Finite Group Databasebeta

← #6536247

Subgroups — #6536247

Order 512.

1825 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34308C211yesmaximal, characteristic
3#30172C212yesmaximal, complemented
4#29980C211yesmaximal, characteristic, complemented
5#29302C212yesmaximal, complemented
6#29208C211yesmaximal, characteristic, complemented
7#28205C211yesmaximal, characteristic, complemented
8#27181C214yesmaximal, complemented
9#26571C214yesmaximal, complemented
10#25016C212yesmaximal, complemented
11#24120C212yesmaximal
12#24064C212yesmaximal, complemented
13#23677C211yesmaximal, characteristic, complemented
14#23574C212yesmaximal, complemented
15#22970C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22671C212yesmaximal, complemented
18C2×#547C2211yescharacteristic
19C2×#533C2212yes—
20C2×#520—21no—
21C2×#484C2214yes—
22C2×#475C2212yes—
23C2×#440C2214yes—
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2552—21no—
28#2542—21no—
29#2538C2212yes—
30#2441—22no—
31#2429—22no—
32#2423C2214yes—
33#2365—21no—
34#2365C2211yescharacteristic
35#2361C2212yes—
36#2360—21no—
37#2359—22no—
38#2354—21no—
39Q8×C4⋊C4C2212yes—
40D4×C4⋊C4C2211yescharacteristic
41D4×C4⋊C4C2211yescharacteristic
42#2344C2212yes—
43#2339—21no—
44#2323—21no—
45#2322—22no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#121C2212yes—
49#2293C2211yescharacteristic
50C2×#389C2212yes—
51C42×D4C2211yescharacteristic
52#2099C2212yes—
53#2093C2212yes—
54#2084C2212yes—
55#2084C2211yescharacteristic
56#2076C2214yes—
57#2071C2211yescharacteristic
58#2066C2212yes—
59#2057C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2055C2212yes—
63#2043C2211yescharacteristic
64#2043C2212yes—
65#2036C2214yes—
66#2034C2214yes—
67#2023C2212yes—
68#2013C2211yescharacteristic
69#2000C2214yes—
70#1990C2212yes—
71#1954C2212yes—
72#1947C2214yes—
73#1947C2212yes—
74#1944C2214yes—
75#1944C2214yes—
76#1927C2214yes—
77#1924C2214yes—
78#1919C2214yes—
79#1872C2214yes—
80#1863C2214yes—
81#1863C2214yes—
82#1862C2214yes—
83#1853C2214yes—
84#1852C2214yes—
85#1845C2212yes—
86#1831C2211yescharacteristic
87#1817C2212yes—
88#1803C2212yes—
89#1802C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C22×#41—21no—
96C22×SD16C2312yes—
97#547—24no—
98#547—24no—
99#533—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125—22no—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2314yes—
111C2×#121C2312yes—
112#520—48no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2314yes—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C2×#117—22no—
128C2×#117—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C42C2312yes—
134#484—48no—
135#475—28no—
136#440—48no—
137C4×SD16—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#399—24no—
157#398—24no—
158#396—24no—
159#391—21no—
160#391—21no—
161#391—22no—
162#391C2312yes—
163#390—22no—
164#390C2311yescharacteristic
165#390C2311yescharacteristic
166#389—21no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—24no—
175#389C2312yes—
176#389C2311yescharacteristic
177#389C2311yescharacteristic
178#389C2312yes—
179#389C2314yes—
180#389C2312yes—
181#388—24no—
182#388—21no—
183#388—22no—
184#388—21no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—24no—
190#386—22no—
191#386—22no—
192#385—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#381—22no—
203#380—22no—
204#380—21no—
205#380C2312yes—
206#380C2314yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222C2×#95—22no—
223C43C2311yescharacteristic
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236C2×#41—28no—
237C2×#41—44no—
238C2×#41—44no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8C2412yes—
242C22×D4—22no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—24no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263C4⋊Q8#4114yes—
264#125—48no—
265#123—216no—
266C22⋊Q8—28no—
267C22⋊Q8—28no—
268C22⋊Q8—44no—
269C22⋊Q8—48no—
270C22⋊Q8—84no—
271#121—24no—
272#121—24no—
273C4×Q8—28no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—48no—
279C4×Q8—44no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—44no—
286C4×D4—48no—
287C4×D4—48no—
288C4×D4—44no—
289C4×D4#4114yes—
290#117—48no—
291#117—48no—
292#117—44no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—44no—
361#107—44no—
362#107—44no—
363#106—24no—
364#106—28no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367#103—28no—
368#103—24no—
369#103—24no—
370#103—24no—
371#103—28no—
372#102—24no—
373#102—44no—
374C22⋊C8—28no—
375C22⋊C8—44no—
376#97—44no—
377C4×C8—24no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—24no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—21no—
395C24C22×D412yes—
396#41—432no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8H(5,𝔽2)14yes—
409C2×D4—28no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4#13714yes—
418C2×D4C2×#4112yes—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C2511yescharacteristic
459SD16—216no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C82−1+414yes—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—84no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4C2×#4112yes—
505C4⋊C4C2×#4114yes—
506C22⋊C4—28no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—81no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C42—22no—
520C42—22no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—42no—
525C42—42no—
526C42—41no—
527C42—48no—
528C42—44no—
529C42—44no—
530C42—44no—
531C42—42no—
532C42AGL(1,ℤ/8)14yes—
533C42C2×#4112yes—
534C42C2×#4112yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×#13714yes—
549C23C23×D411yescharacteristic
550Q8—24no—
551Q8—24no—
552Q8—48no—
553Q8—42no—
554Q8—42no—
555Q8—88no—
556D4—28no—
557D4—24no—
558D4—416no—
559D4—42no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—48no—
584C2×C4—416no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—84no—
610C2×C4—164no—
611C2×C4—164no—
612C2×C4#52512yes—
613C2×C4#52512yes—
614C2×C4C2×AGL(1,ℤ/8)12yes—
615C2×C4C2×AGL(1,ℤ/8)12yes—
616C2×C4C2×AGL(1,ℤ/8)12yes—
617C2×C4C2×#13712yes—
618C2×C4#57812yes—
619C2×C4C22×#4111yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)12yes—
622C2×C4C2×2−1+412yes—
623C8—24no—
624C8—42no—
625C8—44no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22C2×#52511yescharacteristic
641C22#348411yescharacteristic
642C22#349911yescharacteristic
643C22C22×AGL(1,ℤ/8)11yescharacteristic
644C22C22×#13711yescharacteristic
645C22C2×#57811yescharacteristic
646C22#358511yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—41no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—82no—
661C4—84no—
662C4—82no—
663C4#311014yes—
664C4#349212yes—
665C4#352212yes—
666C2—44no—
667C2—82no—
668C2#3569311yescharacteristic
669C2C2×#311011yescharacteristic
670C2#6060011yescharacteristic
671C2#6158711yescharacteristic
672C2#6160211yescharacteristic
673C2#6247911yescharacteristic
674C2#6250411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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. #30172C2semidirect
4. #29980C2semidirect
5. #29302C2semidirect
6. #29208C2semidirect
7. #28205C2semidirect
8. #27181C2semidirect
9. #26571C2semidirect
10. #25016C2semidirect
12. #24064C2semidirect
13. #23677C2semidirect
14. #23574C2semidirect
15. #22970C2semidirect
16. #22839C2semidirect
17. #22671C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#343082
#301722
#299802
#293022
#292082
#282052
#271812
#265712
#250162
#241202
#240642
#236772
#235742
#229702
#228392
#226712