Finite Group Databasebeta

← #6458658

Subgroups — #6458658

Order 512.

1946 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34188C211yesmaximal, characteristic, complemented
2C4×#535C211yesmaximal, characteristic
3#27641C214yesmaximal, complemented
4#27640C214yesmaximal, complemented
5#27627C212yesmaximal, complemented
6#27626C212yesmaximal, complemented
7#27625C212yesmaximal, complemented
8#27624C212yesmaximal, complemented
9#23643C212yesmaximal, complemented
10#23486C212yesmaximal
11#21171C211yesmaximal, characteristic, complemented
12#21169C211yesmaximal, characteristic, complemented
13#21063C212yesmaximal, complemented
14#21039C212yesmaximal, complemented
15#20646C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic, complemented
17#20629C211yesmaximal, characteristic, complemented
18C2×#535C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#418C2212yes—
26C22×#102C2212yes—
27C2×#408C2212yes—
28#2355—21no—
29#2355—21no—
30#2354—21no—
31#2354—21no—
32#2352—22no—
33#2352—22no—
34#2328—21no—
35#2326—21no—
36#2324C2212yes—
37#2324C2212yes—
38#2321—22no—
39#2317C2211yescharacteristic
40#2314C2214yescomplemented
41#2311C2211yescharacteristic
42#2311C2212yes—
43#2309C2211yescharacteristic
44C4×#127—21no—
45C4×#124—21no—
46C4×#121C2212yes—
47C4×#121C2212yes—
48C4×#120—22no—
49#2285C2211yescharacteristic
50#2285C2212yescomplemented
51#2285C2212yescomplemented
52C2×#384C2211yescharacteristic
53#2274C2212yes—
54C4×#117C2211yescharacteristic
55#2077C2214yescomplemented
56#2077C2214yescomplemented
57#2076C2214yescomplemented
58#2075C2214yescomplemented
59#2034C2214yes—
60#2034C2214yes—
61#2033C2214yes—
62#2033C2214yes—
63#1949C2212yescomplemented
64#1949C2212yescomplemented
65#1948C2211yescharacteristic, complemented
66#1948C2211yescharacteristic, complemented
67#1948C2212yescomplemented
68#1948C2212yescomplemented
69#1947C2211yescharacteristic, complemented
70#1947C2211yescharacteristic, complemented
71#1947C2212yescomplemented
72#1947C2212yescomplemented
73#1926C2214yes—
74#1926C2214yes—
75#1925C2214yes—
76#1925C2214yes—
77#1921C2214yescomplemented
78#1920C2214yescomplemented
79#1919C2214yescomplemented
80#1918C2214yescomplemented
81#1804C2214yes—
82#1802C2212yes—
83#1802C2212yes—
84#1802C2214yes—
85#1802C2212yes—
86#1802C2212yes—
87#1798C2214yescomplemented
88#1797C2212yescomplemented
89#1797C2212yescomplemented
90#1736C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93C2×#340C2212yes—
94#1729C2212yes—
95C22×#41C2311yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×M16C2312yes—
99C23×C8C2312yes—
100#535—24no—
101#535—24no—
102C2×#127—21no—
103C2×#124—21no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#120—22no—
107C4×#41—24no—
108C4×#41—24no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—24no—
116C2×C4×D4—24no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D412yes—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D412yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×#117—22no—
124C2×#117—22no—
125C2×#117C2311yescharacteristic
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2312yes—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4—21no—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#428—44no—
138#427—28no—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#107C2314yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×#106C2314yes—
145#423—24no—
146C2×C4⋊C8—24no—
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152#418—28no—
153C2×#102—44no—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158#408—44no—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161#391—21no—
162#391—22no—
163#390—21no—
164#390—22no—
165#390—21no—
166#390—21no—
167#390—22no—
168#390—22no—
169#390C2312yes—
170#390C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#388—24no—
176#388—24no—
177#388—22no—
178#388—22no—
179#387—24no—
180#387—24no—
181#386—22no—
182#386—24no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#382—21no—
198#382—22no—
199#382—22no—
200#382C2312yes—
201#380—21no—
202#380—21no—
203#380—22no—
204#380—22no—
205#380C2314yes—
206#380C2314yes—
207#380C2312yes—
208#380C2312yes—
209#379—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—24no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227#376—21no—
228C43—21no—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#340C2312yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C25—21no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C22×Q8C2×D411yescharacteristic
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4C2×D412yes—
258C22×D4C2×D411yescharacteristic
259C22×D4#4114yes—
260C22×D4C2412yes—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—44no—
266C23×C4—82no—
267C23×C4—82no—
268C23×C4C2×D412yes—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D412yes—
271C23×C4C2×D411yescharacteristic
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C2×M16—24no—
275C2×M16—24no—
276C2×M16—24no—
277C22×C8—24no—
278C22×C8—24no—
279C22×C8—24no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2414yes—
283#127—28no—
284#124—28no—
285#121—24no—
286#121—24no—
287#121—24no—
288#121—24no—
289#120—216no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—84no—
303C4×D4—84no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—24no—
308#117—24no—
309#117—24no—
310#117—24no—
311#117—22no—
312#117—22no—
313#117—84no—
314#117—84no—
315#117#4114yes—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—82no—
333C2×C4⋊C4—82no—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—44no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C42—21no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—24no—
