Finite Group Databasebeta

← #6477851

Subgroups — #6477851

Order 512.

1740 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic
2C2×#2362C211yesmaximal, characteristic
3#32985C212yesmaximal
4#32954C211yesmaximal, characteristic, complemented
5#32906C211yesmaximal, characteristic, complemented
6#29966C211yesmaximal, characteristic, complemented
7#29756C211yesmaximal, characteristic, complemented
8#29599C212yesmaximal
9#27105C214yesmaximal, complemented
10#26937C214yesmaximal, complemented
11#26795C214yesmaximal, complemented
12#26667C214yesmaximal, complemented
13#22044C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic
15C4×#483C212yesmaximal, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#485C2214yes—
20C2×#483C2212yes—
21C2×#475C2212yescomplemented
22C2×#462C2212yescomplemented
23C2×#451C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2848C2212yes—
27#2646—21no—
28#2626—21no—
29#2589—22no—
30#2587—22no—
31#2497C2212yes—
32#2492C2212yes—
33#2429C2212yes—
34#2423C2212yes—
35#2362—22no—
36#2362—22no—
37#2362C2214yes—
38#2362C2214yes—
39#2329—21no—
40#2312—21no—
41C4×C22⋊Q8C2212yes—
42C2×#391C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#388—21no—
46C2×#387—21no—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2099C2212yescomplemented
51#2086C2214yescomplemented
52#2084C2212yescomplemented
53#2066C2214yescomplemented
54#2057C2212yescomplemented
55#2043C2212yescomplemented
56#2035C2214yes—
57#2035C2214yes—
58#2001C2214yes—
59#2001C2214yes—
60#1969C2214yescomplemented
61#1933C2214yescomplemented
62#1928C2214yescomplemented
63#1927C2214yescomplemented
64#1923C2214yes—
65#1923C2214yes—
66#1920C2214yescomplemented
67#1878C2214yescomplemented
68#1878C2214yescomplemented
69#1873C2214yes—
70#1872C2212yes—
71#1872C2212yes—
72#1865C2214yescomplemented
73#1854C2214yes—
74#1853C2212yes—
75#1852C2212yes—
76#1825C2214yes—
77#1823C2212yes—
78#1803C2212yescomplemented
79#1803C2212yescomplemented
80#1797C2211yescharacteristic
81#1797C2211yescharacteristic
82#1787C2212yes—
83#1779C2212yes—
84C4×#106C2214yescomplemented
85C4×#103C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×Q16C2314yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×#117—22no—
107C2×#117—44no—
108C2×#117C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4D411yescharacteristic
118C22×C22⋊C4D411yescharacteristic
119C22×C42—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#485—48no—
125#483—28no—
126#475—44no—
127#462—44no—
128#451—28no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C4×C8C2314yes—
145#400—22no—
146#400—22no—
147#399—22no—
148#398—22no—
149#398—22no—
150#396—22no—
151#396—22no—
152#393—22no—
153#391—21no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2314yes—
158#391C2312yes—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—24no—
174#388—24no—
175#388—24no—
176#388—24no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—24no—
181#387—24no—
182#385—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#380—21no—
189#380—21no—
190#380—21no—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194#379—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—24no—
198C4×C4⋊C4—24no—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4C2312yes—
205C4×C22⋊C4C2312yes—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8#4114yes—
228C22×Q8C2412yes—
229C22×Q8C2412yes—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C2×Q16—28no—
247C2×Q16—44no—
248C2×Q16—44no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—42no—
252C4⋊Q8—42no—
253C22⋊Q8—216no—
254C22⋊Q8—28no—
255C22⋊Q8—28no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×Q8—42no—
264C4×Q8—42no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×Q8—44no—
268#117—28no—
269#117—44no—
270#117—48no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4#4114yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2412yes—
338#107—44no—
339#106—28no—
340#106—28no—
341C4⋊C8—42no—
342C4⋊C8—42no—
343#103—28no—
344#103—28no—
345#103—28no—
346#103—44no—
347#103—44no—
348#103—44no—
349C22⋊C8—24no—
350C22⋊C8—24no—
351C22⋊C8—44no—
352C22⋊C8—44no—
353C4×C8—44no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—24no—
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#95—22no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—41no—
377C24—41no—
378C24—41no—
379C24—41no—
380C24C22×D412yes—
381C24C22×D411yescharacteristic
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—48no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—48no—
396C2×Q8—42no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×Q8C2×#4114yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4#13514yes—
445C22×C4#13714yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2511yescharacteristic
457Q16—88no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C4⋊C4—24no—
465C4⋊C4—216no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4C2×#4114yes—
493C4⋊C4H(5,𝔽2)14yes—
494C22⋊C4—216no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—42no—
510C42—41no—
511C42—41no—
512C42—48no—
513C42—44no—
514C42—44no—
515C42—48no—
516C42—82no—
517C42—82no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23#52512yes—
537C23C2×#13512yes—
538C23C2×#13712yes—
539C23C23×D411yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—48no—
543Q8—42no—
544Q8—42no—
545Q8—44no—
546Q8—44no—
547Q8—162no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
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—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—84no—
606C2×C4#52512yes—
607C2×C4C2×#13512yes—
608C2×C4C2×#13512yes—
609C2×C4C2×#13714yes—
610C2×C4C22×#4111yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4C2×H(5,𝔽2)12yes—
613C2×C4#58512yes—
614C8—44no—
615C8—44no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—88no—
630C22#310714yes—
631C22C2×#52511yescharacteristic
632C22#348411yescharacteristic
633C22#348511yescharacteristic
634C22C22×#13511yescharacteristic
635C22C22×#13711yescharacteristic
636C22#358611yescharacteristic
637C22#358611yescharacteristic
638C4—24no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—44no—
645C4—44no—
646C4—41no—
647C4—41no—
648C4—44no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4#346012yes—
655C4#350512yes—
656C2—24no—
657C2—82no—
658C2#3509111yescharacteristic
659C2C2×#310711yescharacteristic
660C2#6056611yescharacteristic
661C2#6156411yescharacteristic
662C2#6158611yescharacteristic
663C2#6170111yescharacteristic
664C2#6170511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #32954C2semidirect
5. #32906C2semidirect
6. #29966C2semidirect
7. #29756C2semidirect
9. #27105C2semidirect
10. #26937C2semidirect
11. #26795C2semidirect
12. #26667C2semidirect
13. #22044C2semidirect
15. C4×#483C2semidirect
21. C2×#475C22semidirect
22. C2×#462C22semidirect
50. #2099C22semidirect
51. #2086C22semidirect
52. #2084C22semidirect
53. #2066C22semidirect
54. #2057C22semidirect
55. #2043C22semidirect
60. #1969C22semidirect
61. #1933C22semidirect
62. #1928C22semidirect
63. #1927C22semidirect
66. #1920C22semidirect
67. #1878C22semidirect
68. #1878C22semidirect
72. #1865C22semidirect
78. #1803C22semidirect
79. #1803C22semidirect
84. C4×#106C22semidirect
85. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#23622
#329852
#329542
#329062
#299662
#297562
#295992
#271052
#269372
#267952
#266672
#220442
#219022
C4×#4832