Finite Group Databasebeta

← #6593793

Subgroups — #6593793

Order 512.

2200 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#36176C211yesmaximal, characteristic, complemented
2C2×#2366C211yesmaximal, characteristic, complemented
3#33148C211yesmaximal, characteristic, complemented
4#33002C212yesmaximal, complemented
5#27588C212yesmaximal, complemented
6#27080C214yesmaximal, complemented
7#26768C214yesmaximal, complemented
8#26533C212yesmaximal, complemented
9#25855C211yesmaximal, characteristic, complemented
10#25321C212yesmaximal, complemented
11#24506C212yesmaximal, complemented
12#24208C211yesmaximal, characteristic
13#23223C212yesmaximal, complemented
14#23099C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#480C211yesmaximal, characteristic, complemented
17#20739C212yesmaximal, complemented
18C22×#127C2211yescharacteristic
19C22×C4×D4—21no—
20C22×C4×D4C2212yes—
21C2×#486C2212yes—
22C2×#480C2211yescharacteristic
23C2×#459C2212yes—
24C2×#447C2212yes—
25C2×#432C2212yes—
26#2857C2211yescharacteristic
27#2854C2212yes—
28#2571C2211yescharacteristic, complemented
29#2563C2211yescharacteristic, complemented
30#2532—22no—
31#2513C2212yescomplemented
32#2508C2212yescomplemented
33#2495—21no—
34#2488—22no—
35#2486—21no—
36#2369C2211yescharacteristic, complemented
37#2366—21no—
38#2366—21no—
39#2366—22no—
40#2366C2212yes—
41#2366C2214yes—
42#2366C2212yes—
43#2354C2211yescharacteristic, complemented
44#2352—21no—
45#2346—21no—
46C4×#121C2211yescharacteristic
47C2×#390—21no—
48C2×#390C2212yes—
49C2×#386—22no—
50#2283C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2092C2214yescomplemented
54#2073C2214yescomplemented
55#2065C2214yescomplemented
56#2054C2214yescomplemented
57#2041C2212yes—
58#2041C2214yes—
59#2011C2212yes—
60#1987C2212yescomplemented
61#1982C2212yes—
62#1972C2214yescomplemented
63#1957C2212yes—
64#1948C2212yescomplemented
65#1939C2212yescomplemented
66#1936C2214yescomplemented
67#1926C2214yescomplemented
68#1925C2212yescomplemented
69#1925C2214yescomplemented
70#1920C2214yescomplemented
71#1918C2212yescomplemented
72#1917C2214yescomplemented
73#1879C2214yescomplemented
74#1879C2214yescomplemented
75#1874C2211yescharacteristic
76#1862C2214yescomplemented
77#1862C2214yescomplemented
78#1853C2212yes—
79#1852C2211yescharacteristic
80#1823C2211yescharacteristic
81#1823C2212yes—
82#1817C2212yes—
83#1804C2214yescomplemented
84#1804C2214yescomplemented
85#1797C2211yescharacteristic
86#1786C2212yes—
87#1786C2212yes—
88#1786C2212yes—
89C4×#107C2212yescomplemented
90#1770C2212yescomplemented
91#1762C2214yescomplemented
92#1762C2214yescomplemented
93C4×#102C2212yescomplemented
94C4×C22⋊C8C2211yescharacteristic
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C24×C4—22no—
99C22×D8C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127—28no—
102C2×#127—21no—
103C2×#127—21no—
104C2×#127C2312yes—
105C2×#127C2312yes—
106C2×#123—22no—
107C2×C22⋊Q8—21no—
108C2×#121C2311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—24no—
113C2×C4×D4—28no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—44no—
117C2×C4×D4—48no—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2314yes—
120C2×C4×D4C2314yes—
121C2×#117—21no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—22no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#486—28no—
133#480—24no—
134#459—28no—
135#447—44no—
136#432—44no—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2311yescharacteristic
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#400—22no—
158#398—22no—
159#397—22no—
160#397—22no—
161#393—22no—
162#391—21no—
163#391—22no—
164#390—22no—
165#390—22no—
166#390—24no—
167#390—22no—
168#390—22no—
169#390C2312yes—
170#390C2314yes—
171#390C2312yes—
172#390C2314yes—
173#390C2311yescharacteristic
174#389—21no—
175#389—21no—
176#389—21no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2311yescharacteristic
181#388—22no—
182#388—22no—
183#388—22no—
184#388—21no—
185#388—22no—
186#388—22no—
187#387—22no—
188#386—22no—
189#386—21no—
190#386—24no—
191#386—24no—
192#386—24no—
193#386—24no—
194#384C2311yescharacteristic
195#383—22no—
196#382—21no—
197#382—22no—
198#382C2312yes—
199#382C2312yes—
200#381—22no—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380C2312yes—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223#376—22no—
224C2×#95—21no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C25—22no—
238C22×Q8—21no—
239C22×D4—28no—
240C22×D4—28no—
241C22×D4—28no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—24no—
249C22×D4—24no—
250C22×D4—44no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4—82no—
254C22×D4—82no—
255C22×D4C2411yescharacteristic
256C22×D4C2414yes—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—24no—
261C23×C4—24no—
262C23×C4—48no—
263C23×C4—48no—
264C23×C4—84no—
265C23×C4C2×D412yes—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2411yescharacteristic
270C2×D8—28no—
271C2×D8—44no—
272C2×D8—44no—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C4⋊Q8—24no—
277#127—28no—
278#127—28no—
279#127—28no—
280#127—24no—
