Finite Group Databasebeta

← #6470969

Subgroups — #6470969

Order 512.

1744 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#2362C211yesmaximal, characteristic
3#33151C212yesmaximal, complemented
4#32960C212yesmaximal, complemented
5#29972C212yesmaximal, complemented
6#29957C212yesmaximal, complemented
7#26435C214yesmaximal, complemented
8#26433C214yesmaximal, complemented
9#26367C214yesmaximal, complemented
10#26365C214yesmaximal, complemented
11#21934C211yesmaximal, characteristic
12#21765C211yesmaximal, characteristic
13#21765C211yesmaximal, characteristic
14#20846C211yesmaximal, characteristic
15#20846C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#479C2214yescomplemented
20C2×#428C2212yes—
21C2×#428C2212yes—
22C2×#407C2211yescharacteristic
23C2×#407C2211yescharacteristic
24#2851C2212yes—
25#2851C2212yes—
26#2646—21no—
27#2646—21no—
28#2569—22no—
29#2569—22no—
30#2442C2212yes—
31#2442C2212yes—
32#2442C2212yes—
33#2442C2212yes—
34#2362—22no—
35#2362—22no—
36#2362C2214yes—
37#2362C2214yes—
38#2315—21no—
39#2315—21no—
40C2×#391C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#388—21no—
44C2×#387—21no—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2214yescomplemented
51#2097C2214yescomplemented
52#2088C2214yescomplemented
53#2087C2214yescomplemented
54#2068C2214yescomplemented
55#2066C2214yescomplemented
56#2059C2214yescomplemented
57#2026C2214yescomplemented
58#1989C2214yescomplemented
59#1949C2214yescomplemented
60#1921C2214yescomplemented
61#1919C2214yescomplemented
62#1894C2214yes—
63#1894C2214yes—
64#1872C2212yes—
65#1872C2212yes—
66#1872C2212yes—
67#1872C2212yes—
68#1865C2214yescomplemented
69#1863C2214yescomplemented
70#1854C2214yes—
71#1854C2214yes—
72#1853C2212yes—
73#1853C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1851C2214yes—
77#1851C2214yes—
78#1823C2212yes—
79#1823C2212yes—
80#1805C2214yescomplemented
81#1797C2212yes—
82#1797C2212yes—
83#1787C2212yes—
84#1787C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×M16C2312yes—
90C22×M16C2312yes—
91C2×#125C2312yes—
92C2×#125C2312yes—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8Q818yes—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C4×Q8—22no—
100C2×C4×Q8—44no—
101C2×C4×Q8C2314yes—
102C2×#117—21no—
103C2×#117—21no—
104C2×#117—44no—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4D411yescharacteristic
122C22×C22⋊C4D411yescharacteristic
123C22×C42—21no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#479—48no—
129#428—44no—
130#428—44no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141#407—24no—
142#407—24no—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147#400—22no—
148#400—22no—
149#400—22no—
150#400—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#391—22no—
156#391C2314yes—
157#389—22no—
158#389C2314yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#388—24no—
164#388—24no—
165#388—24no—
166#388—24no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—24no—
172#387—24no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#382C2312yes—
182#382C2312yes—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—24no—
197C4×C4⋊C4—24no—
198C4×C4⋊C4C2314yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4C2312yes—
202C4×C22⋊C4C2312yes—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2314yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8—22no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2×Q814yes—
227C22×Q8C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×Q814yes—
246C23×C4#4114yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×M16—24no—
251C2×M16—24no—
252C2×M16—42no—
253C2×M16—42no—
254C2×M16—42no—
255C2×M16—42no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258#125—44no—
259#125—44no—
260C22⋊Q8—28no—
261C22⋊Q8—28no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8C2×Q818yes—
265C22⋊Q8C2×Q818yes—
266C4×Q8—28no—
267C4×Q8—44no—
268C4×Q8—48no—
269#117—24no—
270#117—24no—
271#117—42no—
272#117—42no—
273#117—44no—
274#117—44no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2×Q814yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343#107—28no—
344#107—44no—
345#106—28no—
346#106—44no—
347C4⋊C8—44no—
348C4⋊C8—44no—
349#103—44no—
350#103—44no—
351#103—44no—
352#103—44no—
353C22⋊C8—24no—
354C22⋊C8—24no—
355C22⋊C8—24no—
356C22⋊C8—24no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—22no—
378C24—22no—
379C24—41no—
380C24—41no—
381C24—41no—
382C24—41no—
383C24C22⋊Q814yes—
384C24C22×D411yescharacteristic
385C24C22×D411yescharacteristic
386C24C22×D411yescharacteristic
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—48no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—48no—
395C2×Q8C22×Q814yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
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—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
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—42no—
434C22×C4—42no—
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—42no—
445C22×C4—42no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—81no—
454C22×C4C22⋊Q814yes—
455C22×C4C22⋊Q814yes—
456C22×C4C22⋊Q814yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×Q812yes—
465C22×C4C22×Q812yes—
466C22×C4C22×Q812yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2511yescharacteristic
471M16—48no—
472M16—48no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C4⋊C4—24no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—42no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4C22×Q814yes—
507C4⋊C4H(5,𝔽2)14yes—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C42—24no—
521C42—24no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—44no—
526C42—48no—
527C42—48no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—28no—
545C23—44no—
546C23—44no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23C2×C22⋊Q812yes—
552C23C2×C22⋊Q812yes—
553C23C2×C22⋊Q812yes—
554C23#57512yes—
555C23#57512yes—
556C23C23×D411yescharacteristic
557Q8—28no—
558Q8—48no—
559Q8—44no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—42no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—84no—
619C2×C4—162no—
620C2×C4—162no—
621C2×C4—162no—
622C2×C4—162no—
623C2×C4C2×C22⋊Q812yes—
624C2×C4C2×C22⋊Q812yes—
625C2×C4C2×C22⋊Q812yes—
626C2×C4C23×Q811yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4#58512yes—
630C8—44no—
631C8—44no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—88no—
653C22#308014yes—
654C22C22×C22⋊Q811yescharacteristic
655C22#348411yescharacteristic
656C22#348511yescharacteristic
657C22C2×#57511yescharacteristic
658C22C2×#57511yescharacteristic
659C22#358611yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—44no—
667C4—42no—
668C4—44no—
669C4—42no—
670C4—44no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#345912yes—
677C4#350112yes—
678C2—24no—
679C2—42no—
680C2—42no—
681C2#3498511yescharacteristic
682C2C2×#308011yescharacteristic
683C2#6049111yescharacteristic
684C2#6157511yescharacteristic
685C2#6158311yescharacteristic
686C2#6171211yescharacteristic
687C2#6171611yescharacteristic
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
3. #33151C2semidirect
4. #32960C2semidirect
5. #29972C2semidirect
6. #29957C2semidirect
7. #26435C2semidirect
8. #26433C2semidirect
9. #26367C2semidirect
10. #26365C2semidirect
19. C2×#479C22semidirect
50. #2101C22semidirect
51. #2097C22semidirect
52. #2088C22semidirect
53. #2087C22semidirect
54. #2068C22semidirect
55. #2066C22semidirect
56. #2059C22semidirect
57. #2026C22semidirect
58. #1989C22semidirect
59. #1949C22semidirect
60. #1921C22semidirect
61. #1919C22semidirect
68. #1865C22semidirect
69. #1863C22semidirect
80. #1805C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#23622
#331512
#329602
#299722
#299572
#264352
#264332
#263672
#263652
#219342
#217652
#208462