Finite Group Databasebeta

← #6451368

Subgroups — #6451368

Order 512.

2078 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35554C211yesmaximal, characteristic
2#34254C211yesmaximal, characteristic, complemented
3#27214C214yesmaximal, complemented
4#27207C214yesmaximal, complemented
5#27203C214yesmaximal, complemented
6#27194C214yesmaximal, complemented
7#23735C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic, complemented
9#23619C211yesmaximal, characteristic, complemented
10C2×#2014C211yesmaximal, characteristic
11#21260C211yesmaximal, characteristic
12#21231C211yesmaximal, characteristic, complemented
13#21063C211yesmaximal, characteristic, complemented
14#21032C211yesmaximal, characteristic, complemented
15#21027C211yesmaximal, characteristic, complemented
16#20936C211yesmaximal, characteristic, complemented
17#20936C211yesmaximal, characteristic, complemented
18#20933C211yesmaximal, characteristic, complemented
19#20416C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C22×#127C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2445—22no—
38#2445—22no—
39#2445C2214yes—
40#2445C2214yes—
41#2405—24no—
42#2356C2214yescomplemented
43#2355C2214yescomplemented
44#2343—24no—
45#2320—22no—
46#2317C2211yescharacteristic
47#2310C2211yescharacteristic
48#2309C2211yescharacteristic
49#2309C2211yescharacteristic
50#2309C2211yescharacteristic
51#2296C2211yescharacteristic
52#2296C2211yescharacteristic
53#2295—22no—
54C2×#391C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2094C2212yescomplemented
58#2093C2212yescomplemented
59#2092C2212yescomplemented
60#2092C2212yescomplemented
61#2073C2214yescomplemented
62#2071C2214yescomplemented
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2014C2214yes—
66#2014C2214yes—
67#1982C2214yes—
68#1982C2214yes—
69#1973C2212yescomplemented
70#1973C2212yescomplemented
71#1968C2212yescomplemented
72#1968C2212yescomplemented
73#1926C2214yescomplemented
74#1925C2214yescomplemented
75#1921C2212yescomplemented
76#1920C2212yescomplemented
77#1919C2212yescomplemented
78#1918C2212yescomplemented
79#1917C2214yescomplemented
80#1916C2214yescomplemented
81#1880C2214yescomplemented
82#1880C2214yescomplemented
83#1880C2214yescomplemented
84#1880C2214yescomplemented
85#1877C2214yescomplemented
86#1877C2214yescomplemented
87#1762C2214yescomplemented
88C4×#102C2214yescomplemented
89#1736C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4—22no—
99C23×D4C2311yescharacteristic
100C23×D4C2311yescharacteristic
101C22×SD16—24no—
102C22×SD16—24no—
103C22×SD16—24no—
104C22×SD16—24no—
105C22×SD16—42no—
106C22×SD16—42no—
107C22×SD16C2314yes—
108C22×SD16C2312yes—
109C22×SD16C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114C2×#127—24no—
115C2×#127—21no—
116C2×#127—21no—
117C2×#127—42no—
118C2×#127C2314yes—
119C2×#123—24no—
120C2×#123—24no—
121#519—24no—
122#519—24no—
123#518—24no—
124#518—42no—
125C2×C4×Q8—22no—
126C2×C4×Q8C2314yes—
127C2×C4×D4—22no—
128C2×C4×D4C2314yes—
129C2×#117—22no—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—22no—
138C22×C42C2311yescharacteristic
139#428—42no—
140#427—24no—
141C2×#107C2311yescharacteristic
142C2×#107C2311yescharacteristic
143C2×#107C2312yes—
144C2×#106C2311yescharacteristic
145C2×#106C2311yescharacteristic
146C2×#106C2312yes—
147#423—42no—
148C2×C4⋊C8C2314yes—
149#419—42no—
150#419—42no—
151#418—24no—
152#418—24no—
153#418—42no—
154#418—42no—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#102—24no—
159C2×#102—42no—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169#405—42no—
170C2×#97C2312yes—
171C2×C4×C8C2312yes—
172#394—24no—
173#394—24no—
174#394—24no—
175#394—24no—
176#391—22no—
177#391—21no—
178#391—21no—
179#391C2312yes—
180#391C2312yes—
181#391C2314yes—
182#390—22no—
183#390—22no—
184#390—22no—
185#390C2314yes—
186#390C2312yes—
187#390C2312yes—
188#390C2312yes—
189#390C2312yes—
190#389—22no—
191#389D412yes—
192#389D412yes—
193#389C2312yes—
194#389C2312yes—
195#387—24no—
196#387—24no—
197#387—24no—
198#387—24no—
199#386—24no—
200#386—24no—
201#385—22no—
202#385—22no—
203#385D414yes—
204#384C2312yes—
205#383—24no—
206#382—24no—
207#382C2312yes—
208#381—22no—
209#379—22no—
210#379—22no—
211#379—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222#376—21no—
223#376—24no—
224C2×#95—22no—
225C43—21no—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C25—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8#4114yes—
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—24no—
252C22×D4—24no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—48no—
265C22×D4—84no—
266C22×D4—84no—
267C22×D4#4114yes—
268C22×D4C2412yes—
269C22×D4C2412yes—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—24no—
273C23×C4C2×D412yes—
274C23×C4C2411yescharacteristic
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—28no—
278C2×SD16—28no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—42no—
282C2×SD16—42no—
283C2×SD16—44no—
284C2×SD16—416no—
285C2×SD16—416no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8#4114yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2411yescharacteristic
302C22×C8C2411yescharacteristic
303#127—28no—
304#127—44no—
305#127—44no—
306#127—44no—
307#123—88no—
308#123—88no—
309C4×Q8—216no—
310C4×Q8—28no—
311C4×Q8—28no—
312C4×D4—28no—
313C4×D4—44no—
314C4×D4—48no—
