Finite Group Databasebeta

← #6624975

Subgroups — #6624975

Order 512.

2215 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic, complemented
2C2×#2366C211yesmaximal, characteristic
3#33146C211yesmaximal, characteristic, complemented
4#32959C212yesmaximal, complemented
5#29110C212yesmaximal, complemented
6#28120C212yesmaximal, complemented
7#26913C214yesmaximal, complemented
8#26768C214yesmaximal, complemented
9#26463C212yesmaximal, complemented
10#25855C211yesmaximal, characteristic
11#25325C212yesmaximal, complemented
12#24211C211yesmaximal, characteristic
13#23220C212yesmaximal, complemented
14#23003C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16C4×#483C211yesmaximal, characteristic, complemented
17#21453C212yesmaximal, complemented
18C22×#127C2211yescharacteristic
19C22×C4×D4—21no—
20C22×C4×D4C2212yes—
21C2×#486C2212yes—
22C2×#483C2211yescharacteristic
23C2×#478C2214yescomplemented
24C2×#460C2212yes—
25C2×#450C2212yes—
26C2×#440C2214yescomplemented
27#2857C2211yescharacteristic
28#2851C2212yes—
29#2590—22no—
30#2568C2211yescharacteristic, complemented
31#2552C2211yescharacteristic, complemented
32#2511—21no—
33#2508C2212yescomplemented
34#2505—22no—
35#2502—21no—
36#2453C2212yescomplemented
37#2366—22no—
38#2366—21no—
39#2366—21no—
40#2366C2214yes—
41#2366C2212yes—
42#2366C2212yes—
43#2365C2211yescharacteristic, complemented
44#2360—21no—
45#2355—21no—
46Q8×C4⋊C4C2211yescharacteristic, complemented
47C4×C22⋊Q8C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49C2×#390—21no—
50C2×#390C2212yes—
51C2×#386—22no—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2073C2214yescomplemented
55#2065C2214yescomplemented
56#2055C2214yescomplemented
57#2041C2214yes—
58#2041C2212yes—
59#2025C2214yescomplemented
60#2014C2212yes—
61#1988C2212yescomplemented
62#1982C2212yes—
63#1957C2212yes—
64#1947C2212yescomplemented
65#1939C2212yescomplemented
66#1936C2214yescomplemented
67#1936C2214yescomplemented
68#1927C2212yescomplemented
69#1926C2214yescomplemented
70#1920C2212yescomplemented
71#1918C2214yescomplemented
72#1917C2214yescomplemented
73#1880C2214yescomplemented
74#1877C2214yescomplemented
75#1872C2211yescharacteristic
76#1864C2214yescomplemented
77#1862C2214yescomplemented
78#1862C2214yescomplemented
79#1853C2211yescharacteristic
80#1852C2212yes—
81#1823C2211yescharacteristic
82#1823C2212yes—
83#1804C2214yescomplemented
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1786C2212yes—
87#1786C2212yes—
88#1786C2212yes—
89C4×#106C2212yescomplemented
90#1770C2212yescomplemented
91#1762C2214yescomplemented
92C4×#103C2212yescomplemented
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C24×C4—22no—
99C22×SD16C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127—28no—
102C2×#127—21no—
103C2×#127—21no—
104C2×#127C2312yes—
105C2×#127C2312yes—
106C2×#125C2312yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8C2311yescharacteristic
109C2×C4×Q8—21no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—24no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—48no—
118C2×C4×D4—48no—
119C2×C4×D4C2314yes—
120C2×C4×D4C2314yes—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—22no—
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#486—28no—
132#483—24no—
133#478—48no—
134#460—28no—
135#450—28no—
136#440—48no—
137C2×#107C2314yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#400—22no—
158#400—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391C2312yes—
167#391C2311yescharacteristic
168#390—22no—
169#390—22no—
170#390—24no—
171#390—22no—
172#390—22no—
173#390C2314yes—
174#390C2314yes—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2311yescharacteristic
184#387—22no—
185#387—21no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#386—24no—
192#386—24no—
193#386—24no—
194#386—24no—
195#385—22no—
196#385—22no—
197#385—22no—
198#385—21no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#382—22no—
203#382C2312yes—
204#380—21no—
205#380C2311yescharacteristic
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C2×#95—21no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236#336C2312yes—
237C25—22no—
238C22×Q8—21no—
239C22×Q8C2411yescharacteristic
240C22×D4—28no—
241C22×D4—28no—
242C22×D4—28no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—24no—
249C22×D4—24no—
250C22×D4—44no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4—82no—
254C22×D4—82no—
255C22×D4C2414yes—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—24no—
259C23×C4—24no—
260C23×C4—48no—
261C23×C4—48no—
262C23×C4—84no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D412yes—
266C23×C4C2×D412yes—
267C23×C4C2411yescharacteristic
268C2×SD16—28no—
269C2×SD16—28no—
270C2×SD16—44no—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C4⋊Q8—24no—
275#127—28no—
276#127—28no—
277#127—28no—
278#127—24no—
279#127#4118yes—
280#125—44no—
