Finite Group Databasebeta

← #6616999

Subgroups — #6616999

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#33095C212yesmaximal
4#32954C211yesmaximal, characteristic, complemented
5#27948C211yesmaximal, characteristic, complemented
6#27678C211yesmaximal, characteristic
7#27583C212yesmaximal, complemented
8#27490C214yesmaximal, complemented
9#27362C214yesmaximal, complemented
10#26088C212yesmaximal, complemented
11#24678C212yesmaximal, complemented
12#24503C212yesmaximal
13#23062C212yesmaximal, complemented
14#22805C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16C4×#448C211yesmaximal, characteristic
17#21359C212yesmaximal, complemented
18C2×#552—21no—
19C22×C4×Q8C2211yescharacteristic
20C2×#475C2212yes—
21C2×#448C2211yescharacteristic
22C2×C4×SD16C2212yes—
23C2×#431C2212yes—
24#2857C2211yescharacteristic
25#2854C2212yes—
26#2552C2211yescharacteristic
27#2542C2211yescharacteristic
28#2519—22no—
29#2512C2212yes—
30#2506C2212yes—
31#2498—22no—
32#2489—21no—
33#2417—21no—
34#2365C2211yescharacteristic
35#2362—21no—
36#2355—21no—
37#2355—21no—
38#2355—22no—
39#2355C2212yes—
40#2355C2214yes—
41#2355C2212yes—
42#2354C2211yescharacteristic
43C4×#121C2211yescharacteristic
44C2×#391C2212yes—
45C2×#387—22no—
46#2285C2212yes—
47C2×#384C2211yescharacteristic
48C2×#382—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2212yes—
53#2096C2214yes—
54#2094C2212yes—
55#2084C2214yes—
56#2072C2214yes—
57#2066C2214yes—
58#2055C2214yes—
59#1990C2212yes—
60#1968C2214yes—
61#1948C2212yes—
62#1948C2214yes—
63#1948C2214yes—
64#1939C2212yes—
65#1933C2214yes—
66#1927C2214yes—
67#1927C2214yes—
68#1926C2214yes—
69#1926C2214yes—
70#1919C2214yes—
71#1919C2214yes—
72#1916C2214yes—
73#1880C2212yes—
74#1874C2212yes—
75#1874C2211yescharacteristic
76#1865C2214yes—
77#1865C2212yes—
78#1853C2212yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1844C2212yes—
84#1817C2212yes—
85#1803C2212yes—
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1787C2212yes—
89#1787C2211yescharacteristic
90C4×#106C2214yes—
91C4×#106C2214yes—
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×SD16C2312yes—
97#552—48no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#126—22no—
100C2×#125—21no—
101C2×#125—22no—
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×#121C2311yescharacteristic
107C2×C4×Q8—28no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—21no—
114C2×C4×D4C2311yescharacteristic
115C2×#117—22no—
116C2×#117—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C42—21no—
125C22×C42—22no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#475—28no—
129#448—24no—
130C4×SD16—28no—
131#431—28no—
132C2×#106C2314yes—
133C2×#106C2312yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2311yescharacteristic
149C2×C4×C8C2312yes—
150C2×C4×C8C2314yes—
151#400—22no—
152#399—22no—
153#398—22no—
154#398—22no—
155#397—22no—
156#396—22no—
157#391—21no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391C2314yes—
162#391C2314yes—
163#390C2312yes—
164#390C2312yes—
165#390C2311yescharacteristic
166#389—21no—
167#389—21no—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2311yescharacteristic
172#388—22no—
173#388—22no—
174#388—21no—
175#388—22no—
176#388—22no—
177#387—24no—
178#387—24no—
179#387—24no—
180#387—24no—
181#387—22no—
182#387—22no—
183#386—22no—
184#386—21no—
185#384—21no—
186#384—21no—
187#384C2312yes—
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382—22no—
194#382—24no—
195#382—22no—
196#380—21no—
197#380—22no—
198#380—21no—
199#380C2312yes—
200#380C2311yescharacteristic
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—24no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—24no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—42no—
240C22×Q8C2412yes—
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—42no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259#126—48no—
