Finite Group Databasebeta

← #6537298

Subgroups — #6537298

Order 512.

1985 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34310C211yesmaximal, characteristic
3#29981C211yesmaximal, characteristic, complemented
4#29950C212yesmaximal, complemented
5#29254C211yesmaximal, characteristic, complemented
6#29163C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic, complemented
8#26467C212yesmaximal, complemented
9#26438C214yesmaximal, complemented
10#26025C212yesmaximal, complemented
11#25976C214yesmaximal, complemented
12#24304C212yesmaximal, complemented
13#23774C211yesmaximal, characteristic, complemented
14#23454C212yesmaximal, complemented
15#22713C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22428C212yesmaximal
18C2×D42C2211yescharacteristic
19C2×#533C2212yescomplemented
20C2×#518—21no—
21C2×#478C2214yescomplemented
22C2×#436C2212yes—
23C2×#418C2214yes—
24#2868C2211yescharacteristic, complemented
25#2857C2212yescomplemented
26#2552C2212yescomplemented
27#2542—21no—
28#2538—21no—
29#2441C2214yescomplemented
30#2429—22no—
31#2423—22no—
32#2366C2211yescharacteristic
33#2365—21no—
34#2363—21no—
35#2361C2212yes—
36#2358—22no—
37#2354C2211yescharacteristic
38#2354C2212yes—
39Q8×C4⋊C4—21no—
40D4×C4⋊C4C2211yescharacteristic
41D4×C22⋊C4—21no—
42D4×C22⋊C4C2212yes—
43#2324—21no—
44#2321—22no—
45C4×C4⋊Q8C2211yescharacteristic, complemented
46C4×#127C2211yescharacteristic
47C4×#121C2212yes—
48#2293C2211yescharacteristic
49C2×#390C2212yes—
50C42×D4C2211yescharacteristic
51#2100C2212yescomplemented
52#2099C2211yescharacteristic, complemented
53#2094C2212yes—
54#2093C2211yescharacteristic
55#2093C2212yescomplemented
56#2086C2212yescomplemented
57#2067C2212yes—
58#2067C2214yes—
59#2067C2214yescomplemented
60#2058C2212yescomplemented
61#2057C2211yescharacteristic, complemented
62#2025C2214yes—
63#2025C2214yescomplemented
64#2023C2212yes—
65#2023C2211yescharacteristic
66#2023C2212yescomplemented
67#2023C2212yescomplemented
68#2023C2212yescomplemented
69#1989C2214yescomplemented
70#1982C2212yes—
71#1957C2212yes—
72#1948C2212yescomplemented
73#1947C2212yescomplemented
74#1939C2212yescomplemented
75#1921C2214yescomplemented
76#1895C2214yes—
77#1875C2214yes—
78#1864C2214yescomplemented
79#1862C2214yescomplemented
80#1854C2214yescomplemented
81#1852C2214yescomplemented
82#1851C2214yescomplemented
83#1834C2212yescomplemented
84#1831C2211yescharacteristic, complemented
85#1809C2214yescomplemented
86#1802C2212yes—
87#1802C2214yescomplemented
88#1797C2212yescomplemented
89#1786C2214yes—
90C4×#107C2212yescomplemented
91C4×C4⋊C8C2211yescharacteristic, complemented
92C4×#102C2212yes—
93#1736C2214yescomplemented
94#1728C2212yescomplemented
95C23×D4—21no—
96C23×D4C2312yes—
97C22×M16C2314yes—
98D42—24no—
99D42Q818yescomplemented
100#533—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#125C2314yes—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×#121—21no—
111C2×#121C2312yes—
112C2×#120—22no—
113#518—48no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×#117—21no—
130C2×#117—21no—
131C2×#117C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C42C2312yes—
137#478—48no—
138#436—28no—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2312yes—
146#418—48no—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156#399—24no—
157#398—24no—
158#396—24no—
159#391—21no—
160#391—22no—
161#390—22no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—22no—
166#390—22no—
167#390C2312yes—
168#390C2311yescharacteristic
169#390C2312yes—
170#390C2312yes—
171#390C2311yescharacteristic
172#389—21no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389—24no—
177#389C2312yes—
178#389C2311yescharacteristic
179#389C2314yes—
180#389C2311yescharacteristic
181#388—24no—
182#388—21no—
183#388—22no—
184#388—21no—
185#388—22no—
186#388—22no—
187#387—24no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#384—22no—
196#384—21no—
197#384—21no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#380—22no—
203#380—21no—
204#380C2312yes—
205#380C2314yes—
206#379—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220C43C2311yescharacteristic
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2314yes—
235C25—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×D4—22no—
239C22×D4—28no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—24no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2×Q818yes—
250C22×D4C2412yes—
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C2×M16—28no—
260C2×M16—44no—
261C2×M16—44no—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—82no—
269#127—22no—
270#127C2×Q814yes—
271#125—28no—
272#125—44no—
273C22⋊Q8—48no—
274C22⋊Q8—48no—
275C22⋊Q8—84no—
276#121—24no—
277#121—28no—
278#121C2×Q818yes—
279#120—216no—
280C4×Q8—48no—
281C4×Q8—48no—
282C4×Q8—44no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—42no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—48no—
289C4×D4—44no—
290C4×D4—48no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—82no—
