Finite Group Databasebeta

← #612477

Subgroups — #612477

Order 512.

7571 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2847C211yesmaximal, characteristic
3C2×#2443C213yesmaximal, complemented
4C2×C4×#120C211yesmaximal, characteristic, complemented
5#14777C213yesmaximal, complemented
6#14776C213yesmaximal, complemented
7#14728C213yesmaximal, complemented
8#14719C213yesmaximal, complemented
9#14694C211yesmaximal, characteristic, complemented
10#14694C211yesmaximal, characteristic, complemented
11#14694C211yesmaximal, characteristic, complemented
12#14693C211yesmaximal, characteristic, complemented
13#14019C213yesmaximal, complemented
14#14015C213yesmaximal, complemented
15#13882C213yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C25×C4C2211yescharacteristic
19C22×#123C2211yescharacteristic, complemented
20C22×#121C2213yes—
21C22×#120C2211yescharacteristic
22C22×#120C2211yescharacteristic
23C22×C4×D4C2213yes—
24C23×C4⋊C4—224no—
25C23×C4⋊C4C2213yes—
26C23×C4⋊C4C2213yes—
27C23×C4⋊C4C2213yes—
28C23×C4⋊C4C2213yes—
29#2847—28no—
30#2443—224no—
31C2×#399C2213yescomplemented
32C2×#398C2213yescomplemented
33C2×#397C2213yescomplemented
34C2×#394C2213yescomplemented
35C4×#120—28no—
36C2×#388C2213yescomplemented
37C2×#387C2213yescomplemented
38C2×#386C2213yes—
39C2×#386C2213yes—
40C2×#386C2213yes—
41C2×#386C2213yes—
42C2×#386C2213yes—
43C2×#386C2213yes—
44C2×#386C2213yes—
45C2×#386C2213yes—
46C2×#386C2213yes—
47C2×#385C2213yes—
48C2×#385C2213yes—
49C2×#385C2213yes—
50C2×#385C2213yescomplemented
51C2×#380C2213yes—
52C2×#379C2211yescharacteristic
53C2×#379C2211yescharacteristic
54C2×C4×C22⋊C4C2213yes—
55#1445C2216yescomplemented
56#1444C2216yescomplemented
57#1442C2216yescomplemented
58#1441C2216yescomplemented
59#1439C2216yescomplemented
60#1439C2216yescomplemented
61#1436C2216yescomplemented
62C22⋊#95C2213yes—
63C22⋊#95C2213yes—
64C22⋊#95C2213yes—
65C22⋊#95C2213yes—
66C22⋊#95C2213yes—
67C22⋊#95C2213yes—
68C22⋊#95C2213yes—
69C22⋊#95C2213yes—
70C26C2311yescharacteristic
71C23×D4C2313yes—
72C23×D4C2313yes—
73C23×D4C2311yescharacteristic
74C24×C4—24no—
75C24×C4—212no—
76C24×C4—212no—
77C24×C4—212no—
78C24×C4—212no—
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—46no—
82C24×C4—46no—
83C24×C4C2313yes—
84C24×C4C2313yes—
85C24×C4C2313yes—
86C24×C4C2313yes—
87C24×C4C2311yescharacteristic
88C24×C4C2313yes—
89C24×C4C2313yes—
90C2×#123—212no—
91C2×#121—212no—
92C2×#121—212no—
93C2×#121—46no—
94C2×#120—24no—
95C2×#120—24no—
96C2×#120—24no—
97C2×#120—24no—
98C2×#120—24no—
99C2×#120D418yes—
100C2×C4×D4—212no—
101C2×C4×D4—212no—
102C2×C4×D4—46no—
103C22×C4⋊C4—2128no—
104C22×C4⋊C4—212no—
105C22×C4⋊C4—212no—
