Finite Group Databasebeta

← #6565763

Subgroups — #6565763

Order 512.

1975 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3367C212yesmaximal, complemented
2#36594C211yesmaximal, characteristic, complemented
3#34311C211yesmaximal, characteristic, complemented
4#29800C212yesmaximal
5#29798C211yesmaximal, characteristic, complemented
6#28753C212yesmaximal, complemented
7#28748C211yesmaximal, characteristic, complemented
8#28206C212yesmaximal, complemented
9#28201C211yesmaximal, characteristic, complemented
10#25286C214yesmaximal, complemented
11#25275C212yesmaximal, complemented
12#24640C214yesmaximal, complemented
13#24627C212yesmaximal, complemented
14#23683C212yesmaximal, complemented
15#23676C211yesmaximal, characteristic
16#22736C211yesmaximal, characteristic, complemented
17#22544C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×D4×Q8C2211yescharacteristic
20C2×#519—21no—
21#3367—216no—
22C2×#489C2214yes—
23C2×#487C2212yes—
24C2×#477C2214yes—
25C2×#474C2212yes—
26C2×#463C2214yes—
27C2×#461C2212yes—
28C2×#446C2212yes—
29C2×#440C2214yes—
30C2×C4×D8C2212yes—
31#2868C2211yescharacteristic
32#2858C2212yes—
33#2547C2212yes—
34#2545C2214yes—
35#2538—21no—
36#2538—21no—
37#2512—22no—
38#2512—22no—
39#2365C2211yescharacteristic
40#2365C2212yes—
41#2362—21no—
42#2362—22no—
43#2354—21no—
44#2354—21no—
45#2354C2211yescharacteristic
46#2354C2212yes—
47Q8×C4⋊C4C2211yescharacteristic
48Q8×C22⋊C4—21no—
49#2323—21no—
50#2323—22no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2212yes—
53C4×C4⋊Q8C2212yes—
54C4×#127C2211yescharacteristic
55#2293C2211yescharacteristic
56#2293C2212yes—
57C42×Q8C2211yescharacteristic
58#2083C2212yes—
59#2083C2211yescharacteristic
60#2071C2211yescharacteristic
61#2068C2214yes—
62#2067C2212yes—
63#2055C2214yes—
64#2054C2212yes—
65#2043C2212yes—
66#2043C2212yes—
67#2043C2212yes—
68#2043C2212yes—
69#2042C2214yes—
70#2041C2212yes—
71#2041C2211yescharacteristic
72#2013C2212yes—
73#2013C2212yes—
74#2012C2214yes—
75#2011C2211yescharacteristic
76#1983C2214yes—
77#1983C2214yes—
78#1983C2214yes—
79#1981C2212yes—
80#1981C2214yes—
81#1959C2214yes—
82#1956C2212yes—
83#1940C2214yes—
84#1939C2212yes—
85#1928C2214yes—
86#1928C2214yes—
87#1925C2214yes—
88#1831C2211yescharacteristic
89#1831C2212yes—
90#1803C2214yes—
91#1802C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1763C2214yes—
95C4×#102C2212yes—
96C23×Q8—21no—
97C22×SD16C2314yes—
98C22×D8C2312yes—
99#556—28no—
100#556—28no—
101D4×Q8—24no—
102D4×Q8D418yescomplemented
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127C2311yescharacteristic
109C2×#126—24no—
110C2×#124C2314yes—
111C2×#124C2314yes—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—22no—
115#519—48no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8—21no—
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8C2314yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×#117—22no—
131C2×#117—21no—
132C2×#117—22no—
133C2×#117—22no—
134C2×#117—21no—
135#489—216no—
136#487—28no—
137#477—216no—
138#474—28no—
139#463—216no—
140#461—28no—
141#446—28no—
142#440—216no—
143C4×D8—28no—
144C2×#107C2312yes—
145C2×#107C2314yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#97C2314yes—
164C2×C4×C8C2312yes—
165#397—24no—
166#396—24no—
167#396—24no—
168#391—21no—
169#391—22no—
170#391—21no—
171#391—22no—
172#391C2314yes—
173#390—21no—
174#390—21no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390C2312yes—
180#390C2311yescharacteristic
181#390C2312yes—
182#390C2311yescharacteristic
183#389—21no—
184#389—22no—
185#389—22no—
186#389—21no—
187#389—22no—
188#389—22no—
189#389C2312yes—
190#389C2311yescharacteristic
191#389C2311yescharacteristic
192#389C2312yes—
193#388—22no—
194#388—21no—
195#388—22no—
196#388—21no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—24no—
201#388—24no—
202#387—22no—
203#387—22no—
204#387—22no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384—21no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2312yes—
212#380—21no—
213#380—22no—
214#380—22no—
215#380—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D412yes—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D412yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2312yes—
