Finite Group Databasebeta

← #612306

Subgroups — #612306

Order 512.

2612 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2396C212yesmaximal
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×#2298C211yesmaximal, characteristic
4#17063C211yesmaximal, characteristic, complemented
5#17062C211yesmaximal, characteristic
6#16821C212yesmaximal
7#16820C212yesmaximal, complemented
8#16804C212yesmaximal, complemented
9#16803C212yesmaximal
10#16518C212yesmaximal
11#16517C212yesmaximal, complemented
12#16507C211yesmaximal, characteristic, complemented
13#16506C211yesmaximal, characteristic
14C4×#398C212yesmaximal, complemented
15C4×#398C212yesmaximal, complemented
16C4×#397C212yesmaximal
17C4×#397C212yesmaximal
18#13850C212yesmaximal, complemented
19C4×#382C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C22×#117C2211yescharacteristic
23#2396—216no—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#397C2212yes—
30C2×#397C2212yes—
31C2×#396C2214yes—
32C2×#395C2212yes—
33C2×#393C2212yes—
34C4×C4⋊Q8—28no—
35#2298—28no—
36C2×#391C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#388C2211yescharacteristic
40C2×#388C2211yescharacteristic
41C2×#388C2211yescharacteristic
42C2×#388C2211yescharacteristic
43C2×#387C2211yescharacteristic
44C2×#387C2211yescharacteristic
45C2×#387C2211yescharacteristic
46C2×#387C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#382C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2214yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2214yes—
58C2×#376C2211yescharacteristic
59C2×C43C2211yescharacteristic
60#1446C2212yes—
61#1446C2212yes—
62#1445C2212yes—
63#1445C2212yes—
64#1445C2212yes—
65#1445C2212yes—
66#1445C2212yes—
67#1445C2212yes—
68#1444C2212yes—
69#1444C2212yes—
70#1443C2214yes—
71#1443C2214yes—
72#1442C2212yes—
73#1442C2212yes—
74#1442C2214yes—
75#1442C2214yes—
76#1441C2214yes—
77#1441C2214yes—
78#1441C2212yes—
79#1441C2214yes—
80#1441C2214yes—
81#1441C2212yes—
82#1440C2214yes—
83#1440C2214yes—
84#1439C2212yes—
85#1439C2212yes—
86C4⋊#95C2212yes—
87#1434C2211yescharacteristic
88#1434C2211yescharacteristic
89#1434C2212yes—
90C4×#95C2214yes—
91C4×#95C2214yes—
92C4×#95C2212yes—
93C4×#95C2212yes—
94C23×Q8C2312yes—
95C24×C4C2311yescharacteristic
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—24no—
98C2×C4×Q8—28no—
99C2×C4×Q8—28no—
100C2×C4×Q8—28no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117D418yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#400—28no—
140#399—28no—
141#398—216no—
142#398—28no—
143#398—28no—
144#397—28no—
145#397—28no—
146#396—216no—
147#395—28no—
148#393—28no—
149#391—28no—
150#389—28no—
151#389—28no—
152#388—24no—
153#388—24no—
154#388—24no—
155#388—24no—
156#387—24no—
157#387—24no—
158#387—24no—
159#387—24no—
160#384—24no—
161#382—28no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—216no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—24no—
168C4×C22⋊C4—216no—
169C4×C22⋊C4—24no—
170C4×C22⋊C4—24no—
171#376—24no—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C2×#95C2311yescharacteristic
199C2×#95C2311yescharacteristic
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2312yes—
206C43D418yes—
207C25C2411yescharacteristic
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8—28no—
211C22×Q8—24no—
212C22×Q8—44no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4C2×D414yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C4⋊Q8—232no—
237C4×Q8—432no—
238#117—232no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4#4118yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D414yes—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—28no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—28no—
311C2×C42—28no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42#4114yes—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—28no—
335#95—28no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—24no—
348#95—28no—
349#95—24no—
350#95—28no—
351#95—24no—
352#95—28no—
353#95—28no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95#4118yes—
360C24—21no—
361C24—22no—
362C24—22no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24C22×D412yes—
369C24C2511yescharacteristic
370C2×Q8—216no—
371C2×Q8—416no—
372C2×Q8—416no—
373C2×Q8—48no—
374C2×Q8—48no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—216no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C2×#4118yes—
462C22×C4C2×#4118yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4112yes—
468C22×C42−1+414yes—
469C22×C42−1+414yes—
470C4⋊C4—216no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—216no—
474C4⋊C4—416no—
475C4⋊C4—416no—
476C4⋊C4—416no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C22⋊C4—28no—
486C22⋊C4—28no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C42—28no—
494C42—28no—
495C42—28no—
496C42—28no—
497C42—28no—
498C42—44no—
499C42—44no—
500C42—416no—
501C42—416no—
502C42—44no—
503C42—44no—
504C42#121116yes—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23C23×D411yescharacteristic
523C23C23×D411yescharacteristic
524C23C22×#4111yescharacteristic
525C23C22×#4112yes—
526C23C22×#4112yes—
527C23C2×2−1+412yes—
528C23C2×2−1+412yes—
529C23#58512yes—
530C23#58512yes—
531Q8—432no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—416no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4C2×#12114yes—
615C2×C4C2×#12114yes—
616C2×C4C2×#12114yes—
617C2×C4#52514yes—
618C2×C4#53214yes—
619C2×C4#53614yes—
620C2×C4#53614yes—
621C2×C4#53618yes—
622C2×C4#54118yes—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—48no—
627C22—81no—
628C22—84no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22C22×#12111yescharacteristic
633C22C2×#52511yescharacteristic
634C22C2×#53211yescharacteristic
635C22C2×#53611yescharacteristic
636C22C2×#53611yescharacteristic
637C22C2×#53612yes—
638C22#344911yescharacteristic
639C22#344911yescharacteristic
640C22C2×#54112yes—
641C22#346114yes—
642C22#346114yes—
643C22D4×#4112yes—
644C22D4×#4112yes—
645C22#347512yes—
646C22#348012yes—
647C22#348214yes—
648C22#353114yes—
649C4—24no—
650C4—24no—
651C4—24no—
652C4—24no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—48no—
658C4—42no—
659C4—48no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—84no—
664C4—81no—
665C4—84no—
666C4#262118yes—
667C2—82no—
668C2#3460211yescharacteristic
669C2C2×#262111yescharacteristic
670C2#3463111yescharacteristic
671C2#3493511yescharacteristic
672C2#3493711yescharacteristic
673C2#3497211yescharacteristic
674C2#3669911yescharacteristic
675C2#5216614yes—
676C2#5228314yes—
subgroupG/H
centerC23×C4C24
commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Q8C2semidirect
4. #17063C2semidirect
7. #16820C2semidirect
8. #16804C2semidirect
11. #16517C2semidirect
12. #16507C2semidirect
14. C4×#398C2semidirect
15. C4×#398C2semidirect
18. #13850C2semidirect
19. C4×#382C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23962
C2×C4×C4⋊Q82
C2×#22982
#170632
#170622
#168212
#168202
#168042
#168032
#165182
#165172
#165072
#165062
C4×#3982
C4×#3972
#138502
C4×#3822