106C22×C4⋊C4—212no—
107C22×C4⋊C4—224no—
108C22×C4⋊C4—224no—
109C22×C4⋊C4—224no—
110C22×C4⋊C4—224no—
111C22×C4⋊C4D4124yescomplemented
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2316yes—
114C22×C4⋊C4C2313yes—
115C22×C4⋊C4C2313yes—
116C22×C4⋊C4C2313yes—
117C22×C4⋊C4C2313yes—
118C22×C4⋊C4C2313yes—
119C22×C4⋊C4C2313yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2313yes—
122C22×C22⋊C4C2313yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2313yes—
126C22×C22⋊C4C2313yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2313yes—
129C22×C22⋊C4C2313yes—
130C22×C22⋊C4C2313yes—
131C22×C22⋊C4C2313yes—
132C22×C22⋊C4C2313yes—
133C22×C22⋊C4C2313yes—
134C22×C22⋊C4C2313yes—
135C22×C22⋊C4C2313yes—
136C22×C42C2313yes—
137#399—212no—
138#398—212no—
139#397—212no—
140#394—212no—
141#388—212no—
142#387—212no—
143#386—212no—
144#386—212no—
145#386—212no—
146#386—212no—
147#386—212no—
148#386—212no—
149#386—212no—
150#386—212no—
151#386—46no—
152#385—212no—
153#385—212no—
154#385—212no—
155#385—46no—
156#380—212no—
157#379—24no—
158#379—24no—
159C4×C22⋊C4—212no—
160C2×#95C2313yes—
161C2×#95C2313yes—
162C2×#95C2313yes—
163C2×#95C2313yes—
164C2×#95C2313yes—
165C2×#95C2313yes—
166C2×#95C2313yes—
167C2×#95C2313yes—
168C2×#95C2316yes—
169C2×#95C2316yes—
170C2×#95C2316yes—
171C2×#95C2316yes—
172C2×#95C2316yes—
173C2×#95C2316yes—
174C25—26no—
175C25—26no—
176C25—26no—
177C25—26no—
178C25C2×D414yes—
179C25#4114yes—
180C25#4114yes—
181C25C2411yescharacteristic
182C25C2411yescharacteristic
183C25C2413yes—
184C22×D4—26no—
185C22×D4—26no—
186C22×D4—26no—
187C22×D4—26no—
188C22×D4—26no—
189C22×D4—26no—
190C22×D4—26no—
191C22×D4—22no—
192C22×D4—26no—
193C22×D4—26no—
194C22×D4—26no—
195C22×D4—26no—
196C22×D4—26no—
197C22×D4—26no—
198C22×D4—26no—
199C22×D4—43no—
200C22×D4C2×D4112yes—
201C23×C4—232no—
202C23×C4—296no—
203C23×C4—296no—
204C23×C4—26no—
205C23×C4—26no—
206C23×C4—26no—
207C23×C4—26no—
208C23×C4—212no—
209C23×C4—212no—
210C23×C4—212no—
211C23×C4—212no—
212C23×C4—232no—
213C23×C4—232no—
214C23×C4—26no—
215C23×C4—26no—
216C23×C4—26no—
217C23×C4—26no—
218C23×C4—26no—
219C23×C4—26no—
220C23×C4—212no—
221C23×C4—212no—
222C23×C4—212no—
223C23×C4—212no—
224C23×C4—22no—
225C23×C4—26no—
226C23×C4—26no—
227C23×C4—26no—
228C23×C4—26no—
229C23×C4—26no—
230C23×C4—46no—
231C23×C4—46no—
232C23×C4—46no—
233C23×C4—46no—
234C23×C4—43no—
235C23×C4—448no—
236C23×C4—448no—
237C23×C4—448no—
238C23×C4—448no—
239C23×C4C2×D4112yes—
240C23×C4C2×D4112yes—
241C23×C4C2×D4112yes—
242C23×C4#41112yes—
243C23×C4C2411yescharacteristic
244C23×C4C2413yes—
245C23×C4C2413yes—
246C23×C4C2411yescharacteristic
247C23×C4C2413yes—
