Finite Group Databasebeta

← #6461562

Subgroups — #6461562

Order 512.

1922 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic, complemented
2#34276C211yesmaximal, characteristic, complemented
3#29508C214yesmaximal, complemented
4#29507C214yesmaximal, complemented
5#29503C214yesmaximal, complemented
6#29500C214yesmaximal, complemented
7#23660C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic, complemented
9#23512C211yesmaximal, characteristic, complemented
10#23424C211yesmaximal, characteristic
11#21236C211yesmaximal, characteristic
12C2×#1973C211yesmaximal, characteristic, complemented
13#21094C211yesmaximal, characteristic, complemented
14#21086C211yesmaximal, characteristic, complemented
15#20936C211yesmaximal, characteristic, complemented
16#20932C211yesmaximal, characteristic
17#20881C211yesmaximal, characteristic, complemented
18#20877C211yesmaximal, characteristic, complemented
19#20367C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36C22×#97C2211yescharacteristic
37#2860C2214yes—
38#2644C2214yes—
39#2513—22no—
40#2512—22no—
41#2480—24no—
42#2362—22no—
43#2360—22no—
44#2359—24no—
45#2327C2212yescomplemented
46#2326C2212yescomplemented
47#2317C2211yescharacteristic
48#2317C2211yescharacteristic
49#2316C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52C4×C4⋊Q8C2212yescomplemented
53C4×#124C2212yescomplemented
54#2297C2211yescharacteristic
55#2296C2211yescharacteristic
56C2×#391C2211yescharacteristic
57C2×#390C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2085C2214yescomplemented
60#2083C2214yescomplemented
61#2073C2214yes—
62#2071C2214yes—
63#2043C2214yescomplemented
64#2042C2214yescomplemented
65#2013C2214yes—
66#2012C2214yes—
67#1973C2212yes—
68#1973C2214yes—
69#1973C2214yes—
70#1972C2212yes—
71#1970C2212yes—
72#1968C2212yes—
73#1936C2212yescomplemented
74#1936C2212yescomplemented
75#1935C2212yescomplemented
76#1935C2214yescomplemented
77#1935C2212yescomplemented
78#1933C2214yescomplemented
79#1880C2214yes—
80#1880C2214yes—
81#1880C2214yes—
82#1879C2214yes—
83#1865C2214yescomplemented
84#1863C2214yescomplemented
85#1863C2214yescomplemented
86#1862C2214yescomplemented
87#1746C2214yescomplemented
88#1745C2214yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C2×AGL(1,ℤ/8)—24no—
101C2×AGL(1,ℤ/8)—24no—
102C2×AGL(1,ℤ/8)—42no—
103C22×SD16—24no—
104C22×SD16—24no—
105C22×SD16—42no—
106C22×SD16C2312yes—
107C22×SD16C2314yes—
108C22×D8C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#536—24no—
114#536—24no—
115C2×C4⋊Q8C2312yes—
116C2×#126—24no—
117C2×#124C2312yes—
118C2×#123—22no—
119C2×C22⋊Q8—22no—
120#520—42no—
121#520—42no—
122#519—42no—
123#519—42no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—22no—
130C2×C4×D4C2312yes—
131C2×#117—21no—
132C2×#117—22no—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—21no—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C22⋊C4—22no—
142C22×C42C2311yescharacteristic
143#428—42no—
144C2×#107—42no—
145C2×#107C2312yes—
146C2×#107C2314yes—
147C2×#106C2312yes—
148#419—24no—
149#419—24no—
150#418—42no—
151#417—42no—
152#416—24no—
153#416—24no—
154C2×#103—42no—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#102—42no—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C4×M16—24no—
169C2×#97—24no—
170C2×#97C2312yes—
171C2×#97C2312yes—
172C2×#97C2312yes—
173C2×C4×C8C2312yes—
174#397—24no—
175#397—24no—
176#397—24no—
177#396—24no—
178#391—22no—
179#391C2311yescharacteristic
180#391C2312yes—
181#391C2312yes—
182#391C2311yescharacteristic
183#390—21no—
184#390—22no—
185#390—21no—
186#390—22no—
187#390C2314yes—
188#390C2314yes—
189#390C2311yescharacteristic
190#390C2312yes—
191#390C2312yes—
192#390C2311yescharacteristic
193#389—22no—
194#389—22no—
195#389—24no—
196#389C2314yes—
197#389C2314yes—
198#389C2312yes—
199#389C2312yes—
200#388—24no—
201#388—24no—
202#388—24no—
203#388—24no—
204#388—24no—
205#387—24no—
206#387—24no—
207#386—22no—
208#385—22no—
209#385—22no—
210#385—22no—
211#385—24no—
212#384—21no—
213#384C2312yes—
214#382—21no—
215#382C2312yes—
216#380—22no—
217#379—22no—
218#379—22no—
219C4×C4⋊C4—24no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4D412yes—
222C4×C4⋊C4D412yes—
223C4×C22⋊C4—24no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4D414yes—
226#376C2312yes—
227C2×#95—22no—
228C43C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41#4114yes—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8#4114yes—
253C22×Q8C2412yes—
254C22×Q8C2411yescharacteristic
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4C2412yes—
264C22×D4C2411yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4C2×D412yes—
271C23×C4C2411yescharacteristic
272AGL(1,ℤ/8)—416no—
273C2×SD16—28no—
274C2×SD16—24no—
275C2×SD16—28no—
276C2×SD16—24no—
277C2×SD16—44no—
278C2×SD16—42no—
279C2×SD16—416no—
280C2×D8—24no—
281C2×D8—24no—
282C2×D8—42no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16#4114yes—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8#4114yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—24no—
300C4⋊Q8—24no—
301#126—88no—
302#124—24no—
303#124—24no—
304#123—216no—
305C22⋊Q8—216no—
306C4×Q8—42no—
307C4×Q8—42no—