281#127#4118yes—
282#123—84no—
283C22⋊Q8—82no—
284#121—22no—
285#121—22no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—24no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—48no—
292C4×D4—48no—
293C4×D4—416no—
294C4×D4—48no—
295C4×D4—48no—
296C4×D4—44no—
297#117—44no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4#4114yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—48no—
342C2×C22⋊C4—48no—
343C2×C22⋊C4—84no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—24no—
362#107—42no—
363#106—44no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#102—28no—
367#102—28no—
368#102—28no—
369#102—24no—
370#102—44no—
371#102—44no—
372#102—44no—
373#102—42no—
374#102—44no—
375C22⋊C8—22no—
376C22⋊C8—22no—
377C22⋊C8—24no—
378#97—44no—
379C4×C8—24no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—28no—
400C24—24no—
401C24—21no—
402C24—44no—
403C24—44no—
404C24—44no—
405C24—44no—
406C24—44no—
407C24—44no—
408C24C22×D411yescharacteristic
409C2×Q8—42no—
410C2×Q8—81no—
411C2×Q8—81no—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—416no—
427C2×D4—416no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—432no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—816no—
443C2×D4C2×#4118yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—832no—
480C22×C4AGL(1,ℤ/8)14yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4C2511yescharacteristic
493D8—88no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—81no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4C2×#4114yes—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—82no—
539C22⋊C4—81no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C42—22no—
545C42—22no—
546C42—24no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—44no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—82no—
558C42C2×#4114yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—28no—
564C23—416no—
565C23—416no—
566C23—416no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—88no—
581C23—88no—
582C23—88no—
583C23—88no—
584C23#54612yes—
585C23C2×AGL(1,ℤ/8)12yes—
586C23#57612yes—
587C23C23×D411yescharacteristic
588Q8—162no—
589D4—28no—
590D4—28no—
591D4—416no—
592D4—416no—
593D4—416no—
594D4—44no—
595D4—42no—
596D4—44no—
597D4—42no—
598D4—88no—
599D4—88no—
600D4—84no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
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—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—84no—
650C2×C4—816no—
651C2×C4—816no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4#52512yes—
655C2×C4#52512yes—
656C2×C4#54612yes—
657C2×C4C2×AGL(1,ℤ/8)12yes—
658C2×C4C2×AGL(1,ℤ/8)12yes—
659C2×C4#57512yes—
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4112yes—
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4#58512yes—
664C8—42no—
665C8—42no—
666C8—44no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—48no—
674C22—48no—
675C22—48no—
676C22—832no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22#332514yes—
689C22C2×#52511yescharacteristic
690C22C2×#54611yescharacteristic
691C22#348711yescharacteristic
692C22C22×AGL(1,ℤ/8)11yescharacteristic
693C22C2×#57511yescharacteristic
694C22C2×#57611yescharacteristic
695C22#358511yescharacteristic
696C4—22no—
697C4—22no—
698C4—22no—
699C4—41no—
700C4—42no—
701C4—42no—
702C4—44no—
703C4—42no—
704C4—42no—
705C4—81no—
706C4—82no—
707C4—81no—
708C4—82no—
709C4—82no—
710C4—81no—
711C4#311414yes—
712C4#347512yes—
713C4#351312yes—
714C2—24no—
715C2—88no—
716C2—161no—
717C2#3616211yescharacteristic
718C2C2×#311411yescharacteristic
719C2#6057811yescharacteristic
720C2C2×#332511yescharacteristic
721C2#6116011yescharacteristic
722C2#6189011yescharacteristic
723C2#6189811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#24208C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36176C2semidirect
2. C2×#2366C2semidirect
3. #33148C2semidirect
4. #33002C2semidirect
5. #27588C2semidirect
6. #27080C2semidirect
7. #26768C2semidirect
8. #26533C2semidirect
9. #25855C2semidirect
10. #25321C2semidirect
11. #24506C2semidirect
13. #23223C2semidirect
14. #23099C2semidirect
15. #22603C2semidirect
16. C4×#480C2semidirect
17. #20739C2semidirect
28. #2571C22semidirect
29. #2563C22semidirect
31. #2513C22semidirect
32. #2508C22semidirect
36. #2369C22semidirect
43. #2354C22semidirect
53. #2092C22semidirect
54. #2073C22semidirect
55. #2065C22semidirect
56. #2054C22semidirect
60. #1987C22semidirect
62. #1972C22semidirect
64. #1948C22semidirect
65. #1939C22semidirect
66. #1936C22semidirect
67. #1926C22semidirect
68. #1925C22semidirect
69. #1925C22semidirect
70. #1920C22semidirect
71. #1918C22semidirect
72. #1917C22semidirect
73. #1879C22semidirect
74. #1879C22semidirect
76. #1862C22semidirect
77. #1862C22semidirect
83. #1804C22semidirect
84. #1804C22semidirect
89. C4×#107C22semidirect
90. #1770C22semidirect
91. #1762C22semidirect
92. #1762C22semidirect
93. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#23662
#331482
#330022
#275882
#270802
#267682
#265332
#258552
#253212
#245062
#242082
#232232
#230992
#226032
C4×#4802
#207392