248C23×C4C2413yes—
249C23×C4C2413yes—
250C23×C4C2413yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2413yes—
255#123—416no—
256#121—448no—
257#120—232no—
258#120—416no—
259C4×D4—448no—
260C2×C4⋊C4—296no—
261C2×C4⋊C4—296no—
262C2×C4⋊C4—296no—
263C2×C4⋊C4—212no—
264C2×C4⋊C4—212no—
265C2×C4⋊C4—212no—
266C2×C4⋊C4—212no—
267C2×C4⋊C4—212no—
268C2×C4⋊C4—212no—
269C2×C4⋊C4—296no—
270C2×C4⋊C4—26no—
271C2×C4⋊C4—26no—
272C2×C4⋊C4—26no—
273C2×C4⋊C4—26no—
274C2×C4⋊C4—26no—
275C2×C4⋊C4—26no—
276C2×C4⋊C4—43no—
277C2×C4⋊C4—43no—
278C2×C4⋊C4—46no—
279C2×C4⋊C4C2×D4112yes—
280C2×C4⋊C4C2×D4112yes—
281C2×C4⋊C4C2×D4112yes—
282C2×C4⋊C4C2×D4112yes—
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—26no—
306C2×C22⋊C4—26no—
307C2×C22⋊C4—26no—
308C2×C22⋊C4—26no—
309C2×C22⋊C4—26no—
310C2×C22⋊C4—26no—
311C2×C22⋊C4—212no—
312C2×C22⋊C4—212no—
313C2×C22⋊C4—26no—
314C2×C22⋊C4—43no—
315C2×C22⋊C4—43no—
316C2×C22⋊C4—43no—
317C2×C22⋊C4—46no—
318C2×C22⋊C4C2×D4112yes—
319C2×C42—26no—
320C2×C42—26no—
321C2×C42—43no—
322#95—212no—
323#95—212no—
324#95—26no—
325#95—26no—
326#95—26no—
327#95—26no—
328#95—26no—
329#95—26no—
330#95—26no—
331#95—26no—
332#95—212no—
333#95—212no—
334#95—46no—
335#95—46no—
336C24—28no—
337C24—224no—
338C24—224no—
339C24—224no—
340C24—248no—
341C24—224no—
342C24—224no—
343C24—224no—
344C24—224no—
345C24—224no—
346C24—224no—
347C24—26no—
348C24—26no—
349C24—26no—
350C24—26no—
351C24—23no—
352C24—23no—
353C24—21no—
354C24—21no—
355C24—23no—
356C24—23no—
357C24—23no—
358C24—21no—
359C24—23no—
360C24—21no—
361C24—21no—
362C24—23no—
363C24—23no—
364C24#123116yescomplemented
365C24C22×D412yes—
366C24C22×D416yes—
367C24C22×D416yes—
368C24C2×#4116yes—
369C24C2×#4116yes—
370C24C2×#4116yes—
371C24H(5,𝔽2)16yes—
372C242−1+416yes—
373C24C2511yescharacteristic
374C2×D4—224no—
375C2×D4—224no—
376C2×D4—224no—
377C2×D4—412no—
378C2×D4—412no—
379C2×D4—412no—
380C2×D4—412no—
381C2×D4—412no—
382C2×D4—412no—
383C2×D4—412no—
384C2×D4—412no—
385C2×D4—412no—
386C2×D4—412no—
387C2×D4—412no—
388C2×D4—412no—
389C2×D4—44no—
390C2×D4—86no—
391C22×C4—224no—
392C22×C4—2256no—
393C22×C4—248no—
394C22×C4—248no—
395C22×C4—248no—
396C22×C4—224no—
397C22×C4—224no—
398C22×C4—224no—
399C22×C4—21no—
400C22×C4—26no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—23no—
404C22×C4—23no—
405C22×C4—23no—
406C22×C4—23no—
407C22×C4—23no—
408C22×C4—224no—
409C22×C4—224no—
410C22×C4—248no—
411C22×C4—224no—
412C22×C4—224no—
413C22×C4—224no—
414C22×C4—224no—
415C22×C4—23no—
416C22×C4—23no—
417C22×C4—23no—
418C22×C4—26no—
419C22×C4—26no—
420C22×C4—23no—
421C22×C4—23no—
422C22×C4—23no—
423C22×C4—26no—
424C22×C4—26no—
425C22×C4—23no—
426C22×C4—23no—