294C4×D4C2×Q814yes—
295C4×D4C2×Q814yes—
296C4×D4#4114yes—
297#117—28no—
298#117—44no—
299#117—44no—
300#117—44no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—48no—
348C2×C22⋊C4—84no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—24no—
368#107—42no—
369#107—44no—
370#106—44no—
371C4⋊C8—24no—
372C4⋊C8—42no—
373C4⋊C8—44no—
374#102—24no—
375#102—28no—
376#102—44no—
377#102—42no—
378#102—44no—
379#102—44no—
380#102—44no—
381C22⋊C8—28no—
382C4×C8—24no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24—22no—
397C24—28no—
398C24—44no—
399C24—44no—
400C24—82no—
401C24—82no—
402C24—82no—
403C24—82no—
404C24C22×D412yes—
405C2×Q8—24no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—416no—
424C2×D4—432no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—88no—
430C2×D4—81no—
431C2×D4—81no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4C22⋊Q814yes—
435C2×D4C22×Q812yes—
436C2×D4C22×Q814yes—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×Q814yes—
474C22×C4C2×#4114yes—
475C22×C4C2511yescharacteristic
476M16—88no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—82no—
510C4⋊C4—81no—
511C4⋊C4—84no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4C22⋊Q814yes—
515C4⋊C4C22×Q812yes—
516C4⋊C4H(5,𝔽2)14yes—
517C22⋊C4—24no—
518C22⋊C4—28no—
519C22⋊C4—24no—
520C22⋊C4—42no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—81no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—84no—
536C22⋊C4C22⋊Q818yes—
537C42—22no—
538C42—22no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—42no—
543C42—42no—
544C42—41no—
545C42—48no—
546C42—44no—
547C42—42no—
548C42—44no—
549C42AGL(1,ℤ/8)14yes—
550C42C22×Q812yes—
551C42C2×#4112yes—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—416no—
561C23—41no—
562C23—42no—
563C23—42no—
564C23—41no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—41no—
570C23—42no—
571C23—41no—
572C23—42no—
573C23—88no—
574C23—88no—
575C23—164no—
576C23—164no—
577C23—164no—
578C23—164no—
579C23C2×C22⋊Q814yes—
580C23C23×D411yescharacteristic
581Q8—48no—
582Q8—88no—
583D4—28no—
584D4—24no—
585D4—28no—
586D4—416no—
587D4—48no—
588D4—44no—
589D4—42no—
590D4—88no—
591D4—84no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—48no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—164no—
638C2×C4—164no—
639C2×C4—164no—
640C2×C4C2×C22⋊Q812yes—
641C2×C4C2×C22⋊Q814yes—
642C2×C4C2×C22⋊Q812yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4#57512yes—
647C2×C4#57612yes—
648C2×C4C23×Q811yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)12yes—
651C2×C4#58512yes—
652C8—42no—
653C8—44no—
654C8—42no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—82no—
669C22—81no—
670C22—82no—
671C22—81no—
672C22—82no—
673C22—81no—
674C22—82no—
675C22—81no—
676C22—1616no—
677C22C22×C22⋊Q811yescharacteristic
678C22#348411yescharacteristic
679C22#350011yescharacteristic
680C22C22×AGL(1,ℤ/8)11yescharacteristic
681C22C2×#57511yescharacteristic
682C22C2×#57611yescharacteristic
683C22#358611yescharacteristic
684C4—22no—
685C4—22no—
686C4—24no—
687C4—22no—
688C4—22no—
689C4—42no—
690C4—41no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—82no—
695C4—82no—
696C4—81no—
697C4—81no—
698C4—84no—
699C4#307714yes—
700C4#350112yes—
701C4#351412yes—
702C2—44no—
703C2—84no—
704C2—82no—
705C2#3570211yescharacteristic
706C2C2×#307711yescharacteristic
707C2#6049411yescharacteristic
708C2#6158511yescharacteristic
709C2#6160511yescharacteristic
710C2#6253511yescharacteristic
711C2#6259311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36134C2semidirect
3. #29981C2semidirect
4. #29950C2semidirect
5. #29254C2semidirect
6. #29163C2semidirect
7. #29139C2semidirect
8. #26467C2semidirect
9. #26438C2semidirect
10. #26025C2semidirect
11. #25976C2semidirect
12. #24304C2semidirect
13. #23774C2semidirect
14. #23454C2semidirect
15. #22713C2semidirect
16. #22700C2semidirect
19. C2×#533C22semidirect
21. C2×#478C22semidirect
24. #2868C22semidirect
25. #2857C22semidirect
26. #2552C22semidirect
29. #2441C22semidirect
45. C4×C4⋊Q8C22semidirect
51. #2100C22semidirect
52. #2099C22semidirect
55. #2093C22semidirect
56. #2086C22semidirect
59. #2067C22semidirect
60. #2058C22semidirect
61. #2057C22semidirect
63. #2025C22semidirect
66. #2023C22semidirect
67. #2023C22semidirect
68. #2023C22semidirect
69. #1989C22semidirect
72. #1948C22semidirect
73. #1947C22semidirect
74. #1939C22semidirect
75. #1921C22semidirect
78. #1864C22semidirect
79. #1862C22semidirect
80. #1854C22semidirect
81. #1852C22semidirect
82. #1851C22semidirect
83. #1834C22semidirect
84. #1831C22semidirect
85. #1809C22semidirect
87. #1802C22semidirect
88. #1797C22semidirect
90. C4×#107C22semidirect
91. C4×C4⋊C8C22semidirect
93. #1736C22semidirect
94. #1728C22semidirect
99. D42Q8semidirect

Maximal subgroups

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

subgroupindex
#361342
#343102
#299812
#299502
#292542
#291632
#291392
#264672
#264382
#260252
#259762
#243042
#237742
#234542
#227132
#227002
#224282