Finite Group Databasebeta

← #6563282

Subgroups — #6563282

Order 512.

1777 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36492C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29944C212yesmaximal, complemented
4#29939C211yesmaximal, characteristic, complemented
5#27953C212yesmaximal, complemented
6#27950C211yesmaximal, characteristic, complemented
7#27645C211yesmaximal, characteristic, complemented
8#27644C212yesmaximal, complemented
9#26308C214yesmaximal, complemented
10#26300C212yesmaximal, complemented
11#25022C214yesmaximal, complemented
12#25010C212yesmaximal, complemented
13#23760C212yesmaximal, complemented
14#23755C211yesmaximal, characteristic, complemented
15#22728C212yesmaximal, complemented
16#22532C211yesmaximal, characteristic, complemented
17#22522C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#481C2212yes—
22C2×#480C2214yes—
23C2×#446C2212yes—
24#2857C2211yescharacteristic
25#2851C2212yes—
26#2563C2211yescharacteristic
27#2562C2212yes—
28#2537C2211yescharacteristic
29#2535C2212yes—
30#2477—22no—
31#2463—22no—
32#2421—21no—
33#2407—21no—
34#2361—21no—
35#2355C2211yescharacteristic
36#2352—21no—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4C2212yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—22no—
41D4×C4⋊C4C2212yes—
42D4×C4⋊C4C2214yes—
43#2317—21no—
44#2317—22no—
45C4×#121C2211yescharacteristic
46C4×#121C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51#2100C2211yescharacteristic
52#2100C2212yes—
53#2094C2211yescharacteristic
54#2084C2211yescharacteristic
55#2084C2212yes—
56#2083C2212yes—
57#2083C2212yes—
58#2083C2212yes—
59#2077C2214yes—
60#2077C2214yes—
61#2077C2212yes—
62#2076C2214yes—
63#2075C2214yes—
64#2071C2211yescharacteristic
65#2071C2212yes—
66#2034C2212yes—
67#2033C2214yes—
68#1992C2214yes—
69#1990C2212yes—
70#1989C2214yes—
71#1987C2212yes—
72#1955C2214yes—
73#1954C2212yes—
74#1949C2214yes—
75#1948C2212yes—
76#1947C2212yes—
77#1947C2214yes—
78#1853C2214yes—
79#1852C2214yes—
80#1852C2214yes—
81#1847C2214yes—
82#1845C2212yes—
83#1824C2212yes—
84#1823C2214yes—
85#1802C2212yes—
86#1802C2214yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1776C2214yes—
90C4×#106C2212yes—
91#1762C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41D411yescharacteristic
96#554—28no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#125C2312yes—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×#121C2311yescharacteristic
105#520—48no—
106C4×#41—28no—
107C2×C4×Q8—22no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—24no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D412yes—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C2×#117—21no—
122C2×#117D411yescharacteristic
123C2×#117D412yes—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#481—44no—
134#480—48no—
135#446—28no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2312yes—
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#400—22no—
154#397—22no—
155#394—22no—
156#394—22no—
157#393—22no—
158#392—22no—
159#391—21no—
160#391—21no—
161#390—21no—
162#390C2311yescharacteristic
163#389—22no—
164#389—24no—
165#389—22no—
166#389—24no—
167#389—21no—
168#389C2312yes—
169#389C2311yescharacteristic
170#388—22no—
171#388—22no—
172#388—22no—
173#388—21no—
174#387—22no—
175#387—22no—
176#387—22no—
177#386—22no—
178#386—21no—
179#386—22no—
180#386—22no—
181#385—22no—
182#385—22no—
183#385—24no—
184#385—24no—
185#385—24no—
186#384—22no—
187#384—21no—
188#384—21no—
189#384—22no—
190#384C2312yes—
191#384C2314yes—
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382—21no—
197#382—22no—
198#382—21no—
199#382C2312yes—
200#380—21no—
201#380—22no—
202#380—21no—
203#380—21no—
204#380C2311yescharacteristic
205#380C2312yes—
206#380C2312yes—
207#380C2312yes—
208#380C2312yes—
209#380C2314yes—
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2314yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2311yescharacteristic
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—24no—
237C22×Q8—21no—
238C22×Q8C2×D411yescharacteristic
239C22×D4—22no—
240C22×D4—21no—
241C22×D4C2×D412yes—
242C22×D4C2×D411yescharacteristic
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—24no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258#125—28no—
