Finite Group Databasebeta

← #6458577

Subgroups — #6458577

Order 512.

1859 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#35617C211yesmaximal, characteristic
2#35555C211yesmaximal, characteristic, complemented
3#27764C214yesmaximal, complemented
4#27763C214yesmaximal, complemented
5#27756C214yesmaximal, complemented
6#27753C214yesmaximal, complemented
7#23633C211yesmaximal, characteristic, complemented
8#23630C211yesmaximal, characteristic, complemented
9#23511C211yesmaximal, characteristic
10#23506C211yesmaximal, characteristic, complemented
11#21262C211yesmaximal, characteristic, complemented
12#21206C211yesmaximal, characteristic
13#21205C211yesmaximal, characteristic, complemented
14C2×#1959C211yesmaximal, characteristic, complemented
15#21088C211yesmaximal, characteristic, complemented
16#21086C211yesmaximal, characteristic, complemented
17#20650C211yesmaximal, characteristic, complemented
18#20436C211yesmaximal, characteristic, complemented
19#20416C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#417C2211yescharacteristic
26C2×#417C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C2×#405C2211yescharacteristic
38C2×#405C2211yescharacteristic
39C22×#97C2211yescharacteristic
40#2570C2214yescomplemented
41#2552C2214yescomplemented
42#2547C2214yes—
43#2544C2214yes—
44#2507—22no—
45#2506—22no—
46#2473—24no—
47#2427—22no—
48#2427—22no—
49#2412—24no—
50#2317C2211yescharacteristic
51#2317C2211yescharacteristic
52#2316C2211yescharacteristic
53#2310C2211yescharacteristic
54#2297C2211yescharacteristic
55#2296C2211yescharacteristic
56#2296C2211yescharacteristic
57C2×#384C2211yescharacteristic
58#2274C2211yescharacteristic
59#2074C2214yescomplemented
60#2072C2214yescomplemented
61#2072C2214yescomplemented
62#2071C2214yescomplemented
63#2042C2214yes—
64#2042C2214yes—
65#2042C2214yes—
66#2041C2214yes—
67#1983C2214yescomplemented
68#1981C2214yescomplemented
69#1961C2212yes—
70#1960C2212yescomplemented
71#1959C2214yes—
72#1959C2212yes—
73#1959C2214yes—
74#1959C2212yescomplemented
75#1958C2212yes—
76#1958C2212yescomplemented
77#1957C2212yes—
78#1956C2212yescomplemented
79#1936C2212yescomplemented
80#1936C2212yescomplemented
81#1935C2212yescomplemented
82#1935C2212yescomplemented
83#1935C2214yescomplemented
84#1933C2214yescomplemented
85#1804C2214yes—
86#1802C2214yes—
87#1771C2214yescomplemented
88#1770C2214yescomplemented
89#1762C2214yes—
90C4×#102C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C22×M16C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104#536—24no—
105#536—24no—
106#535—24no—
107#535—42no—
108C2×C4⋊Q8C2312yes—
109C2×#127C2312yes—
110C2×#125—24no—
111C2×#124—24no—
112C2×#124—21no—
113C2×#124—21no—
114C2×#124—42no—
115C2×#124C2312yes—
116C2×#124C2314yes—
117C2×#124C2312yes—
118C2×#123—22no—
119C2×C22⋊Q8—22no—
120C2×#121—22no—
121C2×#120—22no—
122#518—24no—
123#518—24no—
124C2×C4×D4—22no—
125C2×C4×D4C2314yes—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137#424—24no—
138C2×C4⋊C8C2314yes—
139#419—42no—
140#419—42no—
141#418—24no—
142#418—42no—
143#417—42no—
144#417—42no—
145#416—24no—
146#416—42no—
147C2×#103—24no—
148C2×#103—42no—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2311yescharacteristic
152C2×#103C2311yescharacteristic
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#102—24no—
161C2×#102—42no—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2311yescharacteristic
165C2×#102C2311yescharacteristic
166C2×#102C2314yes—
167C2×#102C2312yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C2×#102C2312yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173#405—24no—
174#405—42no—
175C2×#97—42no—
176C2×#97C2312yes—
177C2×#97C2312yes—
178C2×#97C2312yes—
179C2×#97C2312yes—
180C2×C4×C8C2312yes—
181C2×C4×C8C2312yes—
182#400—24no—
183#398—24no—
184#398—24no—
185#398—24no—
186#397—24no—
187#397—24no—
188#394—24no—
189#394—24no—
190#391—22no—
191#391C2314yes—
192#391C2312yes—
193#390C2312yes—
194#389—22no—
195#389—22no—
196#389—22no—
197#389—24no—
198#389D412yes—
199#389D412yes—
200#389C2312yes—
201#389C2314yes—
202#389C2312yes—
203#387—24no—
204#387—24no—
205#386—22no—
206#386—24no—
207#386—24no—
208#385—22no—
209#385—22no—
210#385—22no—
211#385D414yes—
212#384—21no—
213#384—21no—
214#384C2312yes—
215#384C2314yes—
216#384C2312yes—
217#382C2312yes—
218#382C2312yes—
219#381—22no—
220#380—24no—
221#380—24no—
222#380—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4C2314yes—
227C4×C4⋊C4C2314yes—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—21no—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#336C2314yes—
238#336C2314yes—
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×Q8C2412yes—
254C22×Q8C2411yescharacteristic
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—24no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4#4114yes—
264C22×D4C2411yescharacteristic
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—24no—
271C23×C4C2×D412yes—
272C23×C4C2411yescharacteristic
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8#4114yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2411yescharacteristic
289C22×C8C2411yescharacteristic
