Finite Group Databasebeta

← #6460370

Subgroups — #6460370

Order 512.

1952 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35628C211yesmaximal, characteristic, complemented
2#35541C211yesmaximal, characteristic
3#28996C214yesmaximal, complemented
4#28995C214yesmaximal, complemented
5#28988C214yesmaximal, complemented
6#28987C214yesmaximal, complemented
7#23639C211yesmaximal, characteristic, complemented
8#23629C211yesmaximal, characteristic, complemented
9#23454C211yesmaximal, characteristic, complemented
10#23388C211yesmaximal, characteristic
11#21321C211yesmaximal, characteristic, complemented
12C2×#1973C211yesmaximal, characteristic, complemented
13#21088C211yesmaximal, characteristic, complemented
14#21037C211yesmaximal, characteristic, complemented
15#20975C211yesmaximal, characteristic
16#20974C211yesmaximal, characteristic, complemented
17#20933C211yesmaximal, characteristic, complemented
18#20416C211yesmaximal, characteristic, complemented
19#20384C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C22×#131C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#535C2211yescharacteristic
24C2×#533C2211yescharacteristic
25C2×#518C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C2×#408C2211yescharacteristic
34C2×#405C2211yescharacteristic
35C22×#97C2211yescharacteristic
36#2592—22no—
37#2568—22no—
38#2492C2214yescomplemented
39#2489C2214yescomplemented
40#2488—24no—
41#2447—22no—
42#2445—22no—
43#2422C2214yes—
44#2420C2214yes—
45#2412—24no—
46#2316C2211yescharacteristic
47#2311C2211yescharacteristic
48#2311C2211yescharacteristic
49#2310C2211yescharacteristic
50#2297C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#390C2211yescharacteristic
53#2285C2211yescharacteristic
54#2274C2211yescharacteristic
55#2076C2214yescomplemented
56#2075C2214yescomplemented
57#2074C2214yescomplemented
58#2072C2214yescomplemented
59#2025C2214yes—
60#2023C2214yes—
61#2000C2214yes—
62#2000C2214yes—
63#1993C2212yescomplemented
64#1993C2212yescomplemented
65#1992C2212yescomplemented
66#1992C2212yescomplemented
67#1973C2212yes—
68#1973C2212yes—
69#1973C2212yes—
70#1973C2212yes—
71#1935C2214yescomplemented
72#1933C2214yescomplemented
73#1921C2214yescomplemented
74#1918C2214yescomplemented
75#1895C2214yes—
76#1894C2214yescomplemented
77#1893C2214yes—
78#1892C2214yescomplemented
79#1877C2214yes—
80#1877C2214yes—
81#1762C2214yes—
82C4×#102C2214yes—
83#1753C2212yescomplemented
84#1753C2212yescomplemented
85#1752C2212yescomplemented
86#1752C2212yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—24no—
102C22×SD16—42no—
103C22×SD16C2312yes—
104C22×SD16C2312yes—
105C2×#131—24no—
106C2×#131—42no—
107C2×#131—42no—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C22×M16C2312yes—
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2312yes—
114#536—24no—
115#536—42no—
116#535—24no—
117#535—42no—
118#533—24no—
119#533—42no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8C2312yes—
122C2×#127—21no—
123C2×#125C2312yes—
124C2×#124—21no—
125C2×#124—21no—
126C2×#123—22no—
127C2×#123—24no—
128C2×C22⋊Q8—22no—
129C2×C22⋊Q8C2314yes—
130C2×#121C2314yes—
131C2×#120—22no—
132#518—24no—
133#518—42no—
134C2×C4×Q8—22no—
135C2×C4×D4—22no—
136C2×C4×D4C2314yes—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C22⋊C4—22no—
145C22×C22⋊C4—22no—
146C22×C42C2311yescharacteristic
147C2×#107—24no—
148C2×#107C2312yes—
149C2×#107C2312yes—
150#419—42no—
151#419—42no—
152#418—24no—
153#418—42no—
154C2×#103—24no—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#102—24no—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164#408—24no—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2312yes—
167#405—42no—
168C2×#97—42no—
169C2×#97C2311yescharacteristic