259C22⋊Q8—44no—
260C22⋊Q8—48no—
261C22⋊Q8—82no—
262#121—24no—
263#121—22no—
264#121—41no—
265#121—42no—
266#121—44no—
267C4×Q8—24no—
268C4×Q8—48no—
269C4×Q8#4118yes—
270C4×D4—24no—
271C4×D4—28no—
272C4×D4—24no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—28no—
276C4×D4—416no—
277C4×D4—48no—
278C4×D4—44no—
279C4×D4—84no—
280#117—24no—
281#117—28no—
282#117—48no—
283#117—44no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—84no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—24no—
341C2×C42—28no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353#107—44no—
354#107—44no—
355#107—44no—
356#106—24no—
357#106—42no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—28no—
361#102—24no—
362#102—28no—
363#102—24no—
364C22⋊C8—24no—
365C22⋊C8—28no—
366C22⋊C8—44no—
367C22⋊C8—42no—
368#97—28no—
369C4×C8—24no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385C24—21no—
386C24C22×D412yes—
387C24C22×D411yescharacteristic
388#41—232no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—42no—
392C2×Q8—81no—
393C2×Q8—81no—
394C2×Q8#12314yes—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—82no—
408C2×D4—81no—
409C2×D4AGL(1,ℤ/8)14yes—
410C2×D4H(5,𝔽2)12yes—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—84no—
447C22×C4—82no—
448C22×C4—84no—
449C22×C4—82no—
450C22×C4#13714yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C82−1+414yes—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—84no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C4⋊C4—82no—
502C4⋊C4—81no—
503C4⋊C4#12314yes—
504C4⋊C4#12318yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C42—22no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—48no—
527C42—44no—
528C42—42no—
529C42—82no—
530C42C2×#4114yes—
531C42C2×#4118yes—
532C23—22no—
533C23—21no—
534C23—22no—
535C23—21no—
536C23—21no—
537C23—22no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—22no—
542C23—21no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C2×AGL(1,ℤ/8)14yes—
552C23C2×#13712yes—
553C23C23×D411yescharacteristic
554Q8—28no—
555Q8—162no—
556D4—216no—
557D4—28no—
558D4—24no—
559D4—42no—
560D4—84no—
561D4—88no—
562C2×C4—216no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—28no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—84no—
611C2×C4—164no—
612C2×C4—168no—
613C2×C4—164no—
614C2×C4C2×#12312yes—
615C2×C4C2×#12314yes—
616C2×C4C2×#12314yes—
617C2×C4C2×#12312yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4#57612yes—
622C2×C4C22×#4112yes—
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×2−1+412yes—
625C2×C4#58511yescharacteristic
626C8—24no—
627C8—42no—
628C8—44no—
629C22—22no—
630C22—22no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—41no—
641C22—41no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22C22×#12311yescharacteristic
651C22#348811yescharacteristic
652C22#349911yescharacteristic
653C22C22×AGL(1,ℤ/8)11yescharacteristic
654C22C22×#13711yescharacteristic
655C22C2×#57611yescharacteristic
656C22#358611yescharacteristic
657C4—22no—
658C4—24no—
659C4—22no—
660C4—24no—
661C4—22no—
662C4—41no—
663C4—42no—
664C4—81no—
665C4—81no—
666C4—82no—
667C4—84no—
668C4—82no—
669C4—82no—
670C4—81no—
671C4—82no—
672C4#309214yes—
673C4#351612yes—
674C4#352212yes—
675C2—42no—
676C2—44no—
677C2—82no—
678C2#3599611yescharacteristic
679C2C2×#309211yescharacteristic
680C2#6052111yescharacteristic
681C2#6196011yescharacteristic
682C2#6197911yescharacteristic
683C2#6248011yescharacteristic
684C2#6250311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36492C2
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
2. #34183C2semidirect
3. #29944C2semidirect
4. #29939C2semidirect
5. #27953C2semidirect
6. #27950C2semidirect
7. #27645C2semidirect
8. #27644C2semidirect
9. #26308C2semidirect
10. #26300C2semidirect
11. #25022C2semidirect
12. #25010C2semidirect
13. #23760C2semidirect
14. #23755C2semidirect
15. #22728C2semidirect
16. #22532C2semidirect
17. #22522C2semidirect

Maximal subgroups

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

subgroupindex
#364922
#341832
#299442
#299392
#279532
#279502
#276452
#276442
#263082
#263002
#250222
#250102
#237602
#237552
#227282
#225322
#225222