Finite Group Databasebeta

← #6484525

Subgroups — #6484525

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic, complemented
2C2×#2367C211yesmaximal, characteristic
3#32960C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic
5#32953C211yesmaximal, characteristic
6#29956C212yesmaximal
7#29953C211yesmaximal, characteristic, complemented
8#29951C211yesmaximal, characteristic, complemented
9#26537C214yesmaximal, complemented
10#26535C214yesmaximal, complemented
11#26471C214yesmaximal, complemented
12#26469C214yesmaximal, complemented
13#21951C211yesmaximal, characteristic, complemented
14#21806C212yesmaximal
15#21361C212yesmaximal, complemented
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#479C2214yescomplemented
20C2×#476C2212yes—
21C2×#474C2212yes—
22C2×#431C2212yes—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2851C2212yes—
26#2627—21no—
27#2627—21no—
28#2606—22no—
29#2605—22no—
30#2570C2212yescomplemented
31#2570C2212yescomplemented
32#2550C2212yescomplemented
33#2550C2212yescomplemented
34#2367—22no—
35#2367—22no—
36#2367C2214yes—
37#2367C2214yes—
38#2329—21no—
39#2329—21no—
40C2×#391C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#388—21no—
43C2×#388—21no—
44C2×#385—21no—
45#2283C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49#2096C2212yescomplemented
50#2095C2212yescomplemented
51#2092C2214yes—
52#2068C2214yescomplemented
53#2067C2214yes—
54#2066C2214yescomplemented
55#2065C2214yes—
56#2025C2214yes—
57#2024C2212yescomplemented
58#2024C2212yescomplemented
59#1983C2214yes—
60#1983C2214yes—
61#1958C2214yes—
62#1958C2214yes—
63#1949C2214yescomplemented
64#1949C2214yescomplemented
65#1939C2214yescomplemented
66#1939C2214yescomplemented
67#1865C2214yescomplemented
68#1864C2214yes—
69#1863C2214yescomplemented
70#1862C2214yes—
71#1853C2212yes—
72#1853C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1805C2214yescomplemented
76#1802C2212yes—
77#1802C2212yes—
78#1797C2211yescharacteristic
79#1797C2212yes—
80#1797C2211yescharacteristic
81#1787C2212yes—
82#1787C2212yes—
83#1786C2212yes—
84#1786C2212yes—
85C4×#107C2214yescomplemented
86C4×#106C2214yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125C2312yes—
93C2×#124—22no—
94C2×#124—24no—
95C2×#124Q818yescomplemented
96C2×#124C2314yes—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C4×Q8—22no—
100C2×C4×Q8—44no—
101C2×C4×Q8C2314yes—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—44no—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#479—48no—
123#476—44no—
124#474—44no—
125#431—28no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2314yes—
145#400—22no—
146#400—22no—
147#398—22no—
148#398—22no—
149#398—22no—
150#398—22no—
151#395—22no—
152#395—22no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—22no—
158#391—22no—
159#391C2312yes—
160#391C2312yes—
161#390—22no—
162#390C2314yes—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#388—24no—
172#388—24no—
173#388—24no—
174#388—24no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#385—22no—
182#385—22no—
183#385—24no—
184#385—24no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#383—22no—
188#382C2312yes—
189#382C2312yes—
190#382C2312yes—
191#381—22no—
192#380—21no—
193#380—21no—
194#380C2312yes—
195#380C2312yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—24no—
206C4×C22⋊C4—24no—
207#376—22no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2312yes—
219#336C2312yes—
220C25—21no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8C2×Q814yes—
228C22×Q8C2412yes—
229C22×D4—21no—
230C22×D4—22no—
231C22×D4—42no—
232C22×D4—42no—
233C22×D4C2×Q814yes—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C22×C8C2414yes—
251C22×C8C2414yes—
252C4⋊Q8—42no—
253C4⋊Q8—42no—
254#125—44no—
255#124—216no—
256#124—28no—
257#124C2×Q8116yes—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C4×Q8—28no—
261C4×Q8—44no—
262C4×Q8—48no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—42no—
266C4×D4—42no—
267C4×D4—44no—
268C4×D4—44no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—82no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—44no—
