Finite Group Databasebeta

← #6481951

Subgroups — #6481951

Order 512.

1758 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#35239C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#33150C212yesmaximal, complemented
4#33149C212yesmaximal, complemented
5#29956C212yesmaximal
6#29950C212yesmaximal
7#26098C214yesmaximal, complemented
8#26087C212yesmaximal, complemented
9#26086C212yesmaximal, complemented
10#26033C214yesmaximal, complemented
11#26025C212yesmaximal, complemented
12#26024C212yesmaximal, complemented
13#21987C211yesmaximal, characteristic, complemented
14#21982C211yesmaximal, characteristic, complemented
15#21953C211yesmaximal, characteristic, complemented
16#20738C211yesmaximal, characteristic
17#20733C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2857C2212yes—
23#2851C2212yes—
24#2666—22no—
25#2571C2211yescharacteristic, complemented
26#2570C2212yescomplemented
27#2552C2211yescharacteristic, complemented
28#2552C2211yescharacteristic, complemented
29#2551C2211yescharacteristic, complemented
30#2550C2212yescomplemented
31#2442—21no—
32#2440—21no—
33#2438—22no—
34#2354—21no—
35#2354—21no—
36#2354—22no—
37#2354C2212yes—
38#2354C2212yes—
39#2354C2214yes—
40#2331—21no—
41#2330—21no—
42C2×#390C2212yes—
43C2×#388—22no—
44#2285C2211yescharacteristic
45C2×#384C2211yescharacteristic
46#2283C2211yescharacteristic
47C2×#380—21no—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2093C2214yes—
51#2092C2214yes—
52#2067C2214yes—
53#2067C2214yes—
54#2067C2214yes—
55#2065C2214yes—
56#2025C2214yes—
57#2023C2214yes—
58#1989C2214yescomplemented
59#1987C2214yescomplemented
60#1949C2214yescomplemented
61#1948C2212yescomplemented
62#1947C2214yescomplemented
63#1947C2212yescomplemented
64#1947C2214yescomplemented
65#1940C2214yescomplemented
66#1939C2212yescomplemented
67#1939C2212yescomplemented
68#1921C2214yescomplemented
69#1921C2214yescomplemented
70#1920C2214yescomplemented
71#1918C2214yescomplemented
72#1853C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1847C2212yescomplemented
77#1845C2212yescomplemented
78#1836C2212yescomplemented
79#1834C2212yescomplemented
80#1824C2212yes—
81#1824C2211yescharacteristic
82#1824C2211yescharacteristic
83#1804C2212yes—
84#1804C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1797C2212yes—
88#1797C2212yes—
89#1786C2212yes—
90#1786C2211yescharacteristic
91#1786C2211yescharacteristic
92#1762C2214yes—
93C4×#102C2212yes—
94C4×#102C2212yes—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2312yes—
99C2×#125—21no—
100C2×#125C2312yes—
101C2×C22⋊Q8—22no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×C4×D4—24no—
105C2×C4×D4Q818yescomplemented
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—21no—
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×C42C2312yes—
124C22×C42C2311yescharacteristic
125#432—24no—
126#431—24no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×C22⋊C8C2311yescharacteristic
142C2×C22⋊C8C2311yescharacteristic
143C2×C22⋊C8C2312yes—
144C2×#97C2312yes—
145C2×C4×C8C2312yes—
146#400—22no—
147#400—22no—
148#398—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#395—22no—
153#395—22no—
154#391—22no—
155#391—22no—
156#390—22no—
157#390—22no—
158#390C2314yes—
159#390C2314yes—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#388—24no—
171#388—24no—
172#388—24no—
173#388—24no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#385—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#384C2312yes—
188#383—21no—
189#382—21no—
190#382—21no—
191#382C2311yescharacteristic
192#382C2312yes—
193#382C2311yescharacteristic
194#381—21no—
195#380—22no—
196#380—21no—
197#380—22no—
198#380—22no—
199#380—24no—
200#380C2311yescharacteristic
201#380C2312yes—
202#380C2311yescharacteristic
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218#376—21no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233C25—21no—
234C22×Q8—22no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—42no—
238C22×D4—42no—
239C22×D4C2×Q814yes—
240C22×D4C2412yes—
241C23×C4—21no—
242C23×C4—28no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258#125—44no—
259#125—44no—
260C22⋊Q8—84no—
261C4×D4—28no—
262C4×D4—28no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4C2×Q818yes—
266C4×D4C2×Q818yes—
267#117—48no—
268#117—416no—
269#117—48no—
