Finite Group Databasebeta

← #6624351

Subgroups — #6624351

Order 512.

1805 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#33151C212yesmaximal, complemented
4#32953C211yesmaximal, characteristic
5#29110C212yesmaximal
6#28120C212yesmaximal
7#27960C211yesmaximal, characteristic, complemented
8#27647C211yesmaximal, characteristic, complemented
9#27440C214yesmaximal, complemented
10#27374C214yesmaximal, complemented
11#26021C212yesmaximal, complemented
12#24681C212yesmaximal, complemented
13#23066C212yesmaximal, complemented
14#22759C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21579C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#483C2212yes—
21C2×#474C2212yes—
22C2×#450C2211yescharacteristic
23C2×C4×SD16C2212yes—
24#2857C2211yescharacteristic
25#2851C2212yes—
26#2590—22no—
27#2568C2211yescharacteristic, complemented
28#2552C2211yescharacteristic, complemented
29#2511—21no—
30#2508C2212yescomplemented
31#2505—22no—
32#2502—21no—
33#2453C2212yescomplemented
34#2365C2211yescharacteristic, complemented
35#2360—21no—
36#2355—21no—
37#2354—22no—
38#2354—21no—
39#2354—21no—
40#2354C2212yes—
41#2354C2214yes—
42#2354C2212yes—
43Q8×C4⋊C4C2211yescharacteristic, complemented
44C4×C22⋊Q8C2212yes—
45C4×C22⋊Q8C2211yescharacteristic
46C2×#390C2212yes—
47C2×#388—22no—
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2214yescomplemented
53#2084C2212yescomplemented
54#2073C2214yes—
55#2072C2212yescomplemented
56#2065C2214yes—
57#2055C2214yes—
58#2025C2214yes—
59#1990C2212yescomplemented
60#1989C2214yescomplemented
61#1949C2214yescomplemented
62#1948C2212yescomplemented
63#1939C2212yescomplemented
64#1936C2214yes—
65#1936C2214yes—
66#1928C2214yescomplemented
67#1926C2214yes—
68#1921C2214yescomplemented
69#1919C2214yescomplemented
70#1919C2214yescomplemented
71#1917C2214yescomplemented
72#1880C2214yes—
73#1877C2212yes—
74#1872C2211yescharacteristic
75#1864C2212yes—
76#1862C2214yes—
77#1853C2211yescharacteristic
78#1852C2212yes—
79#1834C2212yescomplemented
80#1823C2212yes—
81#1823C2212yes—
82#1823C2212yes—
83#1803C2212yescomplemented
84#1802C2212yes—
85#1797C2212yes—
86#1797C2211yescharacteristic
87#1786C2211yescharacteristic
88#1786C2212yes—
89#1776C2214yescomplemented
90#1770C2212yescomplemented
91#1763C2214yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×SD16C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125C2312yes—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C2311yescharacteristic
102C2×C4×Q8—21no—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×Q8C2312yes—
105C2×C4×D4—28no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117—22no—
111C2×#117—22no—
112C2×#117—24no—
113C2×#117—48no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—22no—
122C22×C22⋊C4—22no—
123C22×C42—22no—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#483—28no—
127#474—28no—
128#450—24no—
129C4×SD16—28no—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2311yescharacteristic
144C2×C22⋊C8C2311yescharacteristic
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147#400—22no—
148#400—22no—
149#400—22no—
150#398—22no—
151#398—22no—
152#398—22no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391C2312yes—
158#391C2311yescharacteristic
159#390—22no—
160#390—22no—
161#390C2314yes—
162#390C2314yes—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#389C2312yes—
172#388—24no—
173#388—24no—
174#388—24no—
175#388—24no—
176#387—22no—
177#387—21no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#385—21no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—22no—
195#382C2312yes—
196#380—21no—
197#380—22no—
198#380—22no—
199#380—24no—
200#380C2311yescharacteristic
201#380C2312yes—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2314yes—
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×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2411yescharacteristic
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—42no—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C4⋊Q8—24no—
262#125—44no—
263C22⋊Q8—22no—
264C22⋊Q8—44no—
265C22⋊Q8#4114yes—
266C4×Q8—24no—
267C4×Q8—44no—
268C4×Q8—44no—
269C4×D4—28no—
270C4×D4—28no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4#4118yes—
