Finite Group Databasebeta

← #6617038

Subgroups — #6617038

Order 512.

1817 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36193C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#33100C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic
5#27950C211yesmaximal, characteristic, complemented
6#27681C211yesmaximal, characteristic, complemented
7#27582C212yesmaximal
8#27509C214yesmaximal, complemented
9#27381C214yesmaximal, complemented
10#26086C212yesmaximal, complemented
11#24680C212yesmaximal, complemented
12#24507C212yesmaximal
13#23067C212yesmaximal, complemented
14#22811C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#450C211yesmaximal, characteristic
17#21365C212yesmaximal, complemented
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#476C2212yes—
21C2×#450C2211yescharacteristic
22C2×C4×SD16C2212yes—
23C2×#432C2212yes—
24#2857C2211yescharacteristic
25#2855C2212yes—
26#2567C2211yescharacteristic, complemented
27#2552C2211yescharacteristic, complemented
28#2520—22no—
29#2511C2212yescomplemented
30#2508—21no—
31#2507C2212yescomplemented
32#2442—21no—
33#2438—22no—
34#2370—21no—
35#2369C2211yescharacteristic, complemented
36#2355C2211yescharacteristic, complemented
37#2354—21no—
38#2354—21no—
39#2354—22no—
40#2354C2212yes—
41#2354C2212yes—
42#2354C2214yes—
43#2344—21no—
44C4×C22⋊Q8C2211yescharacteristic
45C2×#390C2212yes—
46C2×#388—22no—
47#2285C2212yes—
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2212yescomplemented
53#2095C2212yescomplemented
54#2093C2214yes—
55#2083C2214yescomplemented
56#2071C2214yes—
57#2067C2214yes—
58#2055C2214yes—
59#1990C2212yescomplemented
60#1973C2214yes—
61#1948C2214yescomplemented
62#1947C2212yescomplemented
63#1947C2214yescomplemented
64#1939C2212yescomplemented
65#1936C2214yes—
66#1928C2214yescomplemented
67#1927C2214yescomplemented
68#1925C2214yes—
69#1925C2214yes—
70#1919C2214yescomplemented
71#1918C2214yescomplemented
72#1917C2214yescomplemented
73#1877C2212yes—
74#1872C2212yes—
75#1872C2211yescharacteristic
76#1864C2214yes—
77#1864C2212yes—
78#1853C2212yes—
79#1853C2211yescharacteristic
80#1852C2212yes—
81#1852C2212yes—
82#1852C2212yes—
83#1845C2212yescomplemented
84#1817C2212yes—
85#1803C2212yescomplemented
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1786C2212yes—
89#1786C2211yescharacteristic
90#1776C2214yescomplemented
91#1776C2214yescomplemented
92#1770C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×SD16C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C2311yescharacteristic
102C2×C4×Q8C2311yescharacteristic
103C2×C4×D4—28no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—22no—
109C2×#117—22no—
110C2×#117—24no—
111C2×#117—22no—
112C2×#117—48no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—22no—
121C22×C22⋊C4—22no—
122C22×C42—22no—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#476—28no—
126#450—24no—
127C4×SD16—28no—
128#432—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2311yescharacteristic
145C2×C22⋊C8C2311yescharacteristic
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#400—22no—
150#398—22no—
151#398—22no—
152#398—22no—
153#395—22no—
154#391—21no—
155#391—22no—
156#391C2312yes—
157#391C2312yes—
158#391C2311yescharacteristic
159#390—22no—
160#390—22no—
161#390C2314yes—
162#390C2314yes—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2311yescharacteristic
173#388—24no—
174#388—24no—
175#388—24no—
176#388—24no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—21no—
182#387—22no—
183#387—22no—
184#385—22no—
185#385—21no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#382—21no—
194#382—22no—
195#381—22no—
196#380—21no—
197#380—22no—
198#380—22no—
199#380—24no—
200#380—22no—
201#380C2312yes—
202#380C2311yescharacteristic
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
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—
234C25—21no—
235C22×Q8—21no—
236C22×Q8C2411yescharacteristic
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—42no—
242C22×D4C2412yes—
243C23×C4—28no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—42no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262#125—44no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—82no—