281C22⋊Q8—22no—
282C22⋊Q8—44no—
283C22⋊Q8#4114yes—
284C4×Q8—24no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—48no—
292C4×D4—48no—
293C4×D4—416no—
294C4×D4—48no—
295C4×D4—48no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—48no—
339C2×C22⋊C4—48no—
340C2×C22⋊C4—84no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—44no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—28no—
359#106—24no—
360#106—24no—
361C4⋊C8—24no—
362C4⋊C8—44no—
363#103—24no—
364#103—24no—
365#102—28no—
366#102—28no—
367#102—28no—
368#102—28no—
369#102—44no—
370#102—44no—
371#102—44no—
372C22⋊C8—24no—
373C22⋊C8—22no—
374C22⋊C8—22no—
375#97—44no—
376C4×C8—24no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—28no—
399C24—24no—
400C24—44no—
401C24—44no—
402C24—44no—
403C24—44no—
404C24—44no—
405C24—44no—
406C24C22×D411yescharacteristic
407C2×Q8—21no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8C2×#4112yes—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—416no—
425C2×D4—416no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—432no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—816no—
438C2×D4C2×#4118yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—41no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—832no—
473C22×C4AGL(1,ℤ/8)14yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4112yes—
484C22×C42−1+412yes—
485C22×C4C2511yescharacteristic
486SD16—416no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—81no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—81no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4C2×#4112yes—
526C4⋊C4H(5,𝔽2)14yes—
527C22⋊C4—24no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—42no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—24no—
540C42—24no—
541C42—42no—
542C42—42no—
543C42—44no—
544C42—42no—
545C42—44no—
546C42—44no—
547C42—42no—
548C42—82no—
549C42C2×#4114yes—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—28no—
555C23—416no—
556C23—416no—
557C23—416no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—88no—
565C23—88no—
566C23—88no—
567C23—88no—
568C23#54812yes—
569C23C2×AGL(1,ℤ/8)12yes—
570C23#57712yes—
571C23C23×D411yescharacteristic
572Q8—24no—
573Q8—42no—
574Q8—84no—
575D4—28no—
576D4—28no—
577D4—416no—
578D4—416no—
579D4—416no—
580D4—44no—
581D4—44no—
582D4—88no—
583D4—88no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—84no—
636C2×C4—816no—
637C2×C4—816no—
638C2×C4—84no—
639C2×C4#52512yes—
640C2×C4#52512yes—
641C2×C4#54812yes—
642C2×C4C2×AGL(1,ℤ/8)12yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4#57512yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4112yes—
647C2×C4C2×H(5,𝔽2)12yes—
648C2×C4C2×2−1+411yescharacteristic
649C8—42no—
650C8—44no—
651C8—42no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—48no—
659C22—48no—
660C22—48no—
661C22—832no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22#336814yes—
667C22C2×#52511yescharacteristic
668C22C2×#54811yescharacteristic
669C22#348411yescharacteristic
670C22C22×AGL(1,ℤ/8)11yescharacteristic
671C22C2×#57511yescharacteristic
672C22C2×#57711yescharacteristic
673C22#358511yescharacteristic
674C4—22no—
675C4—22no—
676C4—22no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—41no—
684C4—42no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—81no—
689C4—81no—
690C4#311414yes—
691C4#348112yes—
692C4#352012yes—
693C2—24no—
694C2—88no—
695C2#3652811yescharacteristic
696C2C2×#311411yescharacteristic
697C2#6057911yescharacteristic
698C2C2×#336811yescharacteristic
699C2#6132411yescharacteristic
700C2#6158211yescharacteristic
701C2#6159611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. #33146C2semidirect
4. #32959C2semidirect
5. #29110C2semidirect
6. #28120C2semidirect
7. #26913C2semidirect
8. #26768C2semidirect
9. #26463C2semidirect
11. #25325C2semidirect
13. #23220C2semidirect
14. #23003C2semidirect
15. #22629C2semidirect
16. C4×#483C2semidirect
17. #21453C2semidirect
23. C2×#478C22semidirect
26. C2×#440C22semidirect
30. #2568C22semidirect
31. #2552C22semidirect
33. #2508C22semidirect
36. #2453C22semidirect
43. #2365C22semidirect
46. Q8×C4⋊C4C22semidirect
54. #2073C22semidirect
55. #2065C22semidirect
56. #2055C22semidirect
59. #2025C22semidirect
61. #1988C22semidirect
64. #1947C22semidirect
65. #1939C22semidirect
66. #1936C22semidirect
67. #1936C22semidirect
68. #1927C22semidirect
69. #1926C22semidirect
70. #1920C22semidirect
71. #1918C22semidirect
72. #1917C22semidirect
73. #1880C22semidirect
74. #1877C22semidirect
76. #1864C22semidirect
77. #1862C22semidirect
78. #1862C22semidirect
83. #1804C22semidirect
89. C4×#106C22semidirect
90. #1770C22semidirect
91. #1762C22semidirect
92. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#365522
C2×#23662
#331462
#329592
#291102
#281202
#269132
#267682
#264632
#258552
#253252
#242112
#232202
#230032
#226292
C4×#4832
#214532