Finite Group Databasebeta

← #6466541

Subgroups — #6466541

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3137C212yesmaximal
2#35275C211yesmaximal, characteristic, complemented
3C4×#536C211yesmaximal, characteristic
4#28730C214yesmaximal, complemented
5#28729C214yesmaximal, complemented
6#28726C212yesmaximal, complemented
7#28725C212yesmaximal, complemented
8#28721C212yesmaximal, complemented
9#28717C212yesmaximal, complemented
10#23634C212yesmaximal
11#23633C212yesmaximal, complemented
12#21326C212yesmaximal, complemented
13#20648C211yesmaximal, characteristic
14#20647C211yesmaximal, characteristic, complemented
15#20645C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic, complemented
17#20431C211yesmaximal, characteristic, complemented
18C2×#544C2212yes—
19C2×#536C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#3137—216no—
23C2×#491C2214yes—
24C2×#489C2214yes—
25C2×#488C2214yes—
26C2×#487C2214yes—
27C2×#423C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C2×#417C2212yes—
30C2×#419C2212yes—
31C2×#416C2212yes—
32C22×#103C2212yes—
33C2×#405C2212yes—
34#2868C2212yes—
35#2859C2212yes—
36#2660C2214yes—
37#2552—21no—
38#2552—21no—
39#2551—21no—
40#2551—21no—
41#2440—22no—
42#2440—22no—
43#2327C2212yes—
44#2326C2212yes—
45#2323—21no—
46#2323—21no—
47#2322—22no—
48#2317C2211yescharacteristic
49#2317C2212yes—
50#2311C2211yescharacteristic
51#2310C2211yescharacteristic
52C4×C4⋊Q8C2212yes—
53C4×#124C2212yes—
54C4×C22⋊Q8—21no—
55C4×C22⋊Q8—21no—
56C4×#123—22no—
57#2285C2211yescharacteristic
58C2×#384C2211yescharacteristic
59C4×#117C2211yescharacteristic
60#2074C2214yes—
61#2074C2214yes—
62#2074C2214yes—
63#2073C2214yes—
64#2072C2214yes—
65#2072C2214yes—
66#2072C2214yes—
67#2071C2214yes—
68#1993C2214yes—
69#1992C2214yes—
70#1991C2214yes—
71#1990C2214yes—
72#1805C2212yes—
73#1805C2214yes—
74#1804C2212yes—
75#1803C2211yescharacteristic
76#1803C2212yes—
77#1803C2212yes—
78#1803C2212yes—
79#1803C2212yes—
80#1803C2211yescharacteristic
81#1803C2212yes—
82#1803C2212yes—
83#1803C2214yes—
84#1802C2211yescharacteristic
85#1802C2212yes—
86#1802C2211yescharacteristic
87#1802C2212yes—
88#1768C2214yes—
89#1767C2212yes—
90C4×C4⋊C8C2212yes—
91#1736C2212yes—
92#1736C2212yes—
93#1735C2212yes—
94C2×#340C2212yes—
95#1729C2212yes—
96C22×#41C2311yescharacteristic
97C23×Q8C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#544—28no—
101#544—28no—
102#536—24no—
103#536D418yes—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×#125—22no—
110C2×#125C2314yes—
111C2×#124C2312yes—
112C2×#123—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—24no—
116C4×#41—24no—
117C4×#41—24no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—24no—
121C2×C4×Q8—24no—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117—21no—
131C2×#117C2311yescharacteristic
132C2×#117C2312yes—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C42—21no—
141C22×C42—21no—
142C22×C42C2311yescharacteristic
143#491—216no—
144#489—216no—
145#488—216no—
146#487—216no—
147#423—24no—
148C2×C4⋊C8—24no—
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154#419—28no—
155#417—28no—
156#416—28no—
157C2×#103—28no—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168#405—28no—
169C2×#97C2314yes—
170C2×C4×C8C2314yes—
171#398—24no—
172#398—24no—
173#395—24no—
174#395—24no—
175#391—21no—
176#391—21no—
177#391C2312yes—
178#391C2312yes—
179#391C2312yes—
