Finite Group Databasebeta

← #6464265

Subgroups — #6464265

Order 512.

1938 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#35254C211yesmaximal, characteristic
2#34268C211yesmaximal, characteristic, complemented
3#29491C214yesmaximal, complemented
4#29490C214yesmaximal, complemented
5#29483C214yesmaximal, complemented
6#29482C214yesmaximal, complemented
7#23639C211yesmaximal, characteristic, complemented
8#23487C211yesmaximal, characteristic, complemented
9#23486C211yesmaximal, characteristic
10#23388C211yesmaximal, characteristic
11C2×#1973C211yesmaximal, characteristic, complemented
12#21154C211yesmaximal, characteristic, complemented
13#21095C211yesmaximal, characteristic, complemented
14#21094C211yesmaximal, characteristic, complemented
15#21065C211yesmaximal, characteristic, complemented
16#21037C211yesmaximal, characteristic, complemented
17#20936C211yesmaximal, characteristic, complemented
18#20880C211yesmaximal, characteristic, complemented
19#20375C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#535C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×C4×#41C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C2×#408C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×C4×M16C2211yescharacteristic
37C22×#97C2211yescharacteristic
38#2854C2214yes—
39#2624C2214yes—
40#2448—22no—
41#2445—22no—
42#2418—24no—
43#2367—22no—
44#2365—22no—
45#2359—24no—
46#2323C2214yescomplemented
47#2316C2211yescharacteristic
48#2311C2211yescharacteristic
49#2311C2211yescharacteristic
50#2309C2211yescharacteristic
51C4×C22⋊Q8C2214yescomplemented
52#2296C2211yescharacteristic
53#2296C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#390C2211yescharacteristic
56#2274C2211yescharacteristic
57#2076C2214yescomplemented
58#2075C2214yescomplemented
59#2036C2214yescomplemented
60#2035C2214yescomplemented
61#2034C2214yes—
62#2033C2214yes—
63#2000C2214yes—
64#2000C2214yes—
65#1973C2212yes—
66#1973C2212yes—
67#1973C2212yes—
68#1973C2212yes—
69#1944C2212yescomplemented
70#1944C2212yescomplemented
71#1943C2212yescomplemented
72#1943C2212yescomplemented
73#1936C2214yes—
74#1935C2214yescomplemented
75#1934C2214yes—
76#1933C2214yescomplemented
77#1926C2214yes—
78#1925C2214yes—
79#1921C2214yescomplemented
80#1918C2214yescomplemented
81#1880C2214yes—
82#1880C2214yes—
83#1865C2214yescomplemented
84#1863C2214yescomplemented
85#1750C2212yescomplemented
86#1750C2212yescomplemented
87#1749C2212yescomplemented
88#1749C2212yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—24no—
102C22×SD16—42no—
103C22×SD16C2312yes—
104C22×SD16C2312yes—
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#536—24no—
110#536—24no—
111#535—42no—
112#535—42no—
113C2×C4⋊Q8—21no—
114C2×#127—21no—
115C2×#124—21no—
116C2×#124—21no—
117C2×#123—22no—
118C2×#123—24no—
119C2×C22⋊Q8C2314yes—
120C2×#121C2314yes—
121C2×#120—22no—
122#519—42no—
123#519—42no—
124C4×#41—42no—
125C4×#41—42no—
126C2×C4×Q8—22no—
127C2×C4×Q8D412yes—
128C2×C4×Q8C2314yes—
129C2×C4×D4D414yes—
130C2×C4×D4D412yes—
131C2×#117—22no—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2312yes—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C22⋊C4—22no—
141C22×C42C2311yescharacteristic
142C22×C42C2312yes—
143#427—24no—
144#427—42no—
145C2×#107—24no—
146C2×#107C2312yes—
147C2×#107C2312yes—
148C2×#107C2312yes—
149C2×#107C2312yes—
150C2×#106C2312yes—
151C2×#106C2312yes—
152#419—24no—
153#419—42no—
154#418—24no—
155#418—42no—
156C2×#103—24no—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102—24no—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166#408—24no—
167#408—24no—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2312yes—
172C4×M16—42no—
173C2×#97—42no—
174C2×#97C2311yescharacteristic
175C2×#97C2311yescharacteristic
176C2×#97C2311yescharacteristic
177C2×#97C2311yescharacteristic
178C2×#97C2312yes—
179C2×C4×C8C2312yes—
180#399—24no—
181#397—24no—
182#394—24no—
183#394—24no—
184#391—21no—
185#391—21no—
186#391—22no—
187#391C2312yes—
188#391C2312yes—
189#390—21no—
190#390—21no—
191#390—22no—
192#390—22no—
193#390C2314yes—
194#390C2312yes—
195#390C2312yes—
196#389—22no—
197#389—24no—
198#389C2314yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#388—24no—
204#388—24no—
205#388—24no—
206#388—24no—
207#387—24no—
208#387—24no—
209#386—22no—
210#386—22no—
211#386—24no—
212#385—22no—
213#385—24no—
214#385—22no—
215#380—22no—
216#380C2314yes—
217#380C2314yes—
218#379—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—24no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2314yes—
224C4×C22⋊C4—24no—
225C4×C22⋊C4—22no—
226C2×#95—22no—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8C2×D411yescharacteristic
253C22×Q8#4114yes—
254C22×Q8C2412yes—
255C22×D4—24no—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4C2×D412yes—
264C22×D4C2×D411yescharacteristic
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4C2×D412yes—
270C23×C4C2×D414yes—
271C23×C4C2412yes—
272C23×C4C2411yescharacteristic
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—42no—
278C2×SD16—42no—
279C2×SD16—416no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8#4114yes—
291C22×C8#4114yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2411yescharacteristic
295C22×C8C2411yescharacteristic
296C22×C8C2411yescharacteristic
