Finite Group Databasebeta

← #6627944

Subgroups — #6627944

Order 512.

2158 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#32908C212yesmaximal
4#32905C211yesmaximal, characteristic, complemented
5#29336C212yesmaximal, complemented
6#29241C212yesmaximal, complemented
7#27972C211yesmaximal, characteristic, complemented
8#27677C211yesmaximal, characteristic, complemented
9#27566C214yesmaximal, complemented
10#26188C214yesmaximal, complemented
11#26111C212yesmaximal, complemented
12#24727C212yesmaximal, complemented
13#23002C212yesmaximal, complemented
14#22758C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21453C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4—21no—
21C2×#463C2214yes—
22C2×#461C2212yes—
23C2×#450C2212yes—
24C2×#447C2211yescharacteristic
25C2×#440C2214yes—
26C2×C4×D8C2212yes—
27#2857C2211yescharacteristic
28#2850C2212yes—
29#2596—22no—
30#2571C2211yescharacteristic, complemented
31#2547C2211yescharacteristic, complemented
32#2514C2212yescomplemented
33#2510—22no—
34#2494—21no—
35#2487—21no—
36#2448C2212yescomplemented
37#2365C2211yescharacteristic, complemented
38#2354C2211yescharacteristic, complemented
39#2352—21no—
40D4×C4⋊C4—22no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2214yes—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46#2347—21no—
47C4×C22⋊Q8C2212yes—
48C4×#121C2211yescharacteristic
49C2×#389C2212yes—
50C2×#385—22no—
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yescomplemented
55#2099C2212yescomplemented
56#2093C2212yescomplemented
57#2087C2214yescomplemented
58#2072C2214yescomplemented
59#2068C2214yescomplemented
60#2042C2214yescomplemented
61#1956C2212yescomplemented
62#1947C2212yescomplemented
63#1939C2212yescomplemented
64#1936C2214yescomplemented
65#1935C2214yescomplemented
66#1921C2214yescomplemented
67#1917C2214yescomplemented
68#1879C2212yes—
69#1879C2214yes—
70#1877C2214yes—
71#1877C2214yes—
72#1877C2214yes—
73#1874C2212yes—
74#1874C2211yescharacteristic
75#1872C2212yes—
76#1872C2212yes—
77#1872C2212yes—
78#1863C2214yescomplemented
79#1862C2214yes—
80#1862C2212yes—
81#1852C2212yes—
82#1852C2211yescharacteristic
83#1834C2212yescomplemented
84#1804C2214yes—
85#1802C2212yes—
86#1802C2212yescomplemented
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1786C2211yescharacteristic
90#1770C2212yescomplemented
91#1762C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4—22no—
98C22×SD16C2314yes—
99C22×D8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#124C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×#121C2311yescharacteristic
104C2×C4×Q8C2312yes—
105C2×C4×D4—28no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—48no—
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2312yes—
114C2×C4×D4C2314yes—
115C2×C4×D4C2314yes—
116C2×C4×D4C2312yes—
117C2×#117—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4—48no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—22no—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#463—48no—
134#461—28no—
135#450—28no—
136#447—24no—
137#440—48no—
138C4×D8—28no—
139C2×#107C2312yes—
140C2×#107C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×#97C2314yes—
156C2×C4×C8C2312yes—
157#401—22no—
158#400—22no—
159#400—22no—
160#399—22no—
161#397—22no—
162#397—22no—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#390C2312yes—
167#390C2311yescharacteristic
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2314yes—
176#389C2314yes—
177#389C2311yescharacteristic
178#388—22no—
179#388—22no—
180#388—22no—
181#388—21no—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#386—22no—
188#386—22no—
189#386—21no—
190#385—22no—
191#385—24no—
192#385—24no—
193#385—24no—
194#385—24no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#383—22no—
201#382—21no—
202#382C2312yes—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—22no—
207#380—22no—
208#380—24no—
209#380—21no—
210#380—21no—
211#380C2311yescharacteristic
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C2×#95—22no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C22×Q8C2412yes—
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4#4114yes—
245C22×D4C2411yescharacteristic
246C22×D4C2412yes—
247C23×C4—28no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—44no—
256C23×C4—48no—
257C23×C4—44no—
258C23×C4—48no—
259C23×C4—84no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2411yescharacteristic
265C2×SD16—28no—
266C2×SD16—28no—
267C2×SD16—28no—
268C2×D8—24no—
269C2×D8—24no—
270C2×D8—24no—
271C22×C8C2412yes—
272C22×C8C2414yes—
273C22×C8C2412yes—
274C4⋊Q8—24no—
275#124—44no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278#121—22no—
279#121—22no—
280C4×Q8—44no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—48no—
