Finite Group Databasebeta

← #712409

Subgroups — #712409

Order 512.

3622 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2406C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic, complemented
3C2×#2344C211yesmaximal, characteristic, complemented
4C2×#2311C211yesmaximal, characteristic
5#16806C212yesmaximal, complemented
6#16800C212yesmaximal, complemented
7#15512C214yesmaximal, complemented
8#15436C214yesmaximal, complemented
9#15163C212yesmaximal, complemented
10#15097C212yesmaximal, complemented
11#14912C212yesmaximal, complemented
12#14729C212yesmaximal, complemented
13#14342C211yesmaximal, characteristic, complemented
14#14135C212yesmaximal, complemented
15#14019C212yesmaximal, complemented
16#13962C211yesmaximal, characteristic, complemented
17#13956C211yesmaximal, characteristic
18C23×#41C2211yescharacteristic
19C22×#123C2212yes—
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C4⋊C4C2212yes—
25#2406—28no—
26C2×#398—28no—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#398C2212yes—
30C2×#398C2212yes—
31C2×#396C2214yes—
32C2×#395C2211yescharacteristic
33C2×#395C2211yescharacteristic
34C2×#394C2211yescharacteristic
35C2×#394C2211yescharacteristic
36#2344—28no—
37#2311—28no—
38C2×#391C2212yes—
39C2×#390C2212yes—
40C2×#389C2212yes—
41C2×#388C2214yes—
42C2×#387C2212yes—
43C2×#387C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#386C2212yes—
48C2×#386C2212yes—
49C2×#385C2212yes—
50C2×#385C2212yes—
51C2×#385C2212yes—
52C2×#381C2211yescharacteristic
53C2×#381C2211yescharacteristic
54C2×#381C2211yescharacteristic
55C2×#380C2211yescharacteristic
56C2×#380C2214yes—
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C22⋊C4C2211yescharacteristic
59C22×#95C2212yes—
60C22×#95C2211yescharacteristic
61#1446C2212yes—
62#1445C2212yes—
63#1445C2214yes—
64#1445C2212yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1440C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1438C2212yes—
78#1437C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2214yes—
87C22×#41—28no—
88C22×#41—24no—
89C22×#41—24no—
90C22×#41—24no—
91C22×#41—44no—
92C23×Q8C2311yescharacteristic
93C23×D4C2312yes—
94C23×D4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2312yes—
97C24×C4C2312yes—
98C24×C4C2311yescharacteristic
99C24×C4C2311yescharacteristic
100C24×C4C2312yes—
101C2×#123—28no—
102C2×#123—28no—
103C2×#123—44no—
104C2×#117—24no—
105C2×#117—44no—
106C2×#117—44no—
107C2×#117D418yes—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4—24no—
113C22×C4⋊C4—24no—
114C22×C4⋊C4—48no—
115C22×C4⋊C4D418yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2311yescharacteristic
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2311yescharacteristic
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2314yes—
143C22×C42C2312yes—
144C22×C42C2312yes—
145C22×C42C2312yes—
146#398—28no—
147#398—28no—
148#398—28no—
149#398—28no—
150#396—48no—
151#395—24no—
152#395—24no—
153#394—24no—
154#394—24no—
155#391—28no—
156#390—28no—
157#389—28no—
158#388—48no—
159#387—28no—
160#387—28no—
161#386—28no—
162#386—28no—
163#386—28no—
164#386—28no—
165#386—44no—
166#385—28no—
167#385—28no—
168#385—44no—
169#381—24no—
170#381—24no—
171#381—24no—
172#380—216no—
173#380—24no—
174C4×C4⋊C4—28no—
175C4×C22⋊C4—24no—
176C2×#95—24no—
177C2×#95—28no—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C25C2412yes—
205C25C2411yescharacteristic
206C2×#41—232no—
207C2×#41—432no—
208C2×#41—432no—
209C2×#41—416no—
210C2×#41—416no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—24no—
214C22×Q8—22no—
215C22×Q8—42no—
216C22×D4—24no—
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—24no—
222C22×D4—24no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—42no—
227C22×D4#4118yes—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—28no—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—44no—
250C23×C4—42no—
251C23×C4—42no—
252C23×C4—42no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4C2×D414yes—
256C23×C4C2×D414yes—
257C23×C4C2×D414yes—
258C23×C4C2×D414yes—
259C23×C4#4114yes—
260C23×C4C2412yes—
261C23×C4C2412yes—
262C23×C4C2412yes—
263C23×C4C2412yes—
264C23×C4C2414yes—
265C23×C4C2412yes—
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268C23×C4C2411yescharacteristic
269C23×C4C2411yescharacteristic
270C23×C4C2412yes—
271C23×C4C2414yes—
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C23×C4C2411yescharacteristic
275#123—432no—
276#117—416no—
277#117—416no—
278C2×C4⋊C4—232no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—432no—
313C2×C4⋊C4—416no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D418yes—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—42no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4C2×D414yes—
353C2×C22⋊C4C2×D414yes—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—44no—
358C2×C42—44no—
359C2×C42—42no—
