4—296no—
540C2×C4—26no—
541C2×C4—26no—
542C2×C4—26no—
543C2×C4—26no—
544C2×C4—212no—
545C2×C4—212no—
546C2×C4—26no—
547C2×C4—26no—
548C2×C4—26no—
549C2×C4—26no—
550C2×C4—26no—
551C2×C4—26no—
552C2×C4—26no—
553C2×C4—26no—
554C2×C4—26no—
555C2×C4—26no—
556C2×C4—26no—
557C2×C4—212no—
558C2×C4—26no—
559C2×C4—26no—
560C2×C4—26no—
561C2×C4—26no—
562C2×C4—448no—
563C2×C4—448no—
564C2×C4—448no—
565C2×C4—448no—
566C2×C4—41no—
567C2×C4—43no—
568C2×C4—43no—
569C2×C4—43no—
570C2×C4—43no—
571C2×C4—46no—
572C2×C4—46no—
573C2×C4—43no—
574C2×C4—46no—
575C2×C4—46no—
576C2×C4—43no—
577C2×C4—43no—
578C2×C4—46no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—43no—
582C2×C4—43no—
583C2×C4—43no—
584C2×C4—43no—
585C2×C4—46no—
586C2×C4—43no—
587C2×C4—43no—
588C2×C4—46no—
589C2×C4—43no—
590C2×C4—43no—
591C2×C4—43no—
592C2×C4—43no—
593C2×C4—46no—
594C2×C4—46no—
595C2×C4—88no—
596C2×C4C2×#12014yes—
597C2×C4C2×#12014yes—
598C2×C4C2×#12014yes—
599C2×C4#525112yes—
600C22—2128no—
601C22—212no—
602C22—212no—
603C22—224no—
604C22—224no—
605C22—212no—
606C22—224no—
607C22—212no—
608C22—212no—
609C22—224no—
610C22—212no—
611C22—46no—
612C22—42no—
613C22—46no—
614C22—81no—
615C22—83no—
616C22—83no—
617C22—81no—
618C22#2449124yescomplemented
619C22C22×#12011yescharacteristic
620C22C22×C22⋊Q811yescharacteristic
621C22C2×#52513yes—
622C22C2×#52513yes—
623C22C2×#53913yes—
624C22C2×D4213yes—
625C22C2×#54713yes—
626C22C2×D4×Q813yes—
627C22C2×#55413yes—
628C22#348413yes—
629C22#348613yes—
630C22#349216yes—
631C4—24no—
632C4—212no—
633C4—24no—
634C4—24no—
635C4—46no—
636C4—46no—
637C4—46no—
638C4—46no—
639C4—42no—
640C4—81no—
641C4—83no—
642C4—83no—
643C4—81no—
644C2—224no—
645C2—82no—
646C2C2×#244913yes—
647C2#3498913yes—
648C2D4×#12011yescharacteristic
649C2#3610313yes—
650C2Q8×#12011yescharacteristic
651C2#3654613yes—
652C2#3661411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C24×C4⋊C4C2semidirect
2. C2×#2848C2semidirect
3. C2×#2449C2semidirect
5. #14786C2semidirect
6. #14777C2semidirect
7. #14729C2semidirect
8. #14720C2semidirect
11. #14693C2semidirect
12. #14693C2semidirect
13. #14043C2semidirect
14. #14015C2semidirect
15. #13882C2semidirect
20. C22×C22⋊Q8C22semidirect
30. C2×#399C22semidirect
31. C2×#398C22semidirect
32. C2×#396C22semidirect
33. C2×#395C22semidirect
35. C2×#389C22semidirect
36. C2×#387C22semidirect
49. C2×#385C22semidirect
55. #1445C22semidirect
56. #1444C22semidirect
57. #1444C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1439C22semidirect
61. #1436C22semidirect
95. C2×#120Q8semidirect
107. C22×C4⋊C4D4semidirect
350. C24C22⋊Q8semidirect
473. C4⋊C4#120semidirect
618. C22#2449semidirect

Maximal subgroups

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

subgroupindex
C24×C4⋊C42
C2×#28482
C2×#24492
C2×C4×#1202
#147862
#147772
#147292
#147202
#146942
#146932
#140432
#140152
#138822