292C4×D4—48no—
293C4×D4—44no—
294C4×D4#4118yes—
295#117—44no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—416no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—82no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—48no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—44no—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—24no—
364#107—44no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367#103—28no—
368#103—44no—
369#103—44no—
370#102—24no—
371#102—24no—
372#102—28no—
373#102—24no—
374#102—24no—
375#102—44no—
376C22⋊C8—22no—
377C22⋊C8—24no—
378C22⋊C8—22no—
379#97—44no—
380C4×C8—24no—
381#95—24no—
382#95—24no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—24no—
401C24—24no—
402C24—24no—
403C24—22no—
404C24—21no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24C22×D411yescharacteristic
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—216no—
417C2×D4—216no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4C2×#4114yes—
430C2×D4C2×#4114yes—
431C2×D4H(5,𝔽2)14yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—432no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—832no—
475C22×C4AGL(1,ℤ/8)14yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4112yes—
486C22×C42−1+412yes—
487C22×C4C2511yescharacteristic
488SD16—416no—
489D8—48no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—42no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—81no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4C2×#4114yes—
531C22⋊C4—28no—
532C22⋊C4—28no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—81no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—24no—
549C42—24no—
550C42—42no—
551C42—42no—
552C42—44no—
553C42—42no—
554C42—82no—
555C42C2×#4114yes—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—216no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—48no—
569C23—42no—
570C23—48no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23#54812yes—
582C23C2×AGL(1,ℤ/8)12yes—
583C23#57812yes—
584C23C23×D411yescharacteristic
585Q8—44no—
586Q8—44no—
587D4—216no—
588D4—24no—
589D4—24no—
590D4—24no—
591D4—42no—
592D4—42no—
593D4—84no—
594D4#55318yes—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—44no—
621C2×C4—416no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—416no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—84no—
646C2×C4—816no—
647C2×C4—816no—
648C2×C4—88no—
649C2×C4—84no—
650C2×C4#52512yes—
651C2×C4#52512yes—
652C2×C4#54812yes—
653C2×C4C2×AGL(1,ℤ/8)12yes—
654C2×C4C2×AGL(1,ℤ/8)12yes—
655C2×C4#57512yes—
656C2×C4C22×#4111yescharacteristic
657C2×C4C22×#4112yes—
658C2×C4C2×H(5,𝔽2)12yes—
659C2×C4C2×2−1+411yescharacteristic
660C8—42no—
661C8—44no—
662C8—42no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—28no—
670C22—432no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22#336714yes—
685C22C2×#52511yescharacteristic
686C22C2×#54811yescharacteristic
687C22#348411yescharacteristic
688C22C22×AGL(1,ℤ/8)11yescharacteristic
689C22C2×#57511yescharacteristic
690C22C2×#57811yescharacteristic
691C22#358511yescharacteristic
692C4—22no—
693C4—22no—
694C4—22no—
695C4—44no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—41no—
701C4—81no—
702C4—81no—
703C4—82no—
704C4—81no—
705C4—84no—
706C4—82no—
707C4#311414yes—
708C4C2×#55312yes—
709C4#349212yes—
710C2—24no—
711C2—48no—
712C2—161no—
713C2#3658111yescharacteristic
714C2C2×#311411yescharacteristic
715C2#6058911yescharacteristic
716C2C2×#336711yescharacteristic
717C2#6131811yescharacteristic
718C2#6158211yescharacteristic
719C2#6159511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
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. #36586C2semidirect
2. C2×D4×C4⋊C4C2semidirect
4. #32905C2semidirect
5. #29336C2semidirect
6. #29241C2semidirect
7. #27972C2semidirect
8. #27677C2semidirect
9. #27566C2semidirect
10. #26188C2semidirect
11. #26111C2semidirect
12. #24727C2semidirect
13. #23002C2semidirect
14. #22758C2semidirect
15. #22629C2semidirect
16. #21453C2semidirect
30. #2571C22semidirect
31. #2547C22semidirect
32. #2514C22semidirect
36. #2448C22semidirect
37. #2365C22semidirect
38. #2354C22semidirect
54. #2101C22semidirect
55. #2099C22semidirect
56. #2093C22semidirect
57. #2087C22semidirect
58. #2072C22semidirect
59. #2068C22semidirect
60. #2042C22semidirect
61. #1956C22semidirect
62. #1947C22semidirect
63. #1939C22semidirect
64. #1936C22semidirect
65. #1935C22semidirect
66. #1921C22semidirect
67. #1917C22semidirect
78. #1863C22semidirect
83. #1834C22semidirect
86. #1802C22semidirect
90. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×D4×C4⋊C42
#329082
#329052
#293362
#292412
#279722
#276772
#275662
#261882
#261112
#247272
#230022
#227582
#226292
#214532
C4×#4472