Finite Group Databasebeta

← #6561812

Subgroups — #6561812

Order 512.

1653 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36274C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#29922C211yesmaximal, characteristic, complemented
4#29911C212yesmaximal, complemented
5#27972C211yesmaximal, characteristic, complemented
6#27861C212yesmaximal, complemented
7#27648C211yesmaximal, characteristic, complemented
8#27583C212yesmaximal, complemented
9#26319C212yesmaximal, complemented
10#26217C214yesmaximal, complemented
11#24997C212yesmaximal, complemented
12#24880C214yesmaximal, complemented
13#23756C211yesmaximal, characteristic, complemented
14#23629C212yesmaximal, complemented
15#22715C212yesmaximal, complemented
16#22473C211yesmaximal, characteristic
17#22430C212yesmaximal, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#484C2212yes—
21C2×#436C2212yes—
22C2×#419C2214yes—
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2582C2211yescharacteristic
26#2568C2211yescharacteristic
27#2451—21no—
28#2450—22no—
29#2439—21no—
30#2438—22no—
31#2417C2212yes—
32#2409C2212yes—
33#2355—21no—
34#2355C2212yes—
35#2354—21no—
36#2354C2212yes—
37#2352—22no—
38#2352C2214yes—
39#2349—21no—
40D4×C4⋊C4C2211yescharacteristic
41#2339—21no—
42#2314—22no—
43#2311—21no—
44C4×C22⋊Q8C2211yescharacteristic
45C2×#391C2212yes—
46#2285C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51#2101C2214yes—
52#2099C2211yescharacteristic
53#2097C2214yes—
54#2096C2211yescharacteristic
55#2096C2212yes—
56#2084C2211yescharacteristic
57#2083C2212yes—
58#2077C2212yes—
59#2074C2214yes—
60#2074C2214yes—
61#2072C2212yes—
62#2072C2212yes—
63#2072C2212yes—
64#2072C2211yescharacteristic
65#2072C2212yes—
66#2035C2212yes—
67#2035C2214yes—
68#1984C2212yes—
69#1981C2212yes—
70#1956C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1928C2214yes—
74#1927C2214yes—
75#1927C2214yes—
76#1919C2214yes—
77#1894C2214yes—
78#1873C2214yes—
79#1872C2214yes—
80#1870C2214yes—
81#1854C2214yes—
82#1853C2214yes—
83#1834C2212yes—
84#1824C2212yes—
85#1817C2212yes—
86#1809C2214yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1787C2214yes—
90C4×#106C2212yes—
91C4×#103C2212yes—
92C4×C22⋊C8C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8C2312yes—
97C22×M16C2314yes—
98D4×Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2311yescharacteristic
104C2×C22⋊Q8C2312yes—
105C4×#41—28no—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D412yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2311yescharacteristic
115C2×#117—21no—
116C2×#117—22no—
117C2×#117D412yes—
118C2×#117D411yescharacteristic
119C2×#117C2314yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#484—44no—
127#436—28no—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×#106C2312yes—
131C2×#106C2312yes—
132C2×C4⋊C8C2312yes—
133#419—48no—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2312yes—
145#400—22no—
146#398—22no—
147#398—22no—
148#396—22no—
149#395—22no—
150#393—22no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#391C2311yescharacteristic
161#390—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2311yescharacteristic
168#388—24no—
169#388—24no—
170#387—24no—
171#387—24no—
172#387—22no—
173#387—22no—
174#387—21no—
175#386—24no—
176#386—22no—
177#386—22no—
178#385—22no—
179#385—21no—
180#385—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384C2312yes—
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382—21no—
190#382—21no—
191#382—22no—
192#382C2314yes—
193#381—22no—
194#380—22no—
195#380—22no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—22no—
200#380—22no—
201#380C2312yes—
202#380C2314yes—
203#380C2311yescharacteristic
204#380C2312yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—24no—
217C2×#95—22no—
218C2×#95—22no—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—42no—
236C22×Q8—22no—
237C22×Q8—24no—
238C22×Q8—28no—
239C22×Q8—42no—
