Finite Group Databasebeta

← #6447971

Subgroups — #6447971

Order 512.

1856 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34952C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#25004C214yesmaximal, complemented
4#25002C214yesmaximal, complemented
5#24996C212yesmaximal, complemented
6#24994C212yesmaximal, complemented
7#24992C212yesmaximal, complemented
8#24990C212yesmaximal, complemented
9#23745C212yesmaximal, complemented
10#23630C212yesmaximal
11#21303C211yesmaximal, characteristic, complemented
12#21170C211yesmaximal, characteristic, complemented
13#21059C212yesmaximal
14#21033C212yesmaximal, complemented
15#20764C211yesmaximal, characteristic, complemented
16#20648C211yesmaximal, characteristic
17C4×#417C211yesmaximal, characteristic
18C2×C4×#41C2211yescharacteristic
19C22×C4×Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C2×#428C2212yes—
22C2×#423C2211yescharacteristic
23C2×#417C2211yescharacteristic
24C2×#417C2212yes—
25C2×#417C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27#2582C2211yescharacteristic
28#2582C2211yescharacteristic
29#2581C2212yes—
30#2580C2212yes—
31#2579—22no—
32#2538C2211yescharacteristic
33#2538C2211yescharacteristic
34#2487—21no—
35#2409—21no—
36#2396—22no—
37Q8×C4⋊C4—22no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4C2212yes—
41Q8×C4⋊C4C2212yes—
42Q8×C4⋊C4C2214yes—
43C4×C22⋊Q8—21no—
44C4×C22⋊Q8—21no—
45#2297C2212yes—
46#2296C2212yes—
47C2×#389—22no—
48C2×#389—21no—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#384C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2100C2214yes—
55#2100C2214yes—
56#2099C2214yes—
57#2099C2214yes—
58#2072C2214yes—
59#2072C2214yes—
60#2071C2214yes—
61#2071C2214yes—
62#1985C2214yes—
63#1984C2212yes—
64#1984C2212yes—
65#1949C2214yes—
66#1948C2212yes—
67#1947C2212yes—
68#1928C2214yes—
69#1927C2214yes—
70#1926C2214yes—
71#1925C2214yes—
72#1921C2214yes—
73#1920C2214yes—
74#1919C2214yes—
75#1918C2214yes—
76#1836C2214yes—
77#1834C2212yes—
78#1834C2212yes—
79#1803C2212yes—
80#1803C2212yes—
81#1802C2212yes—
82#1802C2212yes—
83#1763C2212yes—
84#1762C2212yes—
85C4×#103C2212yes—
86C4×#102C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1736C2211yescharacteristic
90#1736C2212yes—
91#1736C2212yes—
92#1736C2211yescharacteristic
93#1728C2212yes—
94#1728C2212yes—
95C22×#41C2311yescharacteristic
96C23×Q8D411yescharacteristic
97C22×M16C2311yescharacteristic
98C22×M16C2311yescharacteristic
99C22×M16C2312yes—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C4×#41—24no—
103C4×#41—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—48no—
110C2×C4×Q8D412yes—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D412yes—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C42—22no—
135C22×C42—21no—
136C22×C42—21no—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#428—28no—
140C2×#107C2314yes—
141C2×#106C2314yes—
142#423—24no—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145#417—28no—
146#417—24no—
147#417—24no—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C4×M16—24no—
155C2×#97C2312yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#399—22no—
159#398—22no—
160#398—22no—
161#396—22no—
162#396—22no—
163#395—22no—
164#393—22no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#390C2312yes—
170#390C2312yes—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389—24no—
176#389—24no—
177#389—24no—
178#389—24no—
179#389—24no—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384—22no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384C2312yes—
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2314yes—
201#384C2314yes—
202#382C2312yes—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C2×#95—22no—
222C2×#95—21no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41#4114yes—
245C22×Q8—24no—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8C2×D412yes—
249C22×Q8C2×D411yescharacteristic
250C22×Q8C2×D411yescharacteristic
251C22×Q8C2411yescharacteristic
252C22×D4—21no—
253C22×D4C2411yescharacteristic
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—24no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—22no—
273C2×M16—22no—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2414yes—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—28no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—416no—
293C4×Q8—48no—
294C4×Q8—48no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—44no—
300C4×D4—44no—
301#117—22no—
302#117—22no—
303#117—24no—