427C22×C4—23no—
428C22×C4—23no—
429C22×C4—224no—
430C22×C4—224no—
431C22×C4—23no—
432C22×C4—23no—
433C22×C4—23no—
434C22×C4—23no—
435C22×C4—23no—
436C22×C4—23no—
437C22×C4—23no—
438C22×C4—21no—
439C22×C4—23no—
440C22×C4—26no—
441C22×C4—26no—
442C22×C4—21no—
443C22×C4—23no—
444C22×C4—23no—
445C22×C4—23no—
446C22×C4—23no—
447C22×C4—26no—
448C22×C4—23no—
449C22×C4—23no—
450C22×C4—26no—
451C22×C4—23no—
452C22×C4—23no—
453C22×C4—23no—
454C22×C4—412no—
455C22×C4—412no—
456C22×C4—424no—
457C22×C4—424no—
458C22×C4—424no—
459C22×C4—424no—
460C22×C4—424no—
461C22×C4—424no—
462C22×C4—424no—
463C22×C4—424no—
464C22×C4—412no—
465C22×C4—4128no—
466C22×C4—4128no—
467C22×C4—412no—
468C22×C4—412no—
469C22×C4—412no—
470C22×C4—412no—
471C22×C4—44no—
472C22×C4—86no—
473C22×C4C22×D416yes—
474C22×C4C22×D416yes—
475C22×C4C22×D416yes—
476C22×C4C22×D416yes—
477C22×C4C22×D416yes—
478C22×C4C22×D416yes—
479C22×C4C22×D416yes—
480C22×C4C2×#41112yes—
481C4⋊C4—248no—
482C4⋊C4—224no—
483C4⋊C4—224no—
484C4⋊C4—224no—
485C4⋊C4—412no—
486C4⋊C4—412no—
487C4⋊C4—44no—
488C4⋊C4—86no—
489C4⋊C4#120116yescomplemented
490C22⋊C4—224no—
491C22⋊C4—412no—
492C22⋊C4—412no—
493C22⋊C4—412no—
494C22⋊C4—412no—
495C22⋊C4—412no—
496C22⋊C4—412no—
497C22⋊C4—412no—
498C22⋊C4—412no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—412no—
503C22⋊C4—412no—
504C22⋊C4—86no—
505C42—412no—
506C23—232no—
507C23—232no—
508C23—296no—
509C23—296no—
510C23—232no—
511C23—296no—
512C23—26no—
513C23—26no—
514C23—212no—
515C23—212no—
516C23—212no—
517C23—212no—
518C23—212no—
519C23—212no—
520C23—296no—
521C23—212no—
522C23—26no—
523C23—212no—
524C23—26no—
525C23—212no—
526C23—212no—
527C23—212no—
528C23—212no—
529C23—22no—
530C23—26no—
531C23—26no—
532C23—41no—
533C23—43no—
534C23—43no—
535C23—43no—
536C23—43no—
537C23—43no—
538C23—43no—
539C23—43no—
540C23—46no—
541C23—41no—
542C23—41no—
543C23—43no—
544C23—43no—
545C23—43no—
546C23—43no—
547C23—43no—
548C23—43no—
549C23—43no—
550C23—43no—
551C23—43no—
552C23—46no—
553C23C2×#123112yes—
554C23#525112yes—
555C23#539112yes—
556C23D42112yes—
557C23#547112yes—
558C23#548112yes—
559C23#553112yes—
560C23C23×D413yes—
561C23C23×D411yescharacteristic
562C23C22×#4111yescharacteristic
563C23C22×#4111yescharacteristic
564C23C22×#4113yes—
565C23C2×H(5,𝔽2)13yes—
566C23C2×2−1+413yes—
567D4—448no—
568D4—88no—
569D4—824no—
570C2×C4—296no—
571C2×C4—296no—
572C2×C4—26no—
573C2×C4—26no—
574C2×C4—26no—
575C2×C4—26no—
576C2×C4—212no—
577C2×C4—212no—
578C2×C4—26no—
579C2×C4—26no—
580C2×C4—26no—
581C2×C4—26no—
582C2×C4—26no—
583C2×C4—26no—
584C2×C4—26no—
585C2×C4—26no—
586C2×C4—26no—
587C2×C4—26no—
588C2×C4—26no—
589C2×C4—26no—