297C22×C8C2411yescharacteristic
298C4⋊Q8—28no—
299#127—44no—
300#124—28no—
301#124—44no—
302#123—216no—
303#123—88no—
304C22⋊Q8—28no—
305C22⋊Q8—28no—
306#121—44no—
307#121—44no—
308#120—48no—
309C4×Q8—44no—
310C4×Q8—44no—
311C4×Q8—44no—
312C4×Q8—48no—
313C4×D4—44no—
314C4×D4—48no—
315#117—22no—
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—42no—
325#117—42no—
326#117—48no—
327#117#4114yes—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D414yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—84no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D414yes—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—24no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—24no—
372C2×C42—42no—
373C2×C42—42no—
374C2×C42C2×D412yes—
375C2×C42C2×D414yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—24no—
380#107—24no—
381#107—24no—
382#107—42no—
383#106—24no—
384#106—42no—
385#103—28no—
386#103—24no—
387#103—24no—
388#103—44no—
389#102—24no—
390#102—24no—
391#102—28no—
392#102—44no—
393C22⋊C8—24no—
394C22⋊C8—24no—
395C22⋊C8—24no—
396C22⋊C8—24no—
397#97—41no—
398#97—41no—
399#97—41no—
400#97—41no—
401#97—42no—
402C4×C8—42no—
403#95—24no—
404#95—24no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—24no—
412#95—24no—
413C24—22no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—82no—
418C24—82no—
419C24C22×D412yes—
420C24C22×D411yescharacteristic
421#41—44no—
422#41—44no—
423#41—44no—
424#41—44no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8C2×#4114yes—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—48no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—84no—
462C2×D4—84no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D414yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4C2511yescharacteristic
501SD16—48no—
502SD16—48no—
503M16—44no—
504M16—44no—
505C2×C8—28no—
506C2×C8—28no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8C2×#4112yes—
522C2×C8C2×#4112yes—
523C2×C8C2×#4112yes—
524C2×C8C2×#4112yes—
525C4⋊C4—216no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—48no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—84no—
547C4⋊C4C2×#4114yes—
548C22⋊C4—216no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—44no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—84no—
558C22⋊C4—84no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C22⋊C4—84no—
562C42—24no—
563C42—24no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—22no—
569C42—22no—
570C42—44no—
571C42—48no—
572C42—42no—
573C42—42no—
574C42—42no—
575C42—84no—
576C23—22no—
577C23—22no—
578C23—24no—
579C23—24no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—44no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—88no—
589C23—84no—
590C23—162no—
591C23—162no—
592C23—164no—
593C23#52512yes—
594C23#53212yes—
595C23#53412yes—
596C23C23×D411yescharacteristic
597Q8—24no—
598Q8—24no—
599Q8—48no—
600Q8—44no—
601Q8—42no—
602Q8—42no—
603D4—48no—
604D4—44no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—84no—
610D4—168no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—84no—
648C2×C4—88no—
649C2×C4—162no—
650C2×C4—162no—
651C2×C4#52512yes—
652C2×C4#53212yes—
653C2×C4#53412yes—
654C2×C4#55114yes—
655C2×C4#55314yes—
656C2×C4#55314yes—
657C2×C4#55914yes—
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C8—22no—
665C8—22no—
666C8—22no—
667C8—22no—
668C8—42no—
669C8—42no—
670C22—22no—
671C22—22no—
672C22—24no—
673C22—24no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—44no—
678C22—44no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—84no—
683C22—84no—
684C22—168no—
685C22#260214yes—
686C22C2×#52511yescharacteristic
687C22C2×#53211yescharacteristic
688C22C2×#53411yescharacteristic
689C22#358512yes—
690C22#358612yes—
691C4—42no—
692C4—44no—
693C4—44no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4#260214yes—
703C4C2×#55111yescharacteristic
704C4C2×#55311yescharacteristic
705C4C2×#55311yescharacteristic
706C4C2×#55911yescharacteristic
707C2—24no—
708C2—82no—
709C2—162no—
710C2C2×#260211yescharacteristic
711C2#6057612yes—
712C2#6071911yescharacteristic
713C2#6072011yescharacteristic
714C2#6077212yes—
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
2. #34268C2semidirect
3. #29491C2semidirect
4. #29490C2semidirect
5. #29483C2semidirect
6. #29482C2semidirect
7. #23639C2semidirect
8. #23487C2semidirect
11. C2×#1973C2semidirect
12. #21154C2semidirect
13. #21095C2semidirect
14. #21094C2semidirect
15. #21065C2semidirect
16. #21037C2semidirect
17. #20936C2semidirect
18. #20880C2semidirect
19. #20375C2semidirect
46. #2323C22semidirect
51. C4×C22⋊Q8C22semidirect
57. #2076C22semidirect
58. #2075C22semidirect
59. #2036C22semidirect
60. #2035C22semidirect
69. #1944C22semidirect
70. #1944C22semidirect
71. #1943C22semidirect
72. #1943C22semidirect
74. #1935C22semidirect
76. #1933C22semidirect
79. #1921C22semidirect
80. #1918C22semidirect
83. #1865C22semidirect
84. #1863C22semidirect
85. #1750C22semidirect
86. #1750C22semidirect
87. #1749C22semidirect
88. #1749C22semidirect

Maximal subgroups

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

subgroupindex
#352542
#342682
#294912
#294902
#294832
#294822
#236392
#234872
#234862
#233882
C2×#19732
#211542
#210952
#210942
#210652
#210372
#209362
#208802
#203752