266C4×Q8—24no—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4#4118yes—
273#117—48no—
274#117—416no—
275#117—48no—
276#117—84no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4—84no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—48no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345#107—44no—
346#107—44no—
347#107—44no—
348#106—24no—
349#106—28no—
350C4⋊C8—28no—
351C4⋊C8—24no—
352#103—24no—
353#103—44no—
354#102—24no—
355#102—28no—
356#102—24no—
357#102—44no—
358#102—42no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8#4114yes—
362#97—28no—
363C4×C8—24no—
364#95—24no—
365#95—24no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—24no—
386C24—22no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24C22×D411yescharacteristic
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—42no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4C2×#4114yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—41no—
436C22×C4—44no—
437C22×C4—432no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4#13714yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4112yes—
457C22×C42−1+412yes—
458C22×C4C2511yescharacteristic
459SD16—48no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8C2×#4114yes—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—82no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4AGL(1,ℤ/8)14yes—
503C4⋊C4C2×#4114yes—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—48no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—48no—
523C42—42no—
524C42—48no—
525C42—42no—
526C42—82no—
527C42—82no—
528C42C2×#4114yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—24no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23#54712yes—
548C23C2×#13712yes—
549C23#57612yes—
550C23C23×D411yescharacteristic
551Q8—42no—
552Q8—42no—
553Q8—162no—
554D4—24no—
555D4—24no—
556D4—48no—
557D4—42no—
558D4—42no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—416no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—416no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—88no—
613C2×C4#52512yes—
614C2×C4#52512yes—
615C2×C4#54712yes—
616C2×C4C2×AGL(1,ℤ/8)14yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4112yes—
622C2×C4C2×2−1+411yescharacteristic
623C2×C4#58512yes—
624C8—24no—
625C8—44no—
626C8—42no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—816no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22#335714yes—
642C22C2×#52511yescharacteristic
643C22C2×#54711yescharacteristic
644C22#348711yescharacteristic
645C22C22×AGL(1,ℤ/8)11yescharacteristic
646C22C22×#13711yescharacteristic
647C22C2×#57611yescharacteristic
648C22#358611yescharacteristic
649C4—24no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—41no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—82no—
662C4—81no—
663C4—82no—
664C4—84no—
665C4#311014yes—
666C4#347512yes—
667C4#349512yes—
668C2—24no—
669C2—84no—
670C2#3642711yescharacteristic
671C2C2×#311011yescharacteristic
672C2#6058411yescharacteristic
673C2C2×#335711yescharacteristic
674C2#6125211yescharacteristic
675C2#6188911yescharacteristic
676C2#6190711yescharacteristic
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. #36193C2semidirect
3. #33100C2semidirect
5. #27950C2semidirect
6. #27681C2semidirect
8. #27509C2semidirect
9. #27381C2semidirect
10. #26086C2semidirect
11. #24680C2semidirect
13. #23067C2semidirect
14. #22811C2semidirect
15. #22603C2semidirect
17. #21365C2semidirect
26. #2567C22semidirect
27. #2552C22semidirect
29. #2511C22semidirect
31. #2507C22semidirect
35. #2369C22semidirect
36. #2355C22semidirect
52. #2100C22semidirect
53. #2095C22semidirect
55. #2083C22semidirect
59. #1990C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1947C22semidirect
64. #1939C22semidirect
66. #1928C22semidirect
67. #1927C22semidirect
70. #1919C22semidirect
71. #1918C22semidirect
72. #1917C22semidirect
83. #1845C22semidirect
85. #1803C22semidirect
90. #1776C22semidirect
91. #1776C22semidirect
92. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#361932
C2×#23542
#331002
#329562
#279502
#276812
#275822
#275092
#273812
#260862
#246802
#245072
#230672
#228112
#226032
C4×#4502
#213652