590C2×C4—26no—
591C2×C4—448no—
592C2×C4—448no—
593C2×C4—448no—
594C2×C4—448no—
595C2×C4—43no—
596C2×C4—43no—
597C2×C4—46no—
598C2×C4—46no—
599C2×C4—43no—
600C2×C4—43no—
601C2×C4—46no—
602C2×C4—46no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—43no—
606C2×C4—43no—
607C2×C4—43no—
608C2×C4—43no—
609C2×C4—43no—
610C2×C4—43no—
611C2×C4—43no—
612C2×C4—43no—
613C2×C4—46no—
614C2×C4—43no—
615C2×C4—43no—
616C2×C4—43no—
617C2×C4—43no—
618C2×C4—46no—
619C2×C4—43no—
620C2×C4—43no—
621C2×C4—43no—
622C2×C4—43no—
623C2×C4—43no—
624C2×C4—46no—
625C2×C4—46no—
626C2×C4—88no—
627C2×C4C2×#12014yes—
628C2×C4C2×#12014yes—
629C2×C4C2×#12014yes—
630C2×C4#525112yes—
631C22—2128no—
632C22—212no—
633C22—212no—
634C22—224no—
635C22—224no—
636C22—212no—
637C22—224no—
638C22—212no—
639C22—212no—
640C22—224no—
641C22—212no—
642C22—46no—
643C22—46no—
644C22—42no—
645C22—81no—
646C22—83no—
647C22—81no—
648C22—83no—
649C22—81no—
650C22—83no—
651C22—83no—
652C22—83no—
653C22—81no—
654C22—83no—
655C22—81no—
656C22#2443124yescomplemented
657C22C22×#12011yescharacteristic
658C22C22×#12311yescharacteristic
659C22C2×#52513yes—
660C22C2×#52513yes—
661C22C2×#53913yes—
662C22C2×D4213yes—
663C22C2×#54713yes—
664C22C2×#54813yes—
665C22C2×#55313yes—
666C22#348413yes—
667C22#348613yes—
668C22#349216yes—
669C4—24no—
670C4—212no—
671C4—24no—
672C4—24no—
673C4—46no—
674C4—46no—
675C4—46no—
676C4—83no—
677C4—83no—
678C4—81no—
679C4—83no—
680C4—81no—
681C2—224no—
682C2—82no—
683C2—161no—
684C2C2×#244313yes—
685C2#3498913yes—
686C2D4×#12011yescharacteristic
687C2#3610313yes—
688C2#3657511yescharacteristic
689C2#3658113yes—
690C2#3661411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2847C2
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
3. C2×#2443C2semidirect
4. C2×C4×#120C2semidirect
5. #14777C2semidirect
6. #14776C2semidirect
7. #14728C2semidirect
8. #14719C2semidirect
9. #14694C2semidirect
10. #14694C2semidirect
11. #14694C2semidirect
12. #14693C2semidirect
13. #14019C2semidirect
14. #14015C2semidirect
15. #13882C2semidirect
19. C22×#123C22semidirect
31. C2×#399C22semidirect
32. C2×#398C22semidirect
33. C2×#397C22semidirect
34. C2×#394C22semidirect
36. C2×#388C22semidirect
37. C2×#387C22semidirect
50. C2×#385C22semidirect
55. #1445C22semidirect
56. #1444C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1436C22semidirect
111. C22×C4⋊C4D4semidirect
364. C24#123semidirect
489. C4⋊C4#120semidirect
656. C22#2443semidirect

Maximal subgroups

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

subgroupindex
C24×C4⋊C42
C2×#28472
C2×#24432
C2×C4×#1202
#147772
#147762
#147282
#147192
#146942
#146932
#140192
#140152
#138822