Finite Group Databasebeta

← #6627953

Subgroups — #6627953

Order 512.

2201 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#32908C212yesmaximal, complemented
4#32902C211yesmaximal, characteristic
5#29336C212yesmaximal, complemented
6#29235C212yesmaximal, complemented
7#27973C211yesmaximal, characteristic, complemented
8#27680C211yesmaximal, characteristic, complemented
9#27568C214yesmaximal, complemented
10#26180C214yesmaximal, complemented
11#26107C212yesmaximal, complemented
12#24731C212yesmaximal, complemented
13#23001C212yesmaximal, complemented
14#22762C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21452C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4—21no—
21C2×#463C2214yes—
22C2×#459C2212yes—
23C2×#450C2211yescharacteristic
24C2×#447C2212yes—
25C2×#442C2214yes—
26C2×C4×SD16C2212yes—
27#2856C2211yescharacteristic
28#2850C2212yes—
29#2597—22no—
30#2571C2211yescharacteristic, complemented
31#2548C2211yescharacteristic, complemented
32#2513—21no—
33#2513C2212yescomplemented
34#2504—21no—
35#2447C2212yescomplemented
36#2444—22no—
37#2366C2211yescharacteristic, complemented
38#2361—21no—
39#2357—21no—
40#2355C2211yescharacteristic, complemented
41D4×C4⋊C4—22no—
42D4×C4⋊C4—21no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2214yes—
45D4×C4⋊C4C2212yes—
46D4×C4⋊C4C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48C4×#121C2212yes—
49C2×#389C2212yes—
50C2×#385—22no—
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yescomplemented
55#2099C2212yescomplemented
56#2096C2212yescomplemented
57#2088C2214yescomplemented
58#2071C2214yescomplemented
59#2065C2214yescomplemented
60#2042C2214yescomplemented
61#1957C2212yescomplemented
62#1948C2212yescomplemented
63#1939C2212yescomplemented
64#1936C2214yescomplemented
65#1935C2214yescomplemented
66#1920C2214yescomplemented
67#1917C2214yescomplemented
68#1879C2214yes—
69#1879C2214yes—
70#1879C2214yes—
71#1877C2212yes—
72#1877C2214yes—
73#1874C2212yes—
74#1874C2212yes—
75#1874C2212yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1864C2214yes—
79#1864C2212yes—
80#1864C2214yescomplemented
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1834C2212yescomplemented
84#1804C2214yes—
85#1803C2212yescomplemented
86#1802C2212yes—
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×SD16C2312yes—
99C22×D8C2314yes—
100C2×#127C2311yescharacteristic
101C2×#124C2312yes—
102C2×C22⋊Q8C2311yescharacteristic
103C2×#121—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×D4—28no—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—48no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2314yes—
117C2×C4×D4C2314yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—24no—
126C22×C4⋊C4—48no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#463—48no—
136#459—28no—
137#450—24no—
138#447—28no—
139#442—48no—
140C4×SD16—28no—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2311yescharacteristic
157C2×#97C2314yes—
158C2×C4×C8C2312yes—
159#400—22no—
160#397—22no—
161#397—22no—
162#397—22no—
163#397—22no—
164#394—22no—
165#391C2312yes—
166#391C2311yescharacteristic
167#390—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—21no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2314yes—
180#389C2314yes—
181#389C2311yescharacteristic
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—21no—
188#387—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—21no—
194#385—22no—
195#385—24no—
196#385—24no—
197#385—24no—
198#385—24no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#382—21no—
205#382—21no—
206#382C2312yes—
207#381—22no—
208#380—22no—
209#380—22no—
210#380—24no—
211#380C2311yescharacteristic
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
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—21no—
223C4×C22⋊C4—22no—
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×Q8C2411yescharacteristic
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4#4114yes—
247C22×D4C2412yes—
248C22×D4C2412yes—
249C23×C4—28no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—24no—
255C23×C4—24no—
256C23×C4—22no—
257C23×C4—44no—
258C23×C4—48no—
259C23×C4—44no—
260C23×C4—48no—
261C23×C4—84no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2411yescharacteristic
267C2×SD16—24no—
268C2×SD16—24no—
269C2×SD16—24no—
270C2×D8—28no—
271C2×D8—28no—
272C2×D8—28no—
273C22×C8C2414yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276#127—24no—
277#124—44no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280#121—24no—
281#121—24no—
282#121—44no—
283#121—82no—
284C4×Q8—24no—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—24no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—48no—
297C4×D4—48no—
298C4×D4—84no—