315#117—216no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117#4114yes—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D414yes—
338C2×C4⋊C4C2×D414yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—48no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—24no—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2412yes—
373C2×C42C2412yes—
374C2×C42C2412yes—
375#107—22no—
376#107—22no—
377#107—42no—
378#106—22no—
379#106—22no—
380#106—42no—
381C4⋊C8—44no—
382#103—42no—
383#103—42no—
384#103—44no—
385#102—24no—
386#102—24no—
387#102—28no—
388#102—28no—
389#102—44no—
390#102—42no—
391#102—42no—
392#102—42no—
393#102—42no—
394#97—42no—
395C4×C8—42no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—22no—
405#95—22no—
406#95C2×D414yes—
407#95C2×D414yes—
408C24—22no—
409C24—24no—
410C24—22no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—82no—
417C24—82no—
418C24—82no—
419C24—82no—
420C24C22×D411yescharacteristic
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—28no—
424C2×Q8—28no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8C2×#4114yes—
434C2×D4—28no—
435C2×D4—24no—
436C2×D4—28no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—48no—
449C2×D4—48no—
450C2×D4—48no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—48no—
463C2×D4—84no—
464C2×D4—84no—
465C2×D4—832no—
466C2×D4C2×#4114yes—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—42no—
479C22×C4—48no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4—82no—
486C22×C4C22×D412yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4H(5,𝔽2)12yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4C2511yescharacteristic
498SD16—48no—
499SD16—48no—
500SD16—416no—
501M16—44no—
502M16—44no—
503C2×C8—28no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—44no—
518C2×C8C2×#4112yes—
519C2×C8C2×#4112yes—
520C4⋊C4—216no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—28no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—44no—
541C4⋊C4—84no—
542C4⋊C4C2×#4114yes—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—48no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—216no—
557C42—24no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—42no—
567C42—42no—
568C42—42no—
569C42—44no—
570C42—82no—
571C42—84no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—44no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—48no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—44no—
585C23—88no—
586C23—88no—
587C23—84no—
588C23—164no—
589C23—164no—
590C23—162no—
591C23—162no—
592C23—162no—
593C23—162no—
594C23#52512yes—
595C23#54612yes—
596C23#55012yes—
597C23C23×D411yescharacteristic
598Q8—24no—
599Q8—24no—
600Q8—42no—
601Q8—42no—
602Q8—48no—
603D4—24no—
604D4—24no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—48no—
612D4—816no—
613D4—816no—
614D4—84no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—48no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—48no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—168no—
648C2×C4—162no—
649C2×C4#52512yes—
650C2×C4#52914yes—
651C2×C4#54014yes—
652C2×C4#54612yes—
653C2×C4#55012yes—
654C2×C4#55114yes—
655C2×C4#55114yes—
656C2×C4#57714yes—
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C8—22no—
664C8—22no—
665C8—42no—
666C8—42no—
667C8—42no—
668C22—24no—
669C22—24no—
670C22—22no—
671C22—22no—
672C22—48no—
673C22—44no—
674C22—84no—
675C22—82no—
676C22—82no—
677C22—84no—
678C22—82no—
679C22—84no—
680C22—82no—
681C22—82no—
682C22—168no—
683C22—168no—
684C22#247114yes—
685C22C2×#52511yescharacteristic
686C22C2×#54611yescharacteristic
687C22C2×#55011yescharacteristic
688C22C2×#57712yes—
689C22#358512yes—
690C4—24no—
691C4—42no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—44no—
696C4—82no—
697C4—81no—
698C4—81no—
699C4—81no—
700C4—81no—
701C4—82no—
702C4#247114yes—
703C4C2×#52911yescharacteristic
704C4C2×#54011yescharacteristic
705C4C2×#55111yescharacteristic
706C4C2×#55111yescharacteristic
707C2—24no—
708C2—84no—
709C2—162no—
710C2C2×#247111yescharacteristic
711C2#6059312yes—
712C2#6118412yes—
713C2#6141011yescharacteristic
714C2#6141711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34254C2semidirect
3. #27214C2semidirect
4. #27207C2semidirect
5. #27203C2semidirect
6. #27194C2semidirect
7. #23735C2semidirect
8. #23632C2semidirect
9. #23619C2semidirect
12. #21231C2semidirect
13. #21063C2semidirect
14. #21032C2semidirect
15. #21027C2semidirect
16. #20936C2semidirect
17. #20936C2semidirect
18. #20933C2semidirect
19. #20416C2semidirect
42. #2356C22semidirect
43. #2355C22semidirect
57. #2094C22semidirect
58. #2093C22semidirect
59. #2092C22semidirect
60. #2092C22semidirect
61. #2073C22semidirect
62. #2071C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
69. #1973C22semidirect
70. #1973C22semidirect
71. #1968C22semidirect
72. #1968C22semidirect
73. #1926C22semidirect
74. #1925C22semidirect
75. #1921C22semidirect
76. #1920C22semidirect
77. #1919C22semidirect
78. #1918C22semidirect
79. #1917C22semidirect
80. #1916C22semidirect
81. #1880C22semidirect
82. #1880C22semidirect
83. #1880C22semidirect
84. #1880C22semidirect
85. #1877C22semidirect
86. #1877C22semidirect
87. #1762C22semidirect
88. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#355542
#342542
#272142
#272072
#272032
#271942
#237352
#236322
#236192
C2×#20142
#212602
#212312
#210632
#210322
#210272
#209362
#209332
#204162