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292#127—24no—
293#127—42no—
294#125—88no—
295#124—28no—
296#124—24no—
297#124—24no—
298#124—24no—
299#124—44no—
300#124—42no—
301#124—44no—
302#124—44no—
303#123—48no—
304C22⋊Q8—48no—
305#121—48no—
306#120—48no—
307C4×D4—28no—
308C4×D4—28no—
309C4×D4—48no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117#4114yes—
319#117#4114yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—84no—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D414yes—
337C2×C4⋊C4C2×D414yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—48no—
356C2×C22⋊C4—48no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—24no—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2412yes—
369C2×C42C2412yes—
370C4⋊C8—28no—
371#103—24no—
372#103—24no—
373#103—22no—
374#103—22no—
375#103—44no—
376#103—42no—
377#103—42no—
378#103—42no—
379#103—42no—
380#103—42no—
381#103—44no—
382#102—28no—
383#102—24no—
384#102—24no—
385#102—22no—
386#102—22no—
387#102—44no—
388#102—42no—
389#102—42no—
390#102—42no—
391#102—42no—
392#102—42no—
393#97—24no—
394#97—42no—
395#97—42no—
396#97—42no—
397C4×C8—24no—
398C4×C8—42no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95C2×D414yes—
410#95C2×D414yes—
411C24—22no—
412C24—24no—
413C24—21no—
414C24—42no—
415C24—42no—
416C24—82no—
417C24—82no—
418C24C22×D411yescharacteristic
419#41—44no—
420#41—44no—
421#41—44no—
422#41—44no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—28no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—48no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—44no—
464C2×D4—84no—
465C2×D4—84no—
466C2×D4C2×#4114yes—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—48no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D414yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C4C2511yescharacteristic
503M16—44no—
504M16—44no—
505C2×C8—28no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—44no—
520C2×C8C2×#4112yes—
521C2×C8C2×#4112yes—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—48no—
539C4⋊C4—44no—
540C4⋊C4—48no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4—84no—
546C4⋊C4—84no—
547C4⋊C4#12418yes—
548C22⋊C4—24no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4#12418yes—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—42no—
567C42—42no—
568C42—42no—
569C42—84no—
570C42C2×#4114yes—
571C42C2×#4114yes—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23—48no—
580C23—42no—
581C23—42no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—88no—
586C23—164no—
587C23—162no—
588C23—162no—
589C23C2×#12412yes—
590C23#54612yes—
591C23#55012yes—
592C23C23×D411yescharacteristic
593Q8—42no—
594Q8—41no—
595Q8—42no—
596Q8—41no—
597Q8—42no—
598Q8—42no—
599Q8—41no—
600Q8—41no—
601Q8—84no—
602D4—24no—
603D4—24no—
604D4—41no—
605D4—42no—
606D4—41no—
607D4—42no—
608D4—41no—
609D4—41no—
610D4—48no—
611D4—84no—
612D4—168no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—168no—
648C2×C4—162no—
649C2×C4—162no—
650C2×C4C2×#12418yes—
651C2×C4C2×#12412yes—
652C2×C4#52914yes—
653C2×C4#54014yes—
654C2×C4#54612yes—
655C2×C4#55012yes—
656C2×C4#55314yes—
657C2×C4#55314yes—
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—42no—
667C8—42no—
668C8—42no—
669C22—24no—
670C22—22no—
671C22—22no—
672C22—24no—
673C22—44no—
674C22—48no—
675C22—84no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—82no—
683C22—168no—
684C22#249014yes—
685C22C22×#12411yescharacteristic
686C22C2×#54611yescharacteristic
687C22C2×#55011yescharacteristic
688C22#358512yes—
689C22#358612yes—
690C4—28no—
691C4—42no—
692C4—42no—
693C4—44no—
694C4—82no—
695C4—81no—
696C4—82no—
697C4—81no—
698C4—82no—
699C4—82no—
700C4—81no—
701C4—82no—
702C4#249014yes—
703C4C2×#52911yescharacteristic
704C4C2×#54011yescharacteristic
705C4C2×#55311yescharacteristic
706C4C2×#55311yescharacteristic
707C2—24no—
708C2—84no—
709C2—161no—
710C2C2×#249011yescharacteristic
711C2#6062512yes—
712C2#6117712yes—
713C2#6141711yescharacteristic
714C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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. #35555C2semidirect
3. #27764C2semidirect
4. #27763C2semidirect
5. #27756C2semidirect
6. #27753C2semidirect
7. #23633C2semidirect
8. #23630C2semidirect
10. #23506C2semidirect
11. #21262C2semidirect
13. #21205C2semidirect
14. C2×#1959C2semidirect
15. #21088C2semidirect
16. #21086C2semidirect
17. #20650C2semidirect
18. #20436C2semidirect
19. #20416C2semidirect
40. #2570C22semidirect
41. #2552C22semidirect
59. #2074C22semidirect
60. #2072C22semidirect
61. #2072C22semidirect
62. #2071C22semidirect
67. #1983C22semidirect
68. #1981C22semidirect
70. #1960C22semidirect
74. #1959C22semidirect
76. #1958C22semidirect
78. #1956C22semidirect
79. #1936C22semidirect
80. #1936C22semidirect
81. #1935C22semidirect
82. #1935C22semidirect
83. #1935C22semidirect
84. #1933C22semidirect
87. #1771C22semidirect
88. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#356172
#355552
#277642
#277632
#277562
#277532
#236332
#236302
#235112
#235062
#212622
#212062
#212052
C2×#19592
#210882
#210862
#206502
#204362
#204162