Finite Group Databasebeta

← #6624514

Subgroups — #6624514

Order 512.

1628 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#33110C212yesmaximal
4#32906C211yesmaximal, characteristic, complemented
5#28315C212yesmaximal
6#28121C212yesmaximal, complemented
7#27910C211yesmaximal, characteristic, complemented
8#27645C211yesmaximal, characteristic
9#27506C214yesmaximal, complemented
10#27212C214yesmaximal, complemented
11#26093C212yesmaximal, complemented
12#24629C212yesmaximal, complemented
13#23102C212yesmaximal, complemented
14#22811C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21581C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#484C2212yes—
20C2×#484C2212yes—
21C2×#481C2212yes—
22C2×#462C2212yes—
23C2×#451C2212yes—
24C2×#448C2211yescharacteristic
25C2×C4×SD16C2212yes—
26#2857C2211yescharacteristic
27#2850C2212yes—
28#2596—22no—
29#2571C2211yescharacteristic
30#2547C2211yescharacteristic
31#2514C2212yes—
32#2510—22no—
33#2494—21no—
34#2487—21no—
35#2448C2212yes—
36#2365C2211yescharacteristic
37#2355—22no—
38#2355—21no—
39#2355—21no—
40#2355C2212yes—
41#2355C2212yes—
42#2355C2214yes—
43#2354C2211yescharacteristic
44#2352—21no—
45#2347—21no—
46C4×C22⋊Q8C2212yes—
47C4×#121C2211yescharacteristic
48C2×#391C2212yes—
49C2×#387—22no—
50C2×#384C2211yescharacteristic
51C2×#382—21no—
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2097C2214yes—
56#2095C2214yes—
57#2084C2212yes—
58#2071C2212yes—
59#2068C2214yes—
60#2056C2214yes—
61#2015C2214yes—
62#1983C2214yes—
63#1973C2214yes—
64#1971C2214yes—
65#1956C2212yes—
66#1949C2214yes—
67#1948C2212yes—
68#1939C2212yes—
69#1928C2214yes—
70#1927C2214yes—
71#1927C2214yes—
72#1925C2214yes—
73#1918C2214yes—
74#1917C2214yes—
75#1880C2212yes—
76#1878C2214yes—
77#1878C2214yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1865C2212yes—
81#1853C2211yescharacteristic
82#1845C2212yes—
83#1824C2212yes—
84#1803C2212yes—
85#1802C2212yes—
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1787C2211yescharacteristic
89#1776C2214yes—
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×Q16C2314yes—
97C22×SD16C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#124C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×#121C2311yescharacteristic
102C2×C4×Q8—28no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×D4—21no—
109C2×C4×D4C2311yescharacteristic
110C2×#117—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C42—21no—
119C22×C42—22no—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#484—28no—
123#484—44no—
124#481—44no—
125#462—28no—
126#451—44no—
127#448—24no—
128C4×SD16—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#401—22no—
150#400—22no—
151#400—22no—
152#399—22no—
153#397—22no—
154#397—22no—
155#391—22no—
156#391—22no—
157#391C2312yes—
158#391C2314yes—
159#391C2312yes—
160#391C2312yes—
161#391C2314yes—
162#390C2311yescharacteristic
163#390C2312yes—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389C2312yes—
169#389C2311yescharacteristic
170#388—22no—
171#388—22no—
172#388—22no—
173#388—21no—
174#388—22no—
175#388—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—24no—
180#387—24no—
181#387—24no—
182#387—24no—
183#386—22no—
184#386—22no—
185#386—21no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#383—22no—
193#382—21no—
194#382—22no—
195#382—22no—
196#382—24no—
197#382C2312yes—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380C2311yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—24no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C2×#95—22no—
224C2×#95—22no—
225C2×#95—22no—
226C2×#95D412yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8C2412yes—
242C22×Q8C2412yes—
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C2×Q16—28no—
255C2×Q16—44no—
256C2×Q16—44no—
257C2×SD16—24no—
258C2×SD16—24no—
259C2×SD16—42no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264#124—44no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267#121—22no—
268#121—22no—
269C4×Q8—28no—
270C4×Q8—28no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×Q8#4118yes—
276C4×D4—24no—
277C4×D4—44no—
278#117—44no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C42—28no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—44no—
335C2×C42—84no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344#107—28no—
345#107—44no—
346#106—24no—
347#106—44no—
348C4⋊C8—24no—
349C4⋊C8—44no—
350#103—24no—
351#103—24no—
352#103—28no—
353#103—28no—
354#103—42no—
355#103—44no—
356#103—44no—
357#102—24no—
358#102—44no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8#4114yes—
362#97—44no—
363C4×C8—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—21no—
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8C2×#4114yes—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4#13714yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4C2511yescharacteristic
457Q16—88no—
458SD16—48no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8C2×#4114yes—
468C4⋊C4—24no—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4C2×#4114yes—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—82no—
507C22⋊C4—81no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C42—28no—
511C42—28no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—24no—
517C42—42no—
518C42—42no—
519C42—44no—
520C42—42no—
521C42—84no—
522C42—84no—
523C42—82no—
524C42C2×#4114yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23#55012yes—
537C23C2×#13712yes—
538C23#57812yes—
539C23C23×D411yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—48no—
543Q8—42no—
544Q8—42no—
545Q8—44no—
546Q8—44no—
547D4—42no—
548D4—42no—
549D4—84no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—84no—
603C2×C4—88no—
604C2×C4—84no—
605C2×C4#52512yes—
606C2×C4#52512yes—
607C2×C4#55012yes—
608C2×C4C2×#13712yes—
609C2×C4C2×#13712yes—
610C2×C4#57512yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4112yes—
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4#58512yes—
615C8—24no—
616C8—42no—
617C8—44no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22#339814yes—
632C22C2×#52511yescharacteristic
633C22C2×#55011yescharacteristic
634C22#348511yescharacteristic
635C22C22×#13711yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57811yescharacteristic
638C22#358511yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—41no—
648C4—44no—
649C4—81no—
650C4—81no—
651C4—82no—
652C4—81no—
653C4—82no—
654C4—84no—
655C4#311714yes—
656C4#347512yes—
657C4#349412yes—
658C2—24no—
659C2—161no—
660C2#3651711yescharacteristic
661C2C2×#311711yescharacteristic
662C2#6058911yescharacteristic
663C2C2×#339811yescharacteristic
664C2#6140511yescharacteristic
665C2#6171911yescharacteristic
666C2#6173011yescharacteristic
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×#2355C2semidirect
4. #32906C2semidirect
6. #28121C2semidirect
7. #27910C2semidirect
9. #27506C2semidirect
10. #27212C2semidirect
11. #26093C2semidirect
12. #24629C2semidirect
13. #23102C2semidirect
14. #22811C2semidirect
15. #22629C2semidirect
16. #21581C2semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#23552
#331102
#329062
#283152
#281212
#279102
#276452
#275062
#272122
#260932
#246292
#231022
#228112
#226292
#215812
C4×#4482