Finite Group Databasebeta

← #6593788

Subgroups — #6593788

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33146C211yesmaximal, characteristic, complemented
4#33003C212yesmaximal
5#27586C212yesmaximal, complemented
6#27083C214yesmaximal, complemented
7#26773C214yesmaximal, complemented
8#26532C212yesmaximal, complemented
9#25856C211yesmaximal, characteristic, complemented
10#25320C212yesmaximal, complemented
11#24509C212yesmaximal
12#24209C211yesmaximal, characteristic
13#23222C212yesmaximal, complemented
14#23100C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#480C211yesmaximal, characteristic
17#20740C212yesmaximal, complemented
18C2×#552—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C2×#487C2212yes—
22C2×#480C2211yescharacteristic
23C2×#461C2212yes—
24C2×#448C2212yes—
25C2×#432C2212yes—
26#2857C2211yescharacteristic
27#2854C2212yes—
28#2552C2211yescharacteristic
29#2542C2211yescharacteristic
30#2519—22no—
31#2512C2212yes—
32#2506C2212yes—
33#2498—22no—
34#2489—21no—
35#2417—21no—
36#2365—21no—
37#2365—21no—
38#2365—22no—
39#2365C2211yescharacteristic
40#2365C2212yes—
41#2365C2214yes—
42#2365C2212yes—
43#2362—21no—
44#2354C2211yescharacteristic
45C4×#121C2211yescharacteristic
46C2×#391C2212yes—
47C2×#389—21no—
48C2×#387—22no—
49#2285C2212yes—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2095C2214yes—
54#2072C2214yes—
55#2068C2214yes—
56#2055C2214yes—
57#2043C2214yes—
58#2043C2212yes—
59#2013C2212yes—
60#1988C2212yes—
61#1981C2212yes—
62#1973C2214yes—
63#1956C2212yes—
64#1947C2212yes—
65#1939C2212yes—
66#1935C2214yes—
67#1928C2214yes—
68#1928C2214yes—
69#1925C2212yes—
70#1918C2212yes—
71#1918C2214yes—
72#1917C2214yes—
73#1880C2214yes—
74#1880C2214yes—
75#1874C2211yescharacteristic
76#1863C2214yes—
77#1863C2214yes—
78#1852C2211yescharacteristic
79#1852C2212yes—
80#1823C2211yescharacteristic
81#1823C2212yes—
82#1817C2212yes—
83#1803C2214yes—
84#1803C2214yes—
85#1797C2211yescharacteristic
86#1787C2212yes—
87#1787C2212yes—
88#1787C2212yes—
89C4×#107C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92#1763C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×SD16C2314yes—
97#552—48no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#126—22no—
105C2×#125—21no—
106C2×#125—22no—
107C2×#123—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×#121C2311yescharacteristic
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—28no—
116C2×C4×Q8—44no—
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2314yes—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—22no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#487—28no—
135#480—24no—
136#461—28no—
137#448—44no—
138#432—44no—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2311yescharacteristic
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158#400—22no—
159#399—22no—
160#398—22no—
161#398—22no—
162#397—22no—
163#396—22no—
164#391—21no—
165#391—21no—
166#391—22no—
167#391—22no—
168#391C2314yes—
169#391C2314yes—
170#390C2312yes—
171#390C2312yes—
172#390C2311yescharacteristic
173#389—21no—
174#389—22no—
175#389—22no—
176#389—24no—
177#389—21no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2311yescharacteristic
182#388—22no—
183#388—22no—
184#388—21no—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#387—22no—
193#386—22no—
194#386—21no—
195#384C2312yes—
196#384C2312yes—
197#384C2311yescharacteristic
198#382—22no—
199#380—21no—
200#380—22no—
201#380—21no—
202#380C2312yes—
203#380C2311yescharacteristic
204C4×C4⋊C4—22no—
205C4×C4⋊C4—24no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—21no—
222C2×#95—22no—
223C2×#95—21no—
224C2×#95—22no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—28no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—24no—
241C22×Q8—24no—
242C22×Q8—44no—
243C22×Q8C2414yes—
244C22×D4—21no—
245C22×D4C2411yescharacteristic
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—28no—
256C2×SD16—44no—
257C2×SD16—44no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8#4118yes—
267#126—48no—
268#125—48no—
269#125—44no—
270#123—48no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274#121—22no—
275#121—22no—
276C4×Q8—28no—
277C4×Q8—28no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×Q8—48no—
282C4×Q8—44no—
283C4×D4—24no—
284C4×D4—44no—
285#117—44no—
286#117—48no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4#4114yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—48no—
348C2×C42—44no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—24no—
358#107—42no—
359#107—44no—
360C4⋊C8—24no—
361C4⋊C8—44no—
362#103—28no—
363#103—28no—
364#103—28no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#102—24no—
370#102—42no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—24no—
374#97—44no—
375C4×C8—24no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
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—28no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—416no—
404C2×Q8—416no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8C2×#4118yes—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4AGL(1,ℤ/8)14yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4C2511yescharacteristic
464SD16—88no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—82no—
506C4⋊C4—81no—
507C4⋊C4#13714yes—
508C4⋊C4C2×#4114yes—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—81no—
518C22⋊C4—82no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—48no—
529C42—48no—
530C42—44no—
531C42—44no—
532C42—44no—
533C42—44no—
534C42—42no—
535C42C2×#4114yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23#54612yes—
548C23C2×AGL(1,ℤ/8)12yes—
549C23#57812yes—
550C23C23×D411yescharacteristic
551Q8—28no—
552Q8—28no—
553Q8—416no—
554Q8—44no—
555Q8—44no—
556Q8—84no—
557D4—42no—
558D4—42no—
559D4—84no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—416no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4#52512yes—
615C2×C4#52512yes—
616C2×C4#54612yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C2×#13714yes—
620C2×C4C2×#13712yes—
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4112yes—
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4#58512yes—
625C8—42no—
626C8—42no—
627C8—44no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22#333014yes—
642C22C2×#52511yescharacteristic
643C22C2×#54611yescharacteristic
644C22#348711yescharacteristic
645C22C22×AGL(1,ℤ/8)11yescharacteristic
646C22C22×#13711yescharacteristic
647C22C2×#57811yescharacteristic
648C22#358611yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—24no—
653C4—48no—
654C4—41no—
655C4—42no—
656C4—42no—
657C4—44no—
658C4—42no—
659C4—81no—
660C4—82no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4—81no—
665C4#311214yes—
666C4#347512yes—
667C4#351312yes—
668C2—24no—
669C2—161no—
670C2#3616211yescharacteristic
671C2C2×#311211yescharacteristic
672C2#6058311yescharacteristic
673C2C2×#333011yescharacteristic
674C2#6115711yescharacteristic
675C2#6188811yescharacteristic
676C2#6190011yescharacteristic
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×#2365C2semidirect
3. #33146C2semidirect
5. #27586C2semidirect
6. #27083C2semidirect
7. #26773C2semidirect
8. #26532C2semidirect
9. #25856C2semidirect
10. #25320C2semidirect
13. #23222C2semidirect
14. #23100C2semidirect
15. #22603C2semidirect
17. #20740C2semidirect

Maximal subgroups

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

subgroupindex
#361862
C2×#23652
#331462
#330032
#275862
#270832
#267732
#265322
#258562
#253202
#245092
#242092
#232222
#231002
#226032
C4×#4802
#207402