Finite Group Databasebeta

← #6795033

Subgroups — #6795033

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3#32954C211yesmaximal, characteristic, complemented
4#32453C214yesmaximal, complemented
5#32367C214yesmaximal, complemented
6#29912C212yesmaximal
7#28540C212yesmaximal, complemented
8#27737C212yesmaximal, complemented
9#27586C212yesmaximal, complemented
10#25813C211yesmaximal, characteristic, complemented
11#25456C212yesmaximal, complemented
12#24681C212yesmaximal, complemented
13#24509C212yesmaximal
14#24298C211yesmaximal, characteristic
15#22603C211yesmaximal, characteristic, complemented
16#21986C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic
18C2×#552—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#500C2212yes—
22C2×#475C2212yes—
23C2×#474C2212yes—
24C2×#448C2211yescharacteristic
25C2×C4×SD16C2212yes—
26C2×#432C2212yes—
27#2857C2211yescharacteristic
28#2854C2212yes—
29#2568—21no—
30#2568—21no—
31#2568—22no—
32#2568C2212yes—
33#2568C2212yes—
34#2568C2214yes—
35#2552C2211yescharacteristic
36#2542C2211yescharacteristic
37#2519—22no—
38#2512C2212yes—
39#2506C2212yes—
40#2498—22no—
41#2489—21no—
42#2417—21no—
43C2×#400—22no—
44#2365C2211yescharacteristic
45#2362—21no—
46#2354C2211yescharacteristic
47C4×#121C2211yescharacteristic
48C2×#391—21no—
49C2×#391—21no—
50C2×#391C2212yes—
51#2285C2212yes—
52C2×#382C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yes—
55#2095C2214yes—
56#2092C2214yes—
57#2085C2214yes—
58#2073C2214yes—
59#2072C2214yes—
60#2068C2214yes—
61#2057C2212yes—
62#2057C2214yes—
63#2055C2214yes—
64#2023C2212yes—
65#1990C2212yes—
66#1988C2212yes—
67#1973C2214yes—
68#1949C2214yes—
69#1949C2214yes—
70#1939C2212yes—
71#1935C2214yes—
72#1928C2214yes—
73#1928C2214yes—
74#1917C2214yes—
75#1880C2212yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1865C2212yes—
79#1865C2214yes—
80#1853C2212yes—
81#1853C2211yescharacteristic
82#1852C2212yes—
83#1852C2212yes—
84#1852C2212yes—
85#1847C2214yes—
86#1836C2214yes—
87#1817C2212yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90#1787C2212yes—
91#1787C2211yescharacteristic
92#1770C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98#552—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#126—22no—
106C2×#125—21no—
107C2×#125—22no—
108C2×#123—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×#121C2311yescharacteristic
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—44no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#500—44no—
133#475—44no—
134#474—44no—
135#448—24no—
136C4×SD16—44no—
137#432—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#400—24no—
158#400—24no—
159#400—24no—
160#400—24no—
161#400—22no—
162#399—22no—
163#398—22no—
164#398—22no—
165#397—22no—
166#396—22no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—24no—
171#391—22no—
172#391—24no—
173#391—21no—
174#391—22no—
175#391—22no—
176#391—21no—
177#391C2314yes—
178#391C2314yes—
179#390C2312yes—
180#390C2312yes—
181#390C2311yescharacteristic
182#389—21no—
183#389—21no—
184#389C2312yes—
185#389C2311yescharacteristic
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#388—21no—
191#388—22no—
192#388—22no—
193#387—22no—
194#387—22no—
195#386—22no—
196#386—21no—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#382—22no—
201#382—22no—
202#382—22no—
203#382C2314yes—
204#382C2314yes—
205#380—21no—
206#380—22no—
207#380—21no—
208#380C2312yes—
209#380C2311yescharacteristic
210C4×C4⋊C4—22no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95—22no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—42no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—42no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272#126—48no—
273#125—48no—
274#125—44no—
275#123—48no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279#121—22no—
280#121—22no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—42no—
284C4×Q8—42no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—44no—
290#117—44no—
291#117—48no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355#107—44no—
356#107—44no—
357#107—44no—
358#106—42no—
359#106—44no—
360C4⋊C8—28no—
361C4⋊C8—24no—
362#103—28no—
363#103—24no—
364#103—24no—
365#103—42no—
366#103—44no—
367#102—42no—
368#102—44no—
369C22⋊C8—24no—
370C22⋊C8—22no—
371C22⋊C8#4114yes—
372#97—28no—
373C4×C8—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24C22×D411yescharacteristic
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—816no—
417C2×Q82−1+414yes—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4AGL(1,ℤ/8)14yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4112yes—
469C22×C42−1+412yes—
470C22×C4C2511yescharacteristic
471SD16—48no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8C2×#4114yes—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—81no—
523C22⋊C4—82no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—44no—
534C42—44no—
535C42—44no—
536C42—44no—
537C42—42no—
538C42H(5,𝔽2)14yes—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23#54712yes—
551C23C2×AGL(1,ℤ/8)12yes—
552C23#57812yes—
553C23C23×D411yescharacteristic
554Q8—24no—
555Q8—24no—
556Q8—48no—
557Q8—42no—
558Q8—42no—
559Q8—88no—
560Q8—84no—
561Q8—88no—
562D4—42no—
563D4—42no—
564D4—84no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—88no—
619C2×C4—84no—
620C2×C4#53512yes—
621C2×C4#53512yes—
622C2×C4#54712yes—
623C2×C4C2×AGL(1,ℤ/8)12yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4#57512yes—
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)12yes—
628C2×C4C2×2−1+412yes—
629C2×C4C2×2−1+411yescharacteristic
630C8—24no—
631C8—44no—
632C8—42no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22#335314yes—
647C22C2×#53511yescharacteristic
648C22C2×#54711yescharacteristic
649C22#348611yescharacteristic
650C22C22×AGL(1,ℤ/8)11yescharacteristic
651C22C2×#57511yescharacteristic
652C22C2×#57811yescharacteristic
653C22#358611yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—44no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—41no—
663C4—81no—
664C4—81no—
665C4—84no—
666C4—82no—
667C4—81no—
668C4—82no—
669C4—82no—
670C4#321314yes—
671C4#352012yes—
672C4#353812yes—
673C2—24no—
674C2—161no—
675C2#3891711yescharacteristic
676C2C2×#321311yescharacteristic
677C2#6081411yescharacteristic
678C2C2×#335311yescharacteristic
679C2#6125711yescharacteristic
680C2#6182811yescharacteristic
681C2#6184211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2568C2semidirect
3. #32954C2semidirect
4. #32453C2semidirect
5. #32367C2semidirect
7. #28540C2semidirect
8. #27737C2semidirect
9. #27586C2semidirect
10. #25813C2semidirect
11. #25456C2semidirect
12. #24681C2semidirect
15. #22603C2semidirect
16. #21986C2semidirect

Maximal subgroups

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

subgroupindex
#361862
C2×#25682
#329542
#324532
#323672
#299122
#285402
#277372
#275862
#258132
#254562
#246812
#245092
#242982
#226032
#219862
C4×#4482