170C2×#97C2311yescharacteristic
171C2×#97C2311yescharacteristic
172C2×#97C2311yescharacteristic
173C2×#97C2312yes—
174C2×C4×C8C2312yes—
175#400—24no—
176#400—24no—
177#397—24no—
178#397—24no—
179#397—24no—
180#394—24no—
181#394—24no—
182#393—24no—
183#391—21no—
184#391—21no—
185#391—22no—
186#391—22no—
187#391—24no—
188#391D412yes—
189#391C2314yes—
190#391C2312yes—
191#391C2312yes—
192#390—21no—
193#390—21no—
194#390—22no—
195#390D412yes—
196#390C2312yes—
197#390C2312yes—
198#389—22no—
199#389C2314yes—
200#388—24no—
201#388—24no—
202#386—22no—
203#386—22no—
204#386—24no—
205#386D414yes—
206#385—24no—
207#384C2312yes—
208#382—24no—
209#382C2312yes—
210#381—22no—
211#381—24no—
212#380—22no—
213#380—22no—
214#380C2314yes—
215#380C2314yes—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4C2314yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4C2312yes—
221C4×C22⋊C4C2312yes—
222C2×#95—22no—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
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×Q8—22no—
253C22×Q8—22no—
254C22×Q8C2×D411yescharacteristic
255C22×Q8C2412yes—
256C22×D4—24no—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4C2×D412yes—
264C22×D4C2×D411yescharacteristic
265C22×D4#4114yes—
266C22×D4C2412yes—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—24no—
271C23×C4C2×D412yes—
272C23×C4C2412yes—
273C23×C4C2411yescharacteristic
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—42no—
279C2×SD16—42no—
280C2×SD16—416no—
281#131—416no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—24no—
289C2×M16—42no—
290C2×M16—42no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—24no—
296C22×C8—42no—
297C22×C8—42no—
298C22×C8#4114yes—
299C22×C8#4114yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2411yescharacteristic
303C22×C8C2411yescharacteristic
304C22×C8C2411yescharacteristic
305C22×C8C2411yescharacteristic
306C4⋊Q8—24no—
307C4⋊Q8—42no—
308C4⋊Q8—44no—
309#127—44no—
310#125—24no—
311#125—42no—
312#124—44no—
313#124—44no—
314#123—48no—
315#123—88no—
316C22⋊Q8—28no—
317C22⋊Q8—48no—
318C22⋊Q8—44no—
319#121—28no—
320#121—44no—
321#120—48no—
322C4×Q8—48no—
323C4×D4—28no—
324C4×D4—44no—
325C4×D4—48no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117#4114yes—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—48no—
368C2×C22⋊C4—84no—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D414yes—
371C2×C22⋊C4C2×D414yes—
372C2×C42—21no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—24no—
379C2×C42C2×D412yes—
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—24no—
384#107—24no—
385#103—24no—
386#103—24no—
387#103—44no—
388#103—44no—
389#102—24no—
390#102—24no—
391#102—28no—
392#102—44no—
393C22⋊C8—24no—
394C22⋊C8—24no—
395#97—41no—
396#97—41no—
397#97—41no—
398#97—41no—
399#97—42no—
400C4×C8—42no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95C2×D414yes—
412#95C2×D414yes—
413C24—22no—
414C24—24no—
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—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—82no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—28no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—28no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—48no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—42no—
462C2×D4—44no—
463C2×D4—84no—
464C2×D4—84no—
465C2×D4C2×#4114yes—
466C22×C4—22no—
467C22×C4—24no—
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—22no—
479C22×C4—22no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—48no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—84no—
489C22×C4—84no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4C22×D412yes—
493C22×C4C22×D414yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4C2511yescharacteristic
