Finite Group Databasebeta

← #6563509

Subgroups — #6563509

Order 512.

1805 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36275C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#29939C211yesmaximal, characteristic, complemented
4#29909C212yesmaximal, complemented
5#27950C211yesmaximal, characteristic, complemented
6#27858C212yesmaximal, complemented
7#27648C211yesmaximal, characteristic, complemented
8#27614C212yesmaximal, complemented
9#26282C212yesmaximal, complemented
10#26132C214yesmaximal, complemented
11#25062C212yesmaximal, complemented
12#24939C214yesmaximal, complemented
13#23757C211yesmaximal, characteristic, complemented
14C2×#2083C212yesmaximal, complemented
15D4×#107C212yesmaximal, complemented
16#22487C212yesmaximal, complemented
17#22469C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#484C2212yes—
22C2×C8×D4C2212yes—
23C22×#107C2214yes—
24#2857C2211yescharacteristic
25#2855C2212yes—
26#2582C2211yescharacteristic
27#2567C2211yescharacteristic
28#2487C2212yes—
29#2453—21no—
30#2449—22no—
31#2441—21no—
32#2438—22no—
33#2417C2212yes—
34Q8×C4⋊C4—21no—
35Q8×C4⋊C4C2212yes—
36D4×C4⋊C4—21no—
37D4×C4⋊C4—22no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2214yes—
40D4×C4⋊C4C2211yescharacteristic
41D4×C4⋊C4C2212yes—
42#2339—21no—
43#2317—22no—
44#2317—21no—
45C4×C22⋊Q8C2211yescharacteristic
46C2×#389C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51#2100C2214yes—
52#2100C2211yescharacteristic
53#2100C2212yes—
54#2095C2214yes—
55#2095C2211yescharacteristic
56#2083C2212yes—
57#2083C2214yes—
58#2083C2212yes—
59#2083C2212yes—
60#2083C2211yescharacteristic
61#2083C2214yes—
62#2083C2212yes—
63#2076C2214yes—
64#2076C2212yes—
65#2072C2212yes—
66#2072C2211yescharacteristic
67#2036C2212yes—
68#1990C2212yes—
69#1987C2212yes—
70#1984C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1928C2214yes—
74#1921C2214yes—
75#1921C2214yes—
76#1921C2214yes—
77#1892C2214yes—
78#1872C2214yes—
79#1869C2214yes—
80#1852C2214yes—
81#1852C2214yes—
82#1852C2214yes—
83#1845C2212yes—
84#1824C2212yes—
85#1824C2214yes—
86#1817C2212yes—
87#1808C2214yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91C4×#103C2212yes—
92C4×C22⋊C8C2211yescharacteristic
93C2×#340C2214yes—
94C2×#336C2212yes—
95C22×#41D411yescharacteristic
96C23×C8C2314yes—
97#554—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2311yescharacteristic
104#520—48no—
105C4×#41—28no—
106C2×C4×Q8—22no—
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—24no—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D412yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×#117—22no—
119C2×#117—21no—
120C2×#117D412yes—
121C2×#117D411yescharacteristic
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#484—44no—
134C8×D4—28no—
135C2×#107—48no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#399—22no—
152#398—22no—
153#398—22no—
154#396—22no—
155#395—22no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391C2312yes—
160#391C2311yescharacteristic
161#389—22no—
162#389—24no—
163#389—24no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2311yescharacteristic
175#387—22no—
176#387—22no—
177#387—21no—
178#386—22no—
179#386—22no—
180#385—22no—
181#385—21no—
182#385—24no—
183#385—24no—
184#385—24no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—22no—
189#384—21no—
190#384—21no—
191#384—22no—
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2314yes—
197#384C2311yescharacteristic
198#382—22no—
199#382—21no—
200#381—22no—
201#380—21no—
202#380—22no—
203#380—21no—
204#380—22no—
205#380C2312yes—
206#380C2314yes—
207#380C2312yes—
208#380C2311yescharacteristic
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—21no—
217C2×#95—22no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41Q1618yes—
237C22×Q8—21no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2411yescharacteristic
240C22×D4C2×D412yes—
241C22×D4C2×D411yescharacteristic
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—24no—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C22×C8—28no—
253C22×C8—28no—
254C22×C8—44no—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259#125—44no—
260C22⋊Q8—28no—
261C22⋊Q8—22no—
262C22⋊Q8—41no—
263C22⋊Q8—48no—