275#117—416no—
276#117—48no—
277#117—48no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
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—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—28no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—84no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—44no—
337C2×C42—48no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346#107—28no—
347#106—24no—
348#106—28no—
349C4⋊C8—24no—
350C4⋊C8—44no—
351#103—28no—
352#103—24no—
353#102—24no—
354#102—28no—
355#102—24no—
356#102—44no—
357#102—42no—
358C22⋊C8—22no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361#97—44no—
362C4×C8—24no—
363#95—24no—
364#95—24no—
365#95—21no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—24no—
385C24—22no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24C22×D411yescharacteristic
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8C2×#4112yes—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C2×#4114yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—432no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4AGL(1,ℤ/8)14yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4112yes—
456C22×C42−1+412yes—
457C22×C4C2511yescharacteristic
458SD16—48no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4C2×#4112yes—
502C4⋊C4C2×#4114yes—
503C22⋊C4—28no—
504C22⋊C4—24no—
505C22⋊C4—28no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—24no—
517C42—42no—
518C42—42no—
519C42—48no—
520C42—48no—
521C42—44no—
522C42—42no—
523C42—42no—
524C42—82no—
525C42C2×#4114yes—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—24no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23#54812yes—
545C23C2×AGL(1,ℤ/8)12yes—
546C23#57712yes—
547C23C23×D411yescharacteristic
548Q8—24no—
549Q8—42no—
550Q8—84no—
551D4—24no—
552D4—24no—
553D4—48no—
554D4—42no—
555D4—42no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
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—44no—
584C2×C4—416no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—416no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—84no—
610C2×C4—88no—
611C2×C4#52512yes—
612C2×C4#52512yes—
613C2×C4#54812yes—
614C2×C4C2×AGL(1,ℤ/8)12yes—
615C2×C4C2×AGL(1,ℤ/8)12yes—
616C2×C4#57512yes—
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4112yes—
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58512yes—
621C8—44no—
622C8—42no—
623C8—42no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—816no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22#336814yes—
639C22C2×#52511yescharacteristic
640C22C2×#54811yescharacteristic
641C22#348511yescharacteristic
642C22C22×AGL(1,ℤ/8)11yescharacteristic
643C22C2×#57511yescharacteristic
644C22C2×#57711yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—41no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4—84no—
661C4—81no—
662C4#311414yes—
663C4#347512yes—
664C4#349512yes—
665C2—24no—
666C2—84no—
667C2#3651611yescharacteristic
668C2C2×#311411yescharacteristic
669C2#6058511yescharacteristic
670C2C2×#336811yescharacteristic
671C2#6132311yescharacteristic
672C2#6171711yescharacteristic
673C2#6173111yescharacteristic
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. #36552C2semidirect
3. #33151C2semidirect
7. #27960C2semidirect
8. #27647C2semidirect
9. #27440C2semidirect
10. #27374C2semidirect
11. #26021C2semidirect
12. #24681C2semidirect
13. #23066C2semidirect
14. #22759C2semidirect
15. #22629C2semidirect
16. #21579C2semidirect
27. #2568C22semidirect
28. #2552C22semidirect
30. #2508C22semidirect
33. #2453C22semidirect
34. #2365C22semidirect
43. Q8×C4⋊C4C22semidirect
52. #2099C22semidirect
53. #2084C22semidirect
55. #2072C22semidirect
59. #1990C22semidirect
60. #1989C22semidirect
61. #1949C22semidirect
62. #1948C22semidirect
63. #1939C22semidirect
66. #1928C22semidirect
68. #1921C22semidirect
69. #1919C22semidirect
70. #1919C22semidirect
71. #1917C22semidirect
79. #1834C22semidirect
83. #1803C22semidirect
89. #1776C22semidirect
90. #1770C22semidirect
91. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#365522
C2×#23542
#331512
#329532
#291102
#281202
#279602
#276472
#274402
#273742
#260212
#246812
#230662
#227592
#226292
#215792
C4×#4502