319C2×C42—82no—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×Q814yes—
325C2×C42#4114yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2412yes—
329C2×C42C2412yes—
330#107—28no—
331#107—44no—
332#106—28no—
333#106—44no—
334C4⋊C8—28no—
335C4⋊C8—42no—
336C4⋊C8—42no—
337C4⋊C8—44no—
338#103—44no—
339#103—44no—
340#103—44no—
341#103—44no—
342#102—44no—
343#102—44no—
344#102—44no—
345#102—44no—
346C22⋊C8—24no—
347C22⋊C8—24no—
348C4×C8—28no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24—24no—
373C24—22no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24C22×D411yescharacteristic
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—48no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—48no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q8—81no—
394C2×Q8—81no—
395C2×Q8C22×Q814yes—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—48no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—48no—
406C2×D4C22×Q814yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—816no—
447C22×C4C22⋊Q814yes—
448C22×C4C22⋊Q814yes—
449C22×C4#13514yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×Q812yes—
460C22×C4C2×#4112yes—
461C22×C4C2511yescharacteristic
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—81no—
486C4⋊C4—82no—
487C4⋊C4—81no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4H(5,𝔽2)14yes—
494C22⋊C4—216no—
495C22⋊C4—216no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—84no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—41no—
506C42—41no—
507C42—42no—
508C42—48no—
509C42—82no—
510C42—82no—
511C42—84no—
512C42C22⋊Q818yes—
513C42C22×Q814yes—
514C42C2×#4114yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—24no—
520C23—48no—
521C23—48no—
522C23—42no—
523C23—42no—
524C23—48no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23C2×C22⋊Q812yes—
534C23C2×#13512yes—
535C23#57512yes—
536C23C23×D411yescharacteristic
537Q8—28no—
538Q8—48no—
539Q8—44no—
540Q8—162no—
541Q8—162no—
542D4—24no—
543D4—24no—
544D4—48no—
545D4—42no—
546D4—42no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—88no—
599C2×C4—88no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—84no—
603C2×C4C2×C22⋊Q812yes—
604C2×C4C2×C22⋊Q814yes—
605C2×C4C2×C22⋊Q814yes—
606C2×C4C2×#13512yes—
607C2×C4C2×#13512yes—
608C2×C4C23×Q811yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)12yes—
611C2×C4#58512yes—
612C8—44no—
613C8—44no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—816no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22#307614yes—
629C22C22×C22⋊Q811yescharacteristic
630C22#348411yescharacteristic
631C22#348711yescharacteristic
632C22C22×#13511yescharacteristic
633C22C2×#57511yescharacteristic
634C22#358511yescharacteristic
635C22#358611yescharacteristic
636C4—24no—
637C4—24no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—44no—
642C4—42no—
643C4—44no—
644C4—42no—
645C4—41no—
646C4—41no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4#346012yes—
653C4#350112yes—
654C2—24no—
655C2—84no—
656C2#3517411yescharacteristic
657C2C2×#307611yescharacteristic
658C2#6049211yescharacteristic
659C2#6156111yescharacteristic
660C2#6158311yescharacteristic
661C2#6188211yescharacteristic
662C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
1. #35104C2semidirect
3. #32960C2semidirect
7. #29953C2semidirect
8. #29951C2semidirect
9. #26537C2semidirect
10. #26535C2semidirect
11. #26471C2semidirect
12. #26469C2semidirect
13. #21951C2semidirect
15. #21361C2semidirect
19. C2×#479C22semidirect
30. #2570C22semidirect
31. #2570C22semidirect
32. #2550C22semidirect
33. #2550C22semidirect
49. #2096C22semidirect
50. #2095C22semidirect
52. #2068C22semidirect
54. #2066C22semidirect
57. #2024C22semidirect
58. #2024C22semidirect
63. #1949C22semidirect
64. #1949C22semidirect
65. #1939C22semidirect
66. #1939C22semidirect
67. #1865C22semidirect
69. #1863C22semidirect
75. #1805C22semidirect
85. C4×#107C22semidirect
86. C4×#106C22semidirect
95. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#23672
#329602
#329562
#329532
#299562
#299532
#299512
#265372
#265352
#264712
#264692
#219512
#218062
#213612