504SD16—48no—
505SD16—48no—
506M16—48no—
507M16—44no—
508M16—44no—
509C2×C8—28no—
510C2×C8—28no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—48no—
526C2×C8C2×#4112yes—
527C2×C8C2×#4112yes—
528C2×C8C2×#4112yes—
529C2×C8C2×#4112yes—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—48no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—82no—
550C4⋊C4—84no—
551C4⋊C4C2×#4114yes—
552C22⋊C4—24no—
553C22⋊C4—24no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—48no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C22⋊C4—48no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—84no—
564C22⋊C4—84no—
565C22⋊C4—84no—
566C42—24no—
567C42—24no—
568C42—21no—
569C42—21no—
570C42—22no—
571C42—22no—
572C42—21no—
573C42—21no—
574C42—42no—
575C42—42no—
576C42—42no—
577C42—84no—
578C23—22no—
579C23—22no—
580C23—24no—
581C23—24no—
582C23—22no—
583C23—22no—
584C23—24no—
585C23—22no—
586C23—44no—
587C23—44no—
588C23—42no—
589C23—42no—
590C23—48no—
591C23—88no—
592C23—162no—
593C23—162no—
594C23—164no—
595C23#53212yes—
596C23#54612yes—
597C23#54612yes—
598C23C23×D411yescharacteristic
599Q8—44no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—88no—
605Q8—84no—
606D4—24no—
607D4—24no—
608D4—48no—
609D4—42no—
610D4—42no—
611D4—48no—
612D4—44no—
613D4—168no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
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—42no—
648C2×C4—42no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—168no—
652C2×C4—162no—
653C2×C4—162no—
654C2×C4#52914yes—
655C2×C4#53212yes—
656C2×C4#53814yes—
657C2×C4#54612yes—
658C2×C4#54612yes—
659C2×C4#55114yes—
660C2×C4#55914yes—
661C2×C4C22×#4111yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4111yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C8—22no—
668C8—22no—
669C8—22no—
670C8—22no—
671C8—42no—
672C8—42no—
673C22—24no—
674C22—22no—
675C22—22no—
676C22—24no—
677C22—42no—
678C22—44no—
679C22—48no—
680C22—44no—
681C22—42no—
682C22—42no—
683C22—44no—
684C22—82no—
685C22—82no—
686C22—84no—
687C22—168no—
688C22#255714yes—
689C22C2×#53211yescharacteristic
690C22C2×#54611yescharacteristic
691C22C2×#54611yescharacteristic
692C22#358512yes—
693C22#358612yes—
694C4—42no—
695C4—44no—
696C4—42no—
697C4—42no—
698C4—44no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4#255714yes—
706C4C2×#52911yescharacteristic
707C4C2×#53811yescharacteristic
708C4C2×#55111yescharacteristic
709C4C2×#55911yescharacteristic
710C2—24no—
711C2—82no—
712C2—84no—
713C2C2×#255711yescharacteristic
714C2#6071911yescharacteristic
715C2#6072011yescharacteristic
716C2#6118212yes—
717C2#6121912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35628C2semidirect
3. #28996C2semidirect
4. #28995C2semidirect
5. #28988C2semidirect
6. #28987C2semidirect
7. #23639C2semidirect
8. #23629C2semidirect
9. #23454C2semidirect
11. #21321C2semidirect
12. C2×#1973C2semidirect
13. #21088C2semidirect
14. #21037C2semidirect
16. #20974C2semidirect
17. #20933C2semidirect
18. #20416C2semidirect
19. #20384C2semidirect
38. #2492C22semidirect
39. #2489C22semidirect
55. #2076C22semidirect
56. #2075C22semidirect
57. #2074C22semidirect
58. #2072C22semidirect
63. #1993C22semidirect
64. #1993C22semidirect
65. #1992C22semidirect
66. #1992C22semidirect
71. #1935C22semidirect
72. #1933C22semidirect
73. #1921C22semidirect
74. #1918C22semidirect
76. #1894C22semidirect
78. #1892C22semidirect
83. #1753C22semidirect
84. #1753C22semidirect
85. #1752C22semidirect
86. #1752C22semidirect

Maximal subgroups

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

subgroupindex
#356282
#355412
#289962
#289952
#289882
#289872
#236392
#236292
#234542
#233882
#213212
C2×#19732
#210882
#210372
#209752
#209742
#209332
#204162
#203842