Finite Group Databasebeta

← #6458655

Subgroups — #6458655

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34188C211yesmaximal, characteristic, complemented
2C4×#537C211yesmaximal, characteristic
3#27643C214yesmaximal, complemented
4#27640C214yesmaximal, complemented
5#27631C212yesmaximal, complemented
6#27629C212yesmaximal, complemented
7#27626C212yesmaximal, complemented
8#27624C212yesmaximal, complemented
9#23639C212yesmaximal, complemented
10#23485C212yesmaximal
11#21170C211yesmaximal, characteristic, complemented
12C2×#1947C211yesmaximal, characteristic, complemented
13#21067C212yesmaximal, complemented
14#21036C212yesmaximal, complemented
15#20648C211yesmaximal, characteristic, complemented
16#20645C211yesmaximal, characteristic, complemented
17#20629C211yesmaximal, characteristic, complemented
18C2×#537C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C22×#107C2212yes—
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#417C2212yes—
26C2×#416C2212yes—
27C2×#408C2212yes—
28#2355—21no—
29#2355—21no—
30#2354—21no—
31#2354—21no—
32#2352—22no—
33#2352—22no—
34#2325—22no—
35#2324—21no—
36#2324C2212yes—
37#2323—21no—
38#2323C2212yes—
39#2316C2211yescharacteristic
40#2314C2214yes—
41#2311C2211yescharacteristic
42#2311C2211yescharacteristic
43#2311C2212yes—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8C2212yes—
46C4×#121—22no—
47C4×#121—21no—
48C4×#121C2212yes—
49#2285C2212yes—
50#2285C2212yes—
51#2285C2211yescharacteristic
52C2×#384C2211yescharacteristic
53#2274C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2076C2214yes—
56#2076C2214yes—
57#2075C2214yes—
58#2075C2214yes—
59#2036C2214yes—
60#2035C2214yes—
61#2034C2214yes—
62#2033C2214yes—
63#1949C2214yes—
64#1948C2212yes—
65#1947C2212yes—
66#1947C2212yes—
67#1947C2214yes—
68#1947C2212yes—
69#1928C2214yes—
70#1928C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1921C2214yes—
74#1920C2214yes—
75#1919C2214yes—
76#1918C2214yes—
77#1805C2212yes—
78#1804C2212yes—
79#1803C2211yescharacteristic
80#1803C2211yescharacteristic
81#1803C2212yes—
82#1803C2212yes—
83#1802C2211yescharacteristic
84#1802C2211yescharacteristic
85#1802C2212yes—
86#1802C2212yes—
87#1798C2214yes—
88#1797C2212yes—
89#1797C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93C2×#340C2212yes—
94#1729C2212yes—
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#537—24no—
100#537—24no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C2312yes—
103C2×#121—21no—
104C2×#121—22no—
105C2×#121C2312yes—
106C4×#41—24no—
107C4×#41—24no—
108C4×#41—24no—
109C4×#41—24no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D412yes—
119C2×C4×D4D412yes—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—22no—
125C2×#117—21no—
126C2×#117C2311yescharacteristic
127C2×#117C2312yes—
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C42—21no—
136C22×C42—21no—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#428—28no—
140C2×#107—44no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145#423—24no—
146C2×C4⋊C8—24no—
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2311yescharacteristic
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152#417—44no—
153#416—28no—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160#408—44no—
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2314yes—
163#391—21no—
164#391—22no—
165#391C2312yes—
166#390—21no—
167#390—22no—
168#390—22no—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—24no—
177#388—24no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—24no—
182#387—24no—
183#387—22no—
184#387—22no—
185#386—22no—
186#386—24no—
187#386—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2312yes—
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#382—22no—
199#382—22no—
200#382C2312yes—
201#380—21no—
202#380—21no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380—21no—
207#380—22no—
208#380C2314yes—
209#380C2314yes—
210#380C2312yes—
211#380C2312yes—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4C2314yes—
222C4×C4⋊C4C2314yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—24no—
227C4×C22⋊C4—22no—
228C2×#95—21no—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41#4114yes—
253C22×Q8C2×D411yescharacteristic
254C22×Q8C2411yescharacteristic
255C22×D4—21no—
256C22×D4—22no—
257C22×D4C2×D412yes—
258C22×D4C2×D411yescharacteristic
259C22×D4C2411yescharacteristic
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4C2×D412yes—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2×D411yescharacteristic
270C23×C4C2412yes—
271C23×C4C2411yescharacteristic
272C2×M16—24no—
273C2×M16—24no—
274C2×M16—24no—
275C22×C8—24no—
276C22×C8—24no—
277C22×C8—24no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2414yes—
281C22⋊Q8—28no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284#121—216no—
285#121—28no—
286#121—24no—
287#121—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—84no—
305#117—22no—
306#117—22no—
307#117—24no—
308#117—24no—
309#117—22no—
310#117—22no—
311#117—24no—
312#117—24no—
313#117—44no—
314#117—84no—