373C2×C42—24no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42#4114yes—
381C2×C42#4114yes—
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—24no—
386#107—24no—
387#107—44no—
388#106—24no—
389#106—24no—
390#106—44no—
391C4⋊C8—22no—
392C4⋊C8—22no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396#102—28no—
397#102—28no—
398#102—44no—
399#102—44no—
400C22⋊C8—44no—
401C22⋊C8—44no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—24no—
415#95—24no—
416C24—22no—
417C24—24no—
418C24—42no—
419C24—42no—
420C24—81no—
421C24—81no—
422C24—81no—
423C24—81no—
424C24C22×D412yes—
425C24C22×D411yescharacteristic
426#41—44no—
427#41—44no—
428#41—44no—
429#41—44no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—28no—
437C2×D4—28no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—28no—
443C2×D4—28no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—82no—
454C2×D4—82no—
455C2×D4—82no—
456C2×D4—82no—
457C2×D4C2×#4114yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—41no—
479C22×C4—42no—
480C22×C4—41no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—816no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4#12314yes—
496C22×C4#12314yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4H(5,𝔽2)12yes—
510C22×C42−1+412yes—
511C22×C4C2511yescharacteristic
512M16—48no—
513C2×C8—24no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—24no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—24no—
522C2×C8—48no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—24no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4C2×#4112yes—
554C4⋊C4C2×#4112yes—
555C22⋊C4—28no—
556C22⋊C4—28no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—44no—
562C22⋊C4—42no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—84no—
570C22⋊C4—84no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C42—24no—
574C42—24no—
575C42—24no—
576C42—22no—
577C42—22no—
578C42—44no—
579C42—42no—
580C42—42no—
581C42—42no—
582C42—81no—
583C42—81no—
584C42—84no—
585C42—84no—
586C42#12314yes—
587C42#12314yes—
588C42C2×#4112yes—
589C42C2×#4112yes—
590C42C2×#4112yes—
591C42C2×#4112yes—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—24no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—22no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—48no—
609C23—42no—
610C23—42no—
611C23—84no—
612C23—84no—
613C23—84no—
614C23—84no—
615C23—84no—
616C23—84no—
617C23C2×#12312yes—
618C23#52512yes—
619C23#53712yes—
620C23C23×D411yescharacteristic
621Q8—44no—
622D4—24no—
623D4—24no—
624D4—48no—
625D4—42no—
626D4—42no—
627D4—48no—
628D4—44no—
629D4—88no—
630D4—88no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—28no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—44no—
650C2×C4—48no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—48no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—44no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—44no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—88no—
679C2×C4—84no—
680C2×C4—88no—
681C2×C4—164no—
682C2×C4C2×#12312yes—
683C2×C4C2×#12312yes—
684C2×C4C2×#12312yes—
685C2×C4C2×#12312yes—
686C2×C4C2×#12312yes—
687C2×C4#52512yes—
688C2×C4#53712yes—
689C2×C4#55214yes—
690C2×C4#55314yes—
691C2×C4#55514yes—
692C2×C4#55614yes—
693C2×C4#57612yes—
694C2×C4#57712yes—
695C2×C4C22×#4111yescharacteristic
696C2×C4C22×#4111yescharacteristic
697C2×C4C22×#4111yescharacteristic
698C2×C4C22×#4111yescharacteristic
699C2×C4C2×H(5,𝔽2)11yescharacteristic
700C2×C4C2×2−1+411yescharacteristic
701C8—44no—
702C8—42no—
703C8—42no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—44no—
714C22—42no—
715C22—44no—
716C22—42no—
717C22—44no—
718C22—42no—
719C22—42no—
720C22—816no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22#248514yes—
726C22C22×#12311yescharacteristic
727C22C2×#52511yescharacteristic
728C22C2×#53711yescharacteristic
729C22C2×#57611yescharacteristic
730C22C2×#57711yescharacteristic
731C22#358511yescharacteristic
732C22#358611yescharacteristic
733C4—22no—
734C4—22no—
735C4—22no—
736C4—42no—
737C4—42no—
738C4—42no—
739C4—42no—
740C4—42no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—81no—
745C4—84no—
746C4—81no—
747C4#248514yes—
748C4#309614yes—
749C4C2×#55211yescharacteristic
750C4C2×#55311yescharacteristic
751C4C2×#55511yescharacteristic
752C4C2×#55611yescharacteristic
753C2—24no—
754C2—82no—
755C2—84no—
756C2C2×#248511yescharacteristic
757C2C2×#309611yescharacteristic
758C2#6051711yescharacteristic
759C2#6059611yescharacteristic
760C2#6059711yescharacteristic
761C2#6092211yescharacteristic
762C2#6092311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34188C2semidirect
3. #27641C2semidirect
4. #27640C2semidirect
5. #27627C2semidirect
6. #27626C2semidirect
7. #27625C2semidirect
8. #27624C2semidirect
9. #23643C2semidirect
11. #21171C2semidirect
12. #21169C2semidirect
13. #21063C2semidirect
14. #21039C2semidirect
15. #20646C2semidirect
16. C2×#1802C2semidirect
17. #20629C2semidirect
40. #2314C22semidirect
50. #2285C22semidirect
51. #2285C22semidirect
55. #2077C22semidirect
56. #2077C22semidirect
57. #2076C22semidirect
58. #2075C22semidirect
63. #1949C22semidirect
64. #1949C22semidirect
65. #1948C22semidirect
66. #1948C22semidirect
67. #1948C22semidirect
68. #1948C22semidirect
69. #1947C22semidirect
70. #1947C22semidirect
71. #1947C22semidirect
72. #1947C22semidirect
77. #1921C22semidirect
78. #1920C22semidirect
79. #1919C22semidirect
80. #1918C22semidirect
87. #1798C22semidirect
88. #1797C22semidirect
89. #1797C22semidirect

Maximal subgroups

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

subgroupindex
#341882
C4×#5352
#276412
#276402
#276272
#276262
#276252
#276242
#236432
#234862
#211712
#211692
#210632
#210392
#206462
C2×#18022
#206292