240C22×Q8—44no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C22×D4C2×D412yes—
245C22×D4C2×D411yescharacteristic
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C2×M16—28no—
256C2×M16—44no—
257C2×M16—44no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C22⋊Q8—22no—
264C22⋊Q8—28no—
265C22⋊Q8—41no—
266C22⋊Q8—48no—
267C22⋊Q8—44no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×D4—24no—
272C4×D4—28no—
273C4×D4—28no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4#4118yes—
277#117—24no—
278#117—28no—
279#117—44no—
280#117—44no—
281#117—44no—
282#117—84no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—28no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—44no—
345#106—42no—
346#106—44no—
347#106—42no—
348C4⋊C8—24no—
349#103—24no—
350#103—28no—
351#103—44no—
352#103—42no—
353#103—44no—
354#103—44no—
355#103—44no—
356C22⋊C8—28no—
357C22⋊C8—42no—
358C22⋊C8—42no—
359C4×C8—24no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—21no—
371#95—21no—
372#95—24no—
373#95—22no—
374#95—24no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381C24C22×D412yes—
382C24C22×D411yescharacteristic
383#41—416no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—44no—
395C2×Q8—416no—
396C2×Q8—44no—
397C2×Q8—82no—
398C2×Q8—88no—
399C2×Q8H(5,𝔽2)12yes—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—42no—
405C2×D4#12314yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4#13714yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4118yes—
452C22×C4C2511yescharacteristic
453M16—88no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4#12314yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—81no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—82no—
508C22⋊C4#12318yes—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—22no—
514C42—24no—
515C42—42no—
516C42—44no—
517C42—42no—
518C42—84no—
519C42—84no—
520C42C2×#4114yes—
521C23—21no—
522C23—22no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—21no—
531C23—22no—
532C23—21no—
533C23C2×#12314yes—
534C23C2×#13712yes—
535C23C23×D411yescharacteristic
536Q8—24no—
537Q8—44no—
538Q8—42no—
539Q8—44no—
540Q8—44no—
541Q8—88no—
542D4—28no—
543D4—48no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—88no—
597C2×C4—164no—
598C2×C4—164no—
599C2×C4C2×#12314yes—
600C2×C4C2×#12312yes—
601C2×C4C2×#12312yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4#57512yes—
605C2×C4#57712yes—
606C2×C4C22×#4112yes—
607C2×C4C2×H(5,𝔽2)11yescharacteristic
608C2×C4#58512yes—
609C2×C4#58511yescharacteristic
610C8—44no—
611C8—42no—
612C8—42no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—41no—
624C22—42no—
625C22—41no—
626C22—41no—
627C22C22×#12311yescharacteristic
628C22#348811yescharacteristic
629C22#350011yescharacteristic
630C22C22×#13711yescharacteristic
631C22C2×#57511yescharacteristic
632C22C2×#57711yescharacteristic
633C22#358611yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—24no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—42no—
642C4—82no—
643C4—81no—
644C4—84no—
645C4—82no—
646C4—81no—
647C4—82no—
648C4—81no—
649C4—82no—
650C4#309514yes—
651C4#351312yes—
652C4#352112yes—
653C2—42no—
654C2—44no—
655C2#3597611yescharacteristic
656C2C2×#309511yescharacteristic
657C2#6052211yescharacteristic
658C2#6196611yescharacteristic
659C2#6199711yescharacteristic
660C2#6254011yescharacteristic
661C2#6259311yescharacteristic
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. #29922C2semidirect
4. #29911C2semidirect
5. #27972C2semidirect
6. #27861C2semidirect
7. #27648C2semidirect
8. #27583C2semidirect
9. #26319C2semidirect
10. #26217C2semidirect
11. #24997C2semidirect
12. #24880C2semidirect
13. #23756C2semidirect
14. #23629C2semidirect
15. #22715C2semidirect
17. #22430C2semidirect

Maximal subgroups

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

subgroupindex
#362742
#341882
#299222
#299112
#279722
#278612
#276482
#275832
#263192
#262172
#249972
#248802
#237562
#236292
#227152
#224732
#224302