304#117—24no—
305#117—44no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—48no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—82no—
337C2×C4⋊C4—84no—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4#4114yes—
346C2×C4⋊C4C2414yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—22no—
371C2×C42—48no—
372C2×C42—44no—
373C2×C42—44no—
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42#4114yes—
379C2×C42#4114yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—28no—
385#106—28no—
386C4⋊C8—24no—
387C4⋊C8—24no—
388#103—28no—
389#103—24no—
390#103—24no—
391#102—28no—
392#102—24no—
393#102—24no—
394#97—24no—
395C4×C8—24no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410C24—21no—
411C24C22×D411yescharacteristic
412#41—28no—
413#41—28no—
414#41—44no—
415#41—44no—
416C2×Q8—232no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8AGL(1,ℤ/8)14yes—
430C2×Q8#13714yes—
431C2×Q8C2×#4112yes—
432C2×D4—24no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C2×#4112yes—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4#12314yes—
478C22×C4#12314yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C42−1+412yes—
493C22×C42−1+412yes—
494C22×C4C2511yescharacteristic
495M16—48no—
496M16—44no—
497M16—44no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—22no—
506C2×C8—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—44no—
529C4⋊C4—48no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—48no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—81no—
541C4⋊C4—81no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—84no—
547C4⋊C4—84no—
548C4⋊C4C2×#4114yes—
549C22⋊C4—42no—
550C22⋊C4—44no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C42—24no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—44no—
564C42—48no—
565C42—42no—
566C42—42no—
567C42—42no—
568C42—44no—
569C42—48no—
570C42—44no—
571C42#12314yes—
572C42#12314yes—
573C42C2×#4112yes—
574C42C2×#4112yes—
575C42C2×#4112yes—
576C42C2×#4112yes—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—22no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23C2×#12312yes—
589C23#54812yes—
590C23#54812yes—
591C23C23×D411yescharacteristic
592Q8—216no—
593Q8—28no—
594Q8—28no—
595Q8—22no—
596Q8—22no—
597Q8—41no—
598Q8—41no—
599D4—22no—
600D4—22no—
601D4—48no—
602D4—41no—
603D4—41no—
604C2×C4—216no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—28no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—44no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—84no—
651C2×C4—88no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4C2×#12312yes—
655C2×C4C2×#12312yes—
656C2×C4C2×#12312yes—
657C2×C4C2×#12312yes—
658C2×C4C2×#12312yes—
659C2×C4#52914yes—
660C2×C4#54114yes—
661C2×C4#54812yes—
662C2×C4#54812yes—
663C2×C4#55214yes—
664C2×C4#55614yes—
665C2×C4C2×AGL(1,ℤ/8)14yes—
666C2×C4C2×AGL(1,ℤ/8)12yes—
667C2×C4C2×#13714yes—
668C2×C4C2×#13712yes—
669C2×C4C22×#4111yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C2×2−1+411yescharacteristic
674C2×C4C2×2−1+411yescharacteristic
675C8—44no—
676C8—42no—
677C8—42no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22#242114yes—
692C22C22×#12311yescharacteristic
693C22C2×#54811yescharacteristic
694C22C2×#54811yescharacteristic
695C22C22×AGL(1,ℤ/8)11yescharacteristic
696C22C22×#13711yescharacteristic
697C22#358511yescharacteristic
698C22#358611yescharacteristic
699C4—22no—
700C4—24no—
701C4—24no—
702C4—22no—
703C4—22no—
704C4—42no—
705C4—42no—
706C4—44no—
707C4—42no—
708C4—81no—
709C4—81no—
710C4—82no—
711C4—82no—
712C4—84no—
713C4—82no—
714C4#242114yes—
715C4#309214yes—
716C4C2×#52911yescharacteristic
717C4C2×#54111yescharacteristic
718C4C2×#55211yescharacteristic
719C4C2×#55611yescharacteristic
720C2—24no—
721C2—82no—
722C2C2×#242111yescharacteristic
723C2C2×#309211yescharacteristic
724C2#6051711yescharacteristic
725C2#6131711yescharacteristic
726C2#6131911yescharacteristic
727C2#6132111yescharacteristic
728C2#6132211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
3. #25004C2semidirect
4. #25002C2semidirect
5. #24996C2semidirect
6. #24994C2semidirect
7. #24992C2semidirect
8. #24990C2semidirect
9. #23745C2semidirect
11. #21303C2semidirect
12. #21170C2semidirect
14. #21033C2semidirect
15. #20764C2semidirect

Maximal subgroups

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

subgroupindex
#349522
C2×Q8×C4⋊C42
#250042
#250022
#249962
#249942
#249922
#249902
#237452
#236302
#213032
#211702
#210592
#210332
#207642
#206482
C4×#4172