315#117—84no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—82no—
338C2×C4⋊C4—82no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4#4114yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D412yes—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—24no—
374C2×C42—24no—
375C2×C42—22no—
376C2×C42—44no—
377C2×C42—82no—
378C2×C42C2×D412yes—
379C2×C42C2×D412yes—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42#4114yes—
383C2×C42#4114yes—
384C2×C42C2412yes—
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—28no—
388#107—44no—
389#107—44no—
390#106—28no—
391C4⋊C8—22no—
392C4⋊C8—22no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396#103—24no—
397#103—24no—
398#103—44no—
399#102—24no—
400#102—24no—
401#102—44no—
402C22⋊C8—44no—
403C22⋊C8—44no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—24no—
416#95—24no—
417#95—22no—
418C24—21no—
419C24C22×D412yes—
420C24C22×D411yescharacteristic
421#41—28no—
422#41—28no—
423#41—44no—
424#41—44no—
425#41—44no—
426#41—44no—
427#41—44no—
428#41—44no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8C2×#4112yes—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—28no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4—82no—
455C2×D4C2×#4112yes—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
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—21no—
476C22×C4—21no—
477C22×C4—24no—
478C22×C4—41no—
479C22×C4—42no—
480C22×C4—41no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4#12314yes—
499C22×C4#12314yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C42−1+412yes—
514C22×C4C2511yescharacteristic
515M16—48no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—48no—
526C4⋊C4—28no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4C2×#4114yes—
560C22⋊C4—28no—
561C22⋊C4—28no—
562C22⋊C4—24no—
563C22⋊C4—24no—
564C22⋊C4—24no—
565C22⋊C4—42no—
566C22⋊C4—44no—
567C22⋊C4—42no—
568C22⋊C4—42no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—84no—
573C22⋊C4—84no—
574C22⋊C4—82no—
575C42—24no—
576C42—24no—
577C42—22no—
578C42—22no—
579C42—42no—
580C42—42no—
581C42—42no—
582C42—44no—
583C42—44no—
584C42—82no—
585C42—84no—
586C42—84no—
587C42#12314yes—
588C42#12314yes—
589C42C2×#4112yes—
590C42C2×#4112yes—
591C42C2×#4112yes—
592C42C2×#4112yes—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—41no—
609C23—41no—
610C23—41no—
611C23C2×#12312yes—
612C23#52512yes—
613C23#53712yes—
614C23C23×D411yescharacteristic
615Q8—22no—
616Q8—22no—
617Q8—41no—
618Q8—41no—
619Q8—44no—
620D4—22no—
621D4—22no—
622D4—48no—
623D4—48no—
624D4—41no—
625D4—41no—
626D4—44no—
627D4—88no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—28no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—44no—
647C2×C4—48no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—48no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—48no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4—164no—
681C2×C4C2×#12312yes—
682C2×C4C2×#12312yes—
683C2×C4C2×#12312yes—
684C2×C4C2×#12312yes—
685C2×C4C2×#12312yes—
686C2×C4#52512yes—
687C2×C4#53712yes—
688C2×C4#55214yes—
689C2×C4#55314yes—
690C2×C4#55514yes—
691C2×C4#55614yes—
692C2×C4#57612yes—
693C2×C4#57812yes—
694C2×C4C22×#4111yescharacteristic
695C2×C4C22×#4111yescharacteristic
696C2×C4C22×#4111yescharacteristic
697C2×C4C22×#4111yescharacteristic
698C2×C4C2×H(5,𝔽2)11yescharacteristic
699C2×C4C2×2−1+411yescharacteristic
700C8—42no—
701C8—42no—
702C8—44no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—81no—
720C22—81no—
721C22—81no—
722C22—81no—
723C22#248514yes—
724C22C22×#12311yescharacteristic
725C22C2×#52511yescharacteristic
726C22C2×#53711yescharacteristic
727C22C2×#57611yescharacteristic
728C22C2×#57811yescharacteristic
729C22#358511yescharacteristic
730C22#358611yescharacteristic
731C4—22no—
732C4—22no—
733C4—22no—
734C4—42no—
735C4—42no—
736C4—44no—
737C4—42no—
738C4—42no—
739C4—81no—
740C4—81no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—84no—
745C4#248514yes—
746C4#309814yes—
747C4C2×#55211yescharacteristic
748C4C2×#55311yescharacteristic
749C4C2×#55511yescharacteristic
750C4C2×#55611yescharacteristic
751C2—24no—
752C2—82no—
753C2—82no—
754C2C2×#248511yescharacteristic
755C2C2×#309811yescharacteristic
756C2#6051711yescharacteristic
757C2#6059611yescharacteristic
758C2#6060011yescharacteristic
759C2#6092211yescharacteristic
760C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
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
1. #34188C2semidirect
3. #27643C2semidirect
4. #27640C2semidirect
5. #27631C2semidirect
6. #27629C2semidirect
7. #27626C2semidirect
8. #27624C2semidirect
9. #23639C2semidirect
11. #21170C2semidirect
12. C2×#1947C2semidirect
13. #21067C2semidirect
14. #21036C2semidirect
15. #20648C2semidirect
16. #20645C2semidirect
17. #20629C2semidirect

Maximal subgroups

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

subgroupindex
#341882
C4×#5372
#276432
#276402
#276312
#276292
#276262
#276242
#236392
#234852
#211702
C2×#19472
#210672
#210362
#206482
#206452
#206292