Finite Group Databasebeta

← #6448186

Subgroups — #6448186

Order 512.

1934 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34955C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#26330C214yesmaximal, complemented
4#26328C214yesmaximal, complemented
5#26322C212yesmaximal, complemented
6#26320C212yesmaximal, complemented
7#26318C212yesmaximal, complemented
8#26316C212yesmaximal, complemented
9#23734C212yesmaximal, complemented
10#23630C212yesmaximal, complemented
11#21303C211yesmaximal, characteristic, complemented
12#21258C211yesmaximal, characteristic
13#21200C211yesmaximal, characteristic
14#21171C211yesmaximal, characteristic, complemented
15#21059C212yesmaximal, complemented
16#21059C212yesmaximal, complemented
17C4×#428C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4×Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C2×#428C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#428C2212yes—
24C2×#423C2211yescharacteristic
25C2×#417C2212yes—
26C2×#417C2212yes—
27C2×#417C2212yes—
28C2×#417C2212yes—
29C2×C4×M16C2211yescharacteristic
30#2592C2212yes—
31#2592C2212yes—
32#2591—22no—
33#2568C2211yescharacteristic
34#2568C2211yescharacteristic
35#2548C2211yescharacteristic
36#2548C2211yescharacteristic
37#2504—21no—
38#2436—21no—
39#2398—22no—
40Q8×C4⋊C4—22no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4—21no—
43Q8×C4⋊C4C2212yes—
44Q8×C4⋊C4C2212yes—
45Q8×C4⋊C4C2214yes—
46C4×#124—21no—
47C4×#124—21no—
48#2297C2212yes—
49#2297C2212yes—
50C2×#389—22no—
51C2×#389—21no—
52C2×#384C2212yes—
53C2×#384C2211yescharacteristic
54C4×#117C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2212yes—
57#2096C2214yes—
58#2095C2214yes—
59#2094C2214yes—
60#2093C2214yes—
61#2072C2214yes—
62#2072C2214yes—
63#2071C2214yes—
64#2071C2214yes—
65#1985C2214yes—
66#1984C2212yes—
67#1984C2212yes—
68#1983C2214yes—
69#1982C2212yes—
70#1981C2212yes—
71#1959C2214yes—
72#1957C2212yes—
73#1956C2212yes—
74#1948C2212yes—
75#1948C2212yes—
76#1947C2212yes—
77#1947C2212yes—
78#1928C2214yes—
79#1928C2214yes—
80#1927C2214yes—
81#1927C2214yes—
82#1926C2214yes—
83#1926C2214yes—
84#1925C2214yes—
85#1925C2214yes—
86#1776C2212yes—
87#1776C2212yes—
88C4×#107C2212yes—
89C4×#106C2212yes—
90#1736C2211yescharacteristic
91#1736C2212yes—
92#1736C2211yescharacteristic
93#1728C2212yes—
94#1728C2212yes—
95C22×#41C2312yes—
96C23×Q8D411yescharacteristic
97C22×M16C2311yescharacteristic
98C22×M16C2312yes—
99C22×M16C2311yescharacteristic
100#532—24no—
101#532—24no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104C2×#124—21no—
105C2×#124—21no—
106C2×#124C2312yes—
107C2×#121—21no—
108C2×#121—21no—
109C2×C4×Q8—24no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—24no—
113C2×C4×Q8—24no—
114C2×C4×Q8—48no—
115C2×C4×Q8D412yes—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×#117C2311yescharacteristic
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×C42C2312yes—
137C22×C42C2311yescharacteristic
138#428—24no—
139#428—24no—
140#428—44no—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146C2×#106C2314yes—
147#423—24no—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150#417—28no—
151#417—28no—
152#417—44no—
153#417—44no—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C4×M16—24no—
163C2×#97C2312yes—
164C2×C4×C8C2312yes—
165#400—22no—
166#400—22no—
167#397—22no—
168#397—22no—
169#394—22no—
170#394—22no—
171#393—22no—
172#393—22no—
173#391—21no—
174#391—21no—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#390—21no—
180#390—21no—
181#390C2312yes—
182#390C2312yes—
183#390C2312yes—
184#390C2312yes—
185#389—22no—
186#389—22no—
187#389—24no—
188#389—24no—
189#389—24no—
190#389—24no—
191#389—24no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#386—22no—
197#386—22no—
198#386—22no—
199#386—22no—
200#384—22no—
201#384—22no—
202#384—21no—
203#384—21no—
204#384C2314yes—
205#384C2314yes—
206#384C2312yes—
207#384C2312yes—
208#382—21no—
209#382—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228#376—21no—
229C2×#95—22no—
230C2×#95—21no—
231C43—21no—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#340C2312yes—
238#336C2314yes—
239#336C2314yes—
240#336C2314yes—
241#336C2314yes—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41—24no—
246C2×#41—24no—
247C2×#41—24no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8C2×D412yes—
252C22×Q8C2×D411yescharacteristic
253C22×Q8C2×D411yescharacteristic
254C22×Q8C2412yes—
255C22×D4—22no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4C2412yes—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D412yes—
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16—22no—
274C2×M16—24no—
275C2×M16—22no—
276C2×M16#4114yes—