180#390C2312yes—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#386—22no—
200#385—22no—
201#385—24no—
202#385—22no—
203#385—22no—
204#384—22no—
205#384—22no—
206#384—21no—
207#384—21no—
208#384C2312yes—
209#384C2311yescharacteristic
210#384C2312yes—
211#384C2311yescharacteristic
212#384C2312yes—
213#384C2314yes—
214#382C2312yes—
215#382C2312yes—
216#382C2314yes—
217#381—22no—
218#380—21no—
219#380—21no—
220#380—21no—
221#380—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D411yescharacteristic
229C4×C4⋊C4D411yescharacteristic
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4D412yes—
233C4×C22⋊C4D412yes—
234#376C2312yes—
235C43C2312yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2312yes—
240#340C2314yes—
241#340C2314yes—
242#340C2312yes—
243#340C2312yes—
244#340C2314yes—
245#340C2314yes—
246#336C2314yes—
247#336C2314yes—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41C2×D414yes—
254C22×Q8—21no—
255C22×Q8—24no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8C2×D414yes—
262C22×Q8C2412yes—
263C22×Q8C2411yescharacteristic
264C22×D4—21no—
265C22×D4C2411yescharacteristic
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—22no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2411yescharacteristic
275C2×M16—24no—
276C2×M16—24no—
277C2×M16—24no—
278C22×C8—24no—
279C22×C8—24no—
280C22×C8—24no—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2414yes—
284C4⋊Q8—28no—
285C4⋊Q8—24no—
286C4⋊Q8—28no—
287C4⋊Q8—48no—
288C4⋊Q8—84no—
289C4⋊Q8—84no—
290C4⋊Q8C2×D418yes—
291#125—28no—
292#125—28no—
293#125—48no—
294#124—24no—
295#124C2×D418yes—
296#123—216no—
297C22⋊Q8—28no—
298C22⋊Q8—28no—
299C22⋊Q8—416no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—44no—
315C4×D4—44no—
316#117—22no—
317#117—22no—
318#117—24no—
319#117—22no—
320#117—22no—
321#117—24no—
322#117—44no—
323#117C2×D414yes—
324#117#4114yes—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4—82no—
354C2×C4⋊C4—82no—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2414yes—
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—44no—
382C2×C42—44no—
383C2×C42C2×D412yes—
384C2×C42C2×D412yes—
385C2×C42C2×D412yes—
386C2×C42C2×D412yes—
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42#4114yes—
390C2×C42#4114yes—
391C2×C42C2412yes—
392C2×C42C2412yes—
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C4⋊C8—22no—
396C4⋊C8—22no—
397C4⋊C8—24no—
398C4⋊C8—24no—
399C4⋊C8—24no—
400#103—28no—
401#103—28no—
402#103—24no—
403#103—24no—
404#103—28no—
405#103—28no—
406#103—28no—
407#102—24no—
408#102—24no—
409#102—28no—
410#97—28no—
411C4×C8—28no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—24no—
423#95—24no—
424#95C2×D412yes—
425#95C2×D412yes—
426#95C2×D412yes—
427#95C2×D412yes—
428C24—21no—
429C24C22×D411yescharacteristic
430#41—28no—
431#41—28no—
432#41—44no—
433#41—44no—
434C2×Q8—24no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—28no—
438C2×Q8—28no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—48no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—84no—
453C2×Q8—84no—
454C2×Q8C22×D414yes—
455C2×Q8C22×D412yes—
456C2×D4—24no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4C22×D412yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4#12414yes—
496C22×C4#12414yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C42−1+412yes—
511C22×C42−1+412yes—
512C22×C4C2511yescharacteristic
513M16—48no—
514C2×C8—22no—