360C2×C42C2×D418yes—
361#95—24no—
362#95—24no—
363#95—28no—
364#95—28no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—28no—
369#95—28no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—28no—
379#95—28no—
380#95—44no—
381#95—44no—
382#95—44no—
383#95—44no—
384#95—44no—
385#95—44no—
386#95—44no—
387C24—22no—
388C24—24no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—24no—
395C24—22no—
396C24—22no—
397C24—24no—
398C24—24no—
399C24—24no—
400C24C22×D412yes—
401C24C22×D412yes—
402C24C2×#4114yes—
403C24C2×#4112yes—
404C24C2×#4114yes—
405C24H(5,𝔽2)14yes—
406C24C2511yescharacteristic
407#41—4128no—
408C2×Q8—28no—
409C2×Q8—48no—
410C2×Q8—48no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×D4—28no—
414C2×D4—216no—
415C2×D4—216no—
416C2×D4—216no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—44no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—44no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—216no—
437C22×C4—216no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—216no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—28no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—22no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—44no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—416no—
507C22×C4—416no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4—48no—
512C22×C4—44no—
513C22×C4—44no—
514C22×C4—48no—
515C22×C4—48no—
516C22×C4—48no—
517C22×C4—48no—
518C22×C4—48no—
519C22×C4—48no—
520C22×C4—88no—
521C22×C4—88no—
522C22×C4#121116yes—
523C22×C4C22×D414yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D414yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C22×D412yes—
531C22×C4C22×D412yes—
532C22×C4C22×D412yes—
533C22×C4C2×#4114yes—
534C22×C4C2×#4112yes—
535C22×C4C2×#4112yes—
536C22×C4H(5,𝔽2)14yes—
537C22×C42−1+418yes—
538C4⋊C4—28no—
539C4⋊C4—28no—
540C4⋊C4—28no—
541C4⋊C4—28no—
542C4⋊C4—48no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C4⋊C4—48no—
546C4⋊C4—416no—
547C4⋊C4—48no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C4⋊C4—416no—
551C4⋊C4—88no—
552C22⋊C4—28no—
553C22⋊C4—28no—
554C22⋊C4—416no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—48no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—88no—
563C22⋊C4—88no—
564C22⋊C4—88no—
565C42—48no—
566C42—48no—
567C42—48no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—28no—
576C23—28no—
577C23—28no—
578C23—28no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—28no—
583C23—28no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23C2×#12114yes—
603C23#52514yes—
604C23#53414yes—
605C23#54618yes—
606C23#55118yes—
607C23C23×D411yescharacteristic
608C23C23×D411yescharacteristic
609C23C22×#4111yescharacteristic
610C23C22×#4112yes—
611C23C2×H(5,𝔽2)12yes—
612C23C2×H(5,𝔽2)12yes—
613C23C2×2−1+414yes—
614C23#58512yes—
615Q8—416no—
616D4—416no—
617D4—432no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—232no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—28no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—28no—
649C2×C4—28no—
650C2×C4—416no—
651C2×C4—432no—
652C2×C4—432no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—416no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—816no—
702C2×C4C2×#12114yes—
703C2×C4C2×#12114yes—
704C2×C4#53514yes—
705C22—216no—
706C22—24no—
707C22—28no—
708C22—28no—
709C22—24no—
710C22—28no—
711C22—28no—
712C22—44no—
713C22—48no—
714C22—44no—
715C22—48no—
716C22—44no—
717C22—84no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—84no—
722C22—82no—
723C22#253418yes—
724C22C22×#12111yescharacteristic
725C22C2×#52511yescharacteristic
726C22C2×#53411yescharacteristic
727C22C2×#53511yescharacteristic
728C22#344911yescharacteristic
729C22C2×#54612yes—
730C22C2×#55112yes—
731C22#348612yes—
732C22#348812yes—
733C22#349412yes—
734C22#349912yes—
735C22#350912yes—
736C22#351514yes—
737C22#352014yes—
738C22#353614yes—
739C22#355914yes—
740C4—24no—
741C4—24no—
742C4—24no—
743C4—42no—
744C4—42no—
745C4—44no—
746C4—42no—
747C4—44no—
748C4—44no—
749C4—48no—
750C4—82no—
751C4—82no—
752C4—84no—
753C4—82no—
754C4—82no—
755C2—28no—
756C2—84no—
757C2C2×#253411yescharacteristic
758C2#3525211yescharacteristic
759C2#3566711yescharacteristic
760C2#3600112yes—
761C2#3802912yes—
762C2#3890512yes—
763C2#4193712yes—
764C2#4404314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#398C2semidirect
3. C2×#2344C2semidirect
5. #16806C2semidirect
6. #16800C2semidirect
7. #15512C2semidirect
8. #15436C2semidirect
9. #15163C2semidirect
10. #15097C2semidirect
11. #14912C2semidirect
12. #14729C2semidirect
13. #14342C2semidirect
14. #14135C2semidirect
15. #14019C2semidirect
16. #13962C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24062
C22×#3982
C2×#23442
C2×#23112
#168062
#168002
#155122
#154362
#151632
#150972
#149122
#147292
#143422
#141352
#140192
#139622
#139562