260#125—48no—
261#125—44no—
262#123—48no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266#121—22no—
267#121—22no—
268C4×Q8—28no—
269C4×Q8—28no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—44no—
274C4×Q8#4118yes—
275C4×D4—24no—
276C4×D4—44no—
277#117—48no—
278#117—44no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
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—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4SD1618yescomplemented
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C42—28no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—84no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348#106—24no—
349#106—28no—
350#106—44no—
351#106—44no—
352#106—44no—
353C4⋊C8—28no—
354C4⋊C8—24no—
355#103—24no—
356#103—28no—
357#103—24no—
358#103—44no—
359#103—42no—
360#102—24no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—22no—
364C22⋊C8#4114yes—
365C4×C8—28no—
366C4×C8—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—21no—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8C2×#4114yes—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4C2×SD1618yes—
442C22×C4C2×SD1614yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4112yes—
453C22×C42−1+412yes—
454C22×C4C2511yescharacteristic
455SD16—48no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8C2×#4114yes—
465C4⋊C4—24no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4C2×SD1614yes—
497C4⋊C4C2×SD1614yes—
498C4⋊C4C2×SD1618yes—
499C4⋊C4C2×#4114yes—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—81no—
509C22⋊C4—82no—
510C42—28no—
511C42—28no—
512C42—22no—
513C42—22no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—44no—
522C42—44no—
523C42—42no—
524C42—84no—
525C42—84no—
526C42C2×#4114yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23#54712yes—
539C23C22×SD1612yes—
540C23#57712yes—
541C23C23×D411yescharacteristic
542Q8—24no—
543Q8—24no—
544Q8—48no—
545Q8—42no—
546Q8—42no—
547Q8—84no—
548D4—42no—
549D4—42no—
550D4—84no—
551C2×C4—21no—
552C2×C4—216no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—24no—
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—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—84no—
602C2×C4—88no—
603C2×C4—84no—
604C2×C4#52512yes—
605C2×C4#52512yes—
606C2×C4#54712yes—
607C2×C4C22×SD1614yes—
608C2×C4C22×SD1614yes—
609C2×C4C22×SD1612yes—
610C2×C4C22×SD1612yes—
611C2×C4#57512yes—
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4112yes—
614C2×C4C2×2−1+411yescharacteristic
615C2×C4#58512yes—
616C8—24no—
617C8—44no—
618C8—42no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22#333514yes—
633C22C2×#52511yescharacteristic
634C22C2×#54711yescharacteristic
635C22#348711yescharacteristic
636C22C23×SD1611yescharacteristic
637C22C2×#57511yescharacteristic
638C22C2×#57711yescharacteristic
639C22#358611yescharacteristic
640C4—24no—
641C4—24no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—41no—
649C4—44no—
650C4—81no—
651C4—81no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4—84no—
656C4#310314yes—
657C4#347512yes—
658C4#349512yes—
659C2—24no—
660C2—161no—
661C2#3642711yescharacteristic
662C2C2×#310311yescharacteristic
663C2#6058511yescharacteristic
664C2C2×#333511yescharacteristic
665C2#6125711yescharacteristic
666C2#6188311yescharacteristic
667C2#6190411yescharacteristic
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. #32954C2semidirect
5. #27948C2semidirect
7. #27583C2semidirect
8. #27490C2semidirect
9. #27362C2semidirect
10. #26088C2semidirect
11. #24678C2semidirect
13. #23062C2semidirect
14. #22805C2semidirect
15. #22599C2semidirect
17. #21359C2semidirect
304. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#361862
C2×#23552
#330952
#329542
#279482
#276782
#275832
#274902
#273622
#260882
#246782
#245032
#230622
#228052
#225992
C4×#4482
#213592