515C2×C8—24no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—24no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—24no—
523C2×C8—48no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—44no—
544C4⋊C4—48no—
545C4⋊C4—48no—
546C4⋊C4—48no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—84no—
560C4⋊C4—84no—
561C4⋊C4#12114yes—
562C4⋊C4#12114yes—
563C22⋊C4—24no—
564C22⋊C4—28no—
565C22⋊C4—28no—
566C22⋊C4—44no—
567C22⋊C4—42no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4#12118yes—
573C42—24no—
574C42—24no—
575C42—22no—
576C42—22no—
577C42—42no—
578C42—42no—
579C42—44no—
580C42—42no—
581C42—42no—
582C42—42no—
583C42—44no—
584C42—44no—
585C42#12414yes—
586C42#12414yes—
587C42C22×D414yes—
588C42C2×#4114yes—
589C42C2×#4112yes—
590C42C2×#4112yes—
591C42C2×#4112yes—
592C42C2×#4112yes—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—22no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23—41no—
604C23C2×#12112yes—
605C23C2×#12412yes—
606C23#53612yes—
607C23C23×D411yescharacteristic
608Q8—24no—
609Q8—24no—
610Q8—22no—
611Q8—22no—
612Q8—42no—
613Q8—42no—
614Q8—41no—
615Q8—41no—
616Q8—48no—
617Q8—168no—
618D4—22no—
619D4—22no—
620D4—48no—
621D4—41no—
622D4—41no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—28no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—48no—
648C2×C4—48no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—84no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—164no—
671C2×C4C2×#12114yes—
672C2×C4C2×#12114yes—
673C2×C4C2×#12112yes—
674C2×C4C2×#12412yes—
675C2×C4C2×#12412yes—
676C2×C4C2×#12412yes—
677C2×C4C2×#12412yes—
678C2×C4C2×#12412yes—
679C2×C4#53612yes—
680C2×C4#54714yes—
681C2×C4#54714yes—
682C2×C4#55614yes—
683C2×C4#55614yes—
684C2×C4#57712yes—
685C2×C4#57812yes—
686C2×C4C23×D411yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C22×#4111yescharacteristic
689C2×C4C22×#4111yescharacteristic
690C2×C4C2×2−1+411yescharacteristic
691C2×C4C2×2−1+411yescharacteristic
692C8—42no—
693C8—42no—
694C8—44no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22—81no—
708C22#254714yes—
709C22C22×#12111yescharacteristic
710C22C22×#12411yescharacteristic
711C22C2×#53611yescharacteristic
712C22C2×#57711yescharacteristic
713C22C2×#57811yescharacteristic
714C22#358511yescharacteristic
715C22#358611yescharacteristic
716C4—24no—
717C4—24no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—42no—
722C4—44no—
723C4—42no—
724C4—42no—
725C4—84no—
726C4—82no—
727C4—82no—
728C4—81no—
729C4—82no—
730C4—81no—
731C4#254714yes—
732C4#313714yescomplemented
733C4C2×#54711yescharacteristic
734C4C2×#54711yescharacteristic
735C4C2×#55611yescharacteristic
736C4C2×#55611yescharacteristic
737C2—24no—
738C2—82no—
739C2C2×#254711yescharacteristic
740C2#6043111yescharacteristic
741C2#6043211yescharacteristic
742C2C2×#313711yescharacteristic
743C2#6062111yescharacteristic
744C2#6088911yescharacteristic
745C2#6089311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35275C2semidirect
4. #28730C2semidirect
5. #28729C2semidirect
6. #28726C2semidirect
7. #28725C2semidirect
8. #28721C2semidirect
9. #28717C2semidirect
11. #23633C2semidirect
12. #21326C2semidirect
14. #20647C2semidirect
16. C2×#1803C2semidirect
17. #20431C2semidirect
732. C4#3137semidirect

Maximal subgroups

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

subgroupindex
C2×#31372
#352752
C4×#5362
#287302
#287292
#287262
#287252
#287212
#287172
#236342
#236332
#213262
#206482
#206472
#206452
C2×#18032
#204312