264C22⋊Q8—82no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—48no—
269C4×D4—24no—
270C4×D4—28no—
271C4×D4—28no—
272C4×D4—416no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—48no—
276C4×D4#4118yes—
277#117—24no—
278#117—28no—
279#117—48no—
280#117—44no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4—84no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—28no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—24no—
354#107—24no—
355#107—28no—
356#107—44no—
357#107—44no—
358#107—44no—
359C4⋊C8—24no—
360#103—24no—
361#103—42no—
362#103—44no—
363C22⋊C8—28no—
364C22⋊C8—24no—
365C22⋊C8—42no—
366C4×C8—24no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—24no—
374#95—21no—
375#95—21no—
376#95—24no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384C24C22×D411yescharacteristic
385C24C22×D412yes—
386#41—232no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—42no—
394C2×Q8—81no—
395C2×Q8—81no—
396C2×Q8C2×Q1614yes—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4#12314yes—
400C2×D4C2×Q1618yes—
401C2×D4C2×Q1614yes—
402C2×D4AGL(1,ℤ/8)14yes—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—82no—
435C22×C4—84no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—84no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4C2×Q1618yes—
442C22×C4C2×Q1614yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4118yes—
452C22×C4C2511yescharacteristic
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—416no—
462C2×C82−1+414yes—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—48no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—82no—
496C4⋊C4—84no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—84no—
502C4⋊C4—81no—
503C4⋊C4#12314yes—
504C4⋊C4H(5,𝔽2)12yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—81no—
513C22⋊C4—82no—
514C22⋊C4#12318yes—
515C42—24no—
516C42—22no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—48no—
522C42—42no—
523C42—44no—
524C42—42no—
525C42C2×#4114yes—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23C2×#12314yes—
537C23C22×Q1612yes—
538C23C22×Q1614yes—
539C23C2×AGL(1,ℤ/8)14yes—
540C23C23×D411yescharacteristic
541Q8—28no—
542Q8—42no—
543Q8—42no—
544Q8—162no—
545D4—216no—
546D4#484116yescomplemented
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—216no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—88no—
599C2×C4—168no—
600C2×C4—164no—
601C2×C4—164no—
602C2×C4C2×#12314yes—
603C2×C4C2×#12312yes—
604C2×C4C2×#12312yes—
605C2×C4C22×Q1612yes—
606C2×C4C22×Q1614yes—
607C2×C4C22×Q1612yes—
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4#57612yes—
610C2×C4C22×#4112yes—
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4C2×2−1+412yes—
613C2×C4#58511yescharacteristic
614C8—24no—
615C8—42no—
616C8—44no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22C2×#48418yes—
631C22C22×#12311yescharacteristic
632C22#348811yescharacteristic
633C22#349911yescharacteristic
634C22C23×Q1611yescharacteristic
635C22C22×AGL(1,ℤ/8)11yescharacteristic
636C22C2×#57611yescharacteristic
637C22#358611yescharacteristic
638C4—22no—
639C4—24no—
640C4—22no—
641C4—22no—
642C4—24no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—82no—
647C4—82no—
648C4—84no—
649C4—81no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4—81no—
654C4C2×#48414yes—
655C4#351512yes—
656C4#352012yes—
657C2—28no—
658C2—42no—
659C2#3599811yescharacteristic
660C2C22×#48411yescharacteristic
661C2#6052111yescharacteristic
662C2#6195211yescharacteristic
663C2#6197911yescharacteristic
664C2#6247211yescharacteristic
665C2#6250311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
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
3. #29939C2semidirect
4. #29909C2semidirect
5. #27950C2semidirect
6. #27858C2semidirect
7. #27648C2semidirect
8. #27614C2semidirect
9. #26282C2semidirect
10. #26132C2semidirect
11. #25062C2semidirect
12. #24939C2semidirect
13. #23757C2semidirect
14. C2×#2083C2semidirect
15. D4×#107C2semidirect
16. #22487C2semidirect
546. D4#484semidirect

Maximal subgroups

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

subgroupindex
#362752
#341832
#299392
#299092
#279502
#278582
#276482
#276142
#262822
#261322
#250622
#249392
#237572
C2×#20832
D4×#1072
#224872
#224692