270#117—82no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×Q814yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4—84no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42—44no—
329C2×C42—48no—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×Q814yes—
335C2×C42#4114yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—44no—
341#107—44no—
342#107—44no—
343#106—44no—
344C4⋊C8—24no—
345C4⋊C8—24no—
346C4⋊C8—44no—
347C4⋊C8—44no—
348#102—42no—
349#102—44no—
350#102—42no—
351#102—42no—
352#102—44no—
353#102—42no—
354C22⋊C8—22no—
355C22⋊C8—22no—
356C22⋊C8—24no—
357#97—24no—
358C4×C8—24no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—24no—
382C24—22no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24C22×D411yescharacteristic
390C2×Q8—44no—
391C2×Q8—82no—
392C2×Q8—82no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—48no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—48no—
403C2×D4C22×Q814yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—432no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4C22⋊Q814yes—
438C22×C4C22⋊Q814yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×Q812yes—
449C22×C4C2×#4112yes—
450C22×C4C2511yescharacteristic
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4AGL(1,ℤ/8)14yes—
494C4⋊C4C22×Q814yes—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—48no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—24no—
505C42—24no—
506C42—42no—
507C42—42no—
508C42—48no—
509C42—48no—
510C42—42no—
511C42—42no—
512C42—82no—
513C42C22⋊Q814yes—
514C42C22⋊Q814yes—
515C42C22×Q812yes—
516C42C22×Q812yes—
517C42C2×#4112yes—
518C42C2×#4112yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—24no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23C2×C22⋊Q812yes—
538C23#57612yes—
539C23#57812yes—
540C23C23×D411yescharacteristic
541Q8—164no—
542D4—24no—
543D4—24no—
544D4—48no—
545D4—42no—
546D4—42no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—416no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—416no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—88no—
600C2×C4C2×C22⋊Q812yes—
601C2×C4C2×C22⋊Q812yes—
602C2×C4C2×C22⋊Q812yes—
603C2×C4C2×C22⋊Q812yes—
604C2×C4C2×C22⋊Q812yes—
605C2×C4C2×AGL(1,ℤ/8)14yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4#57512yes—
608C2×C4C23×Q811yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4#58512yes—
611C2×C4#58512yes—
612C8—42no—
613C8—42no—
614C8—44no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—816no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22#308314yes—
630C22C22×C22⋊Q811yescharacteristic
631C22#348511yescharacteristic
632C22#348711yescharacteristic
633C22C22×AGL(1,ℤ/8)11yescharacteristic
634C22C2×#57511yescharacteristic
635C22C2×#57611yescharacteristic
636C22C2×#57811yescharacteristic
637C4—24no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—44no—
646C4—81no—
647C4—82no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—82no—
652C4—84no—
653C4#307714yes—
654C4#345812yes—
655C4#349312yes—
656C2—24no—
657C2—84no—
658C2#3515111yescharacteristic
659C2C2×#307711yescharacteristic
660C2C2×#308311yescharacteristic
661C2#6173711yescharacteristic
662C2#6174011yescharacteristic
663C2#6190511yescharacteristic
664C2#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. #35239C2semidirect
3. #33150C2semidirect
4. #33149C2semidirect
7. #26098C2semidirect
8. #26087C2semidirect
9. #26086C2semidirect
10. #26033C2semidirect
11. #26025C2semidirect
12. #26024C2semidirect
13. #21987C2semidirect
14. #21982C2semidirect
15. #21953C2semidirect
25. #2571C22semidirect
26. #2570C22semidirect
27. #2552C22semidirect
28. #2552C22semidirect
29. #2551C22semidirect
30. #2550C22semidirect
58. #1989C22semidirect
59. #1987C22semidirect
60. #1949C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1947C22semidirect
64. #1947C22semidirect
65. #1940C22semidirect
66. #1939C22semidirect
67. #1939C22semidirect
68. #1921C22semidirect
69. #1921C22semidirect
70. #1920C22semidirect
71. #1918C22semidirect
76. #1847C22semidirect
77. #1845C22semidirect
78. #1836C22semidirect
79. #1834C22semidirect
105. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#352392
C2×#23542
#331502
#331492
#299562
#299502
#260982
#260872
#260862
#260332
#260252
#260242
#219872
#219822
#219532
#207382
#207332