308C4×Q8—42no—
309C4×Q8—42no—
310C4×Q8—42no—
311C4×Q8—42no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×D4—42no—
315C4×D4—42no—
316C4×D4—48no—
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—48no—
328#117—44no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—84no—
368C2×C22⋊C4C2×D412yes—
369C2×C22⋊C4C2×D412yes—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42C2×D412yes—
379C2×C42C2×D414yes—
380C2×C42C2×D414yes—
381C2×C42C2412yes—
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—44no—
386#107—42no—
387#106—42no—
388#103—24no—
389#103—28no—
390#103—24no—
391#103—28no—
392#103—44no—
393#103—42no—
394#102—24no—
395#102—24no—
396#102—44no—
397#102—42no—
398#102—42no—
399#102—42no—
400#97—24no—
401#97—24no—
402#97—24no—
403C4×C8—24no—
404#95—24no—
405#95—24no—
406#95—22no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—24no—
412#95—24no—
413#95C2×D414yes—
414#95C2×D414yes—
415C24—21no—
416C24—22no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—82no—
421C24—82no—
422C24C22×D411yescharacteristic
423#41—28no—
424#41—28no—
425#41—44no—
426#41—44no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—21no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8C2×#4114yes—
445C2×Q8C2×#4112yes—
446C2×D4—24no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—44no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—44no—
467C2×D4C2×#4112yes—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—84no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D414yes—
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—416no—
505SD16—48no—
506D8—48no—
507M16—28no—
508M16—44no—
509C2×C8—28no—
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—22no—
520C2×C8—44no—
521C2×C8C2×#4114yes—
522C2×C8C2×#4114yes—
523C4⋊C4—216no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—84no—
546C4⋊C4—82no—
547C4⋊C4—84no—
548C4⋊C4—84no—
549C22⋊C4—216no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—44no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—82no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C22⋊C4—84no—
562C22⋊C4—84no—
563C22⋊C4—84no—
564C42—24no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—44no—
574C42—44no—
575C42—44no—
576C42—44no—
577C42—44no—
578C42—42no—
579C42—42no—
580C42—42no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—42no—
585C23—42no—
586C23—41no—
587C23—41no—
588C23—44no—
589C23—44no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—88no—
594C23—84no—
595C23—164no—
596C23—162no—
597C23—162no—
598C23#52512yes—
599C23#53212yes—
600C23#53512yes—
601C23C23×D411yescharacteristic
602Q8—24no—
603Q8—22no—
604Q8—24no—
605Q8—22no—
606Q8—48no—
607Q8—42no—
608Q8—41no—
609Q8—42no—
610Q8—41no—
611D4—22no—
612D4—22no—
613D4—48no—
614D4—41no—
615D4—42no—
616D4—42no—
617D4—41no—
618D4—42no—
619D4—42no—
620D4—84no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—48no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—48no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—84no—
657C2×C4—164no—
658C2×C4—162no—
659C2×C4—162no—
660C2×C4#52512yes—
661C2×C4#53212yes—
662C2×C4#53512yes—
663C2×C4#55114yes—
664C2×C4#55114yes—
665C2×C4#55314yes—
666C2×C4#55314yes—
667C2×C4C22×#4111yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C8—24no—
674C8—24no—
675C8—42no—
676C8—42no—
677C22—24no—
678C22—22no—
679C22—22no—
680C22—24no—
681C22—42no—
682C22—44no—
683C22—82no—
684C22—81no—
685C22—81no—
686C22—84no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—84no—
691C22—168no—
692C22#260314yes—
693C22C2×#52511yescharacteristic
694C22C2×#53211yescharacteristic
695C22C2×#53511yescharacteristic
696C22#358512yes—
697C22#358612yes—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4—82no—
706C4—81no—
707C4—82no—
708C4—82no—
709C4—81no—
710C4—82no—
711C4#260314yes—
712C4C2×#55111yescharacteristic
713C4C2×#55111yescharacteristic
714C4C2×#55311yescharacteristic
715C4C2×#55311yescharacteristic
716C2—24no—
717C2—82no—
718C2—162no—
719C2C2×#260311yescharacteristic
720C2#6057612yes—
721C2#6070912yes—
722C2#6082511yescharacteristic
723C2#6082811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35266C2semidirect
2. #34276C2semidirect
3. #29508C2semidirect
4. #29507C2semidirect
5. #29503C2semidirect
6. #29500C2semidirect
7. #23660C2semidirect
8. #23632C2semidirect
9. #23512C2semidirect
12. C2×#1973C2semidirect
13. #21094C2semidirect
14. #21086C2semidirect
15. #20936C2semidirect
17. #20881C2semidirect
18. #20877C2semidirect
19. #20367C2semidirect
45. #2327C22semidirect
46. #2326C22semidirect
52. C4×C4⋊Q8C22semidirect
53. C4×#124C22semidirect
59. #2085C22semidirect
60. #2083C22semidirect
63. #2043C22semidirect
64. #2042C22semidirect
73. #1936C22semidirect
74. #1936C22semidirect
75. #1935C22semidirect
76. #1935C22semidirect
77. #1935C22semidirect
78. #1933C22semidirect
83. #1865C22semidirect
84. #1863C22semidirect
85. #1863C22semidirect
86. #1862C22semidirect
87. #1746C22semidirect
88. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#342762
#295082
#295072
#295032
#295002
#236602
#236322
#235122
#234242
#212362
C2×#19732
#210942
#210862
#209362
#209322
#208812
#208772
#203672