277C22×C8C2412yes—
278C22×C8C2414yes—
279C22×C8C2412yes—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282#127—22no—
283#127—22no—
284#124—24no—
285#124—24no—
286#124—44no—
287#124—44no—
288#121—44no—
289#121—44no—
290C4×Q8—28no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8—416no—
300C4×Q8—48no—
301C4×Q8—48no—
302C4×D4—44no—
303C4×D4—44no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—22no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—48no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4—84no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4#4114yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—48no—
372C2×C42—44no—
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42#4114yes—
378C2×C42#4114yes—
379C2×C42C2412yes—
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—24no—
384#107—24no—
385#107—44no—
386#106—24no—
387#106—24no—
388#106—44no—
389C4⋊C8—24no—
390C4⋊C8—24no—
391#103—28no—
392#103—28no—
393#103—44no—
394#103—44no—
395#102—28no—
396#102—28no—
397#102—44no—
398#102—44no—
399#97—24no—
400C4×C8—24no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417C24—22no—
418C24C22×D411yescharacteristic
419#41—48no—
420#41—48no—
421#41—48no—
422#41—48no—
423C2×Q8—232no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8AGL(1,ℤ/8)14yes—
438C2×Q8#13714yes—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—81no—
453C2×D4—81no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4#12314yes—
485C22×C4#12314yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C42−1+412yes—
501C22×C4C2511yescharacteristic
502M16—28no—
503M16—44no—
504M16—48no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—22no—
513C2×C8C2×#4114yes—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—48no—
542C4⋊C4—44no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—84no—
546C4⋊C4—84no—
547C4⋊C4C2×#4114yes—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C42—24no—
561C42—24no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—44no—
567C42—48no—
568C42—42no—
569C42—42no—
570C42—48no—
571C42—44no—
572C42—42no—
573C42—42no—
574C42—81no—
575C42—81no—
576C42#12314yes—
577C42#12314yes—
578C42C2×#4112yes—
579C42C2×#4112yes—
580C42C2×#4112yes—
581C42C2×#4112yes—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23C2×#12312yes—
594C23#54812yes—
595C23#55012yes—
596C23C23×D411yescharacteristic
597Q8—216no—
598Q8—28no—
599Q8—28no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—84no—
609D4—88no—
610D4—84no—
611C2×C4—216no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—28no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—88no—
657C2×C4—84no—
658C2×C4—88no—
659C2×C4C2×#12312yes—
660C2×C4C2×#12312yes—
661C2×C4C2×#12312yes—
662C2×C4C2×#12312yes—
663C2×C4C2×#12312yes—
664C2×C4#52914yes—
665C2×C4#54214yes—
666C2×C4#54812yes—
667C2×C4#55012yes—
668C2×C4#55614yes—
669C2×C4#55914yes—
670C2×C4C2×AGL(1,ℤ/8)14yes—
671C2×C4C2×AGL(1,ℤ/8)12yes—
672C2×C4C2×#13714yes—
673C2×C4C2×#13712yes—
674C2×C4C22×#4111yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C22×#4111yescharacteristic
677C2×C4C22×#4111yescharacteristic
678C2×C4C2×H(5,𝔽2)11yescharacteristic
679C2×C4C2×2−1+411yescharacteristic
680C8—24no—
681C8—42no—
682C8—44no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22#244814yes—
697C22C22×#12311yescharacteristic
698C22C2×#54811yescharacteristic
699C22C2×#55011yescharacteristic
700C22C22×AGL(1,ℤ/8)11yescharacteristic
701C22C22×#13711yescharacteristic
702C22#358511yescharacteristic
703C22#358611yescharacteristic
704C4—22no—
705C4—24no—
706C4—24no—
707C4—22no—
708C4—22no—
709C4—42no—
710C4—42no—
711C4—44no—
712C4—81no—
713C4—81no—
714C4—82no—
715C4—81no—
716C4—84no—
717C4—82no—
718C4—81no—
719C4#244814yes—
720C4#309214yes—
721C4C2×#52911yescharacteristic
722C4C2×#54211yescharacteristic
723C4C2×#55611yescharacteristic
724C4C2×#55911yescharacteristic
725C2—24no—
726C2—162no—
727C2C2×#244811yescharacteristic
728C2C2×#309211yescharacteristic
729C2#6051711yescharacteristic
730C2#6131711yescharacteristic
731C2#6131911yescharacteristic
732C2#6140211yescharacteristic
733C2#6140411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #26330C2semidirect
4. #26328C2semidirect
5. #26322C2semidirect
6. #26320C2semidirect
7. #26318C2semidirect
8. #26316C2semidirect
9. #23734C2semidirect
10. #23630C2semidirect
11. #21303C2semidirect
14. #21171C2semidirect
15. #21059C2semidirect
16. #21059C2semidirect
17. C4×#428C2semidirect

Maximal subgroups

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

subgroupindex
#349552
C2×Q8×C4⋊C42
#263302
#263282
#263222
#263202
#263182
#263162
#237342
#236302
#213032
#212582
#212002
#211712
#210592
C4×#4282