299C4×D4#4118yes—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—416no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—82no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—48no—
354C2×C42—24no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—44no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2412yes—
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#106—24no—
368#106—44no—
369C4⋊C8—24no—
370C4⋊C8—44no—
371#103—24no—
372#103—44no—
373#102—24no—
374#102—28no—
375#102—24no—
376#102—24no—
377#102—28no—
378#102—44no—
379#102—44no—
380C22⋊C8—24no—
381C22⋊C8—22no—
382C22⋊C8—22no—
383#97—44no—
384C4×C8—24no—
385#95—24no—
386#95—24no—
387#95—21no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—24no—
403C24—24no—
404C24—24no—
405C24—22no—
406C24—22no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24C22×D411yescharacteristic
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—216no—
419C2×D4—216no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—81no—
436C2×D4—81no—
437C2×D4C2×#4114yes—
438C2×D4C2×#4114yes—
439C2×D4H(5,𝔽2)14yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—28no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—432no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—832no—
483C22×C4AGL(1,ℤ/8)14yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4112yes—
494C22×C42−1+412yes—
495C22×C4C2511yescharacteristic
496SD16—48no—
497D8—416no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—22no—
506C2×C8—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—81no—
536C4⋊C4C2×#4114yes—
537C22⋊C4—28no—
538C22⋊C4—28no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—24no—
557C42—24no—
558C42—42no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42—82no—
563C42C2×#4114yes—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—216no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—48no—
573C23—48no—
574C23—48no—
575C23—48no—
576C23—48no—
577C23—42no—
578C23—48no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23#54812yes—
590C23C2×AGL(1,ℤ/8)12yes—
591C23#57712yes—
592C23C23×D411yescharacteristic
593Q8—42no—
594Q8—42no—
595D4—216no—
596D4—24no—
597D4—24no—
598D4—24no—
599D4—44no—
600D4—44no—
601D4—88no—
602D4—84no—
603D4—162no—
604D4#55318yes—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—44no—
631C2×C4—416no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—416no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—84no—
656C2×C4—816no—
657C2×C4—816no—
658C2×C4—88no—
659C2×C4—164no—
660C2×C4#52512yes—
661C2×C4#52512yes—
662C2×C4#54812yes—
663C2×C4C2×AGL(1,ℤ/8)12yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4#57512yes—
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4112yes—
668C2×C4C2×H(5,𝔽2)12yes—
669C2×C4C2×2−1+411yescharacteristic
670C8—44no—
671C8—42no—
672C8—42no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—28no—
680C22—432no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22#337314yes—
695C22C2×#52511yescharacteristic
696C22C2×#54811yescharacteristic
697C22#348411yescharacteristic
698C22C22×AGL(1,ℤ/8)11yescharacteristic
699C22C2×#57511yescharacteristic
700C22C2×#57711yescharacteristic
701C22#358511yescharacteristic
702C4—22no—
703C4—22no—
704C4—22no—
705C4—44no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—41no—
711C4—82no—
712C4—81no—
713C4—84no—
714C4—81no—
715C4—81no—
716C4—81no—
717C4#311414yes—
718C4C2×#55312yes—
719C4#349212yes—
720C2—24no—
721C2—48no—
722C2—162no—
723C2#3658111yescharacteristic
724C2C2×#311411yescharacteristic
725C2#6058811yescharacteristic
726C2C2×#337311yescharacteristic
727C2#6131811yescharacteristic
728C2#6158211yescharacteristic
729C2#6160411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32902C2
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. #36587C2semidirect
2. C2×D4×C4⋊C4C2semidirect
3. #32908C2semidirect
5. #29336C2semidirect
6. #29235C2semidirect
7. #27973C2semidirect
8. #27680C2semidirect
9. #27568C2semidirect
10. #26180C2semidirect
11. #26107C2semidirect
12. #24731C2semidirect
13. #23001C2semidirect
14. #22762C2semidirect
15. #22629C2semidirect
16. #21452C2semidirect
17. C4×#450C2semidirect
30. #2571C22semidirect
31. #2548C22semidirect
33. #2513C22semidirect
35. #2447C22semidirect
37. #2366C22semidirect
40. #2355C22semidirect
54. #2101C22semidirect
55. #2099C22semidirect
56. #2096C22semidirect
57. #2088C22semidirect
58. #2071C22semidirect
59. #2065C22semidirect
60. #2042C22semidirect
61. #1957C22semidirect
62. #1948C22semidirect
63. #1939C22semidirect
64. #1936C22semidirect
65. #1935C22semidirect
66. #1920C22semidirect
67. #1917C22semidirect
80. #1864C22semidirect
83. #1834C22semidirect
85. #1803C22semidirect
90. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×D4×C4⋊C42
#329082
#329022
#293362
#292352
#279732
#276802
#275682
#261802
#261072
#247312
#230012
#227622
#226292
#214522
C4×#4502