Finite Group Databasebeta

← #460498

Subgroups — #460498

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2101C212yesmaximal, complemented
3#14682C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic, complemented
5#11384C211yesmaximal, characteristic, complemented
6#11267C211yesmaximal, characteristic, complemented
7#11267C211yesmaximal, characteristic, complemented
8#11208C212yesmaximal, complemented
9#11200C212yesmaximal
10#11199C212yesmaximal
11#10712C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C23×C42C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16#2857—28no—
17C2×#391—21no—
18C2×#389—21no—
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380—21no—
24C2×#380C2212yes—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27#2101—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42#1434—21no—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C23×Q8—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C22⋊Q8—28no—
56C2×C22⋊Q8—44no—
57C2×C22⋊Q8—44no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—24no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—42no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×C22⋊C8—24no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—42no—
82#391—28no—
83#389—28no—
84#389—24no—
85#389D418yes—
86#384—24no—
87#382—24no—
88#382—42no—
89#380—28no—
90#380—24no—
91#380—24no—
92#380—24no—
93#380—24no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340—42no—
114#336—24no—
115#336—22no—
116#336—22no—
117C25C2×D411yescharacteristic
118C22×Q8—24no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—41no—
151C23×C4—41no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4114yes—
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C4⋊Q8—48no—
176C4⋊Q8—48no—
177C22⋊Q8—432no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×D414yes—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—48no—
237C2×C42—48no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42C2×D414yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C22⋊C8—48no—
244#95—28no—
245#95—28no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—42no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24#12412yes—
269C24AGL(1,ℤ/8)12yes—
270C24#13712yes—
271C24C22×D411yescharacteristic
272C2×Q8—48no—
273C2×Q8—48no—
274C2×Q8—48no—
275C2×Q8—84no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8—84no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4#12012yes—
378C22×C4#12012yes—
379C22×C4#12312yes—
380C22×C4#12312yes—
381C22×C4#12412yes—
382C22×C4#13512yes—
383C22×C4#13512yes—
384C22×C4AGL(1,ℤ/8)12yes—
385C22×C4#13712yes—
386C22×C4C22×D412yes—
387C22×C4C2×#4112yes—
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)12yes—
392C22×C42−1+411yescharacteristic
393C22×C42−1+412yes—
394C22×C42−1+411yescharacteristic
395C22×C42−1+411yescharacteristic
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—416no—
417C4⋊C4—48no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C22⋊C4—416no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C42—24no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—44no—
438C42—44no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23#48914yes—
474C23C2×#12011yescharacteristic
475C23C2×#12311yescharacteristic
476C23C2×#12411yescharacteristic
477C23#53411yescharacteristic
478C23#53611yescharacteristic
479C23#53611yescharacteristic
480C23#53711yescharacteristic
481C23C2×#13511yescharacteristic
482C23C2×#13511yescharacteristic
483C23C2×AGL(1,ℤ/8)11yescharacteristic
484C23C2×AGL(1,ℤ/8)11yescharacteristic
485C23C2×#13711yescharacteristic
486C23C2×#13711yescharacteristic
487C23#57511yescharacteristic
488C23#57511yescharacteristic
489Q8—816no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—416no—
512C2×C4—416no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4#48214yes—
572C2×C4#48514yes—
573C2×C4#53612yes—
574C2×C4#53712yes—
575C2×C4#53812yes—
576C2×C4#53912yes—
577C2×C4#54111yescharacteristic
578C2×C4#54111yescharacteristic
579C2×C4#54211yescharacteristic
580C2×C4#54211yescharacteristic
581C2×C4#54612yes—
582C2×C4#54612yes—
583C2×C4#54812yes—
584C2×C4#54812yes—
585C2×C4#55212yes—
586C2×C4#55212yes—
587C2×C4#56312yes—
588C2×C4#56312yes—
589C8—82no—
590C8—82no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22#240511yescharacteristic
607C22#243611yescharacteristic
608C22#250311yescharacteristic
609C22#261611yescharacteristic
610C22#262511yescharacteristic
611C22#263911yescharacteristic
612C22#281711yescharacteristic
613C22#300211yescharacteristic
614C22#300211yescharacteristic
615C22#300311yescharacteristic
616C22#300411yescharacteristic
617C22C2×#48211yescharacteristic
618C22C2×#48511yescharacteristic
619C22#309411yescharacteristic
620C22#309511yescharacteristic
621C22#313111yescharacteristic
622C22C2×#48911yescharacteristic
623C22#314011yescharacteristic
624C22#314111yescharacteristic
625C22#318811yescharacteristic
626C22#319011yescharacteristic
627C22#319411yescharacteristic
628C22#319611yescharacteristic
629C22#322811yescharacteristic
630C22#323111yescharacteristic
631C22#323311yescharacteristic
632C22#323311yescharacteristic
633C22#323411yescharacteristic
634C22#323611yescharacteristic
635C22#323811yescharacteristic
636C22#323911yescharacteristic
637C22#324711yescharacteristic
638C22#324811yescharacteristic
639C22#325111yescharacteristic
640C22#325211yescharacteristic
641C4—24no—
642C4—24no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—84no—
647C4—84no—
648C4—161no—
649C4—161no—
650C4—161no—
651C4—161no—
652C4#269012yes—
653C4#269012yes—
654C4#270212yes—
655C4#270212yes—
656C2—44no—
657C2#1636611yescharacteristic
658C2#2400911yescharacteristic
659C2#2401011yescharacteristic
660C2#2590611yescharacteristic
661C2#2590811yescharacteristic
662C2#2796211yescharacteristic
663C2#2796911yescharacteristic
664C2#2969411yescharacteristic
665C2#2969511yescharacteristic
666C2#2994111yescharacteristic
667C2#2994211yescharacteristic
668C2#3018811yescharacteristic
669C2#3020011yescharacteristic
670C2#3278711yescharacteristic
671C2#3278911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2101C2semidirect
4. #11384C2semidirect
5. #11384C2semidirect
6. #11267C2semidirect
7. #11267C2semidirect
8. #11208C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#21012
#146822
#113842
#112672
#112082
#112002
#111992
#107122