231C4×C22⋊C4—21no—
232C4×C22⋊C4—22no—
233C43C2311yescharacteristic
234C43C2312yes—
235#340C2312yes—
236#340C2314yes—
237#340C2312yes—
238#336C2311yescharacteristic
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2312yes—
242#336C2314yes—
243C22×Q8—28no—
244C22×Q8—22no—
245C22×Q8—21no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2414yes—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4C2412yes—
253C22×D4C2411yescharacteristic
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—22no—
257C2×SD16—28no—
258C2×SD16—44no—
259C2×SD16—44no—
260C2×D8—24no—
261C2×D8—42no—
262C2×D8—42no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—22no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8C2×D414yes—
273C4⋊Q8C2×D418yes—
274#127—42no—
275#126—416no—
276#124—28no—
277#124—28no—
278#124—44no—
279#124—44no—
280C22⋊Q8—28no—
281C22⋊Q8—216no—
282C22⋊Q8—48no—
283C4×Q8—24no—
284C4×Q8—28no—
285C4×Q8—28no—
286C4×Q8—42no—
287C4×Q8—48no—
288C4×Q8—44no—
289C4×Q8—48no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8C2×D414yes—
293C4×Q8#4114yes—
294C4×D4—24no—
295C4×D4—22no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4C2×D414yes—
301C4×D4C2×D418yes—
302#117—48no—
303#117—44no—
304#117—48no—
305#117—48no—
306#117—44no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—24no—
359C2×C42C2×D412yes—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372#107—24no—
373#107—28no—
374C4⋊C8—24no—
375C4⋊C8—28no—
376C4⋊C8—44no—
377C4⋊C8—42no—
378#103—28no—
379#103—28no—
380#103—28no—
381#103—44no—
382#103—44no—
383#103—44no—
384#103—44no—
385#102—24no—
386#102—24no—
387#102—28no—
388#102—42no—
389#102—44no—
390#102—42no—
391#97—28no—
392C4×C8—24no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—21no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407C24—21no—
408C24—22no—
409C2×Q8—28no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—432no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8#12114yes—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C22×D412yes—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2511yescharacteristic
468SD16—416no—
469D8—48no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—22no—
479C4⋊C4—28no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—82no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4#12114yes—
510C4⋊C4#12118yes—
511C4⋊C4C22×D412yes—
512C4⋊C4C22×D414yes—
513C22⋊C4—28no—
514C22⋊C4—24no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—82no—
525C22⋊C4—81no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—24no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—42no—
536C42—41no—
537C42—42no—
538C42—44no—
539C42—44no—
540C42—44no—
541C42—48no—
542C42—44no—
543C42—42no—
544C42AGL(1,ℤ/8)14yes—
545C42C22×D412yes—
546C42C22×D414yes—
547C42C2×#4112yes—
548C42C2×#4114yes—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23C23×D411yescharacteristic
564Q8—28no—
565Q8—416no—
566Q8—48no—
567Q8—44no—
568Q8—44no—
569Q8—44no—
570Q8—44no—
571D4—24no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—84no—
578D4—84no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—416no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—88no—
624C2×C4C2×#12114yes—
625C2×C4C2×#12112yes—
626C2×C4C2×#12114yes—
627C2×C4C2×#12112yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4C2×AGL(1,ℤ/8)12yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4#57512yes—
632C2×C4#57812yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4#58512yes—
636C2×C4#58512yes—
637C8—42no—
638C8—42no—
639C8—44no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22C22×#12111yescharacteristic
655C22#348711yescharacteristic
656C22#350011yescharacteristic
657C22C22×AGL(1,ℤ/8)11yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57811yescharacteristic
660C22#358611yescharacteristic
661C4—24no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—24no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—44no—
671C4—42no—
672C4—81no—
673C4—84no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4#303614yes—
678C4#348712yes—
679C4#352412yes—
680C2—82no—
681C2—162no—
682C2#3601811yescharacteristic
683C2C2×#303611yescharacteristic
684C2#6043211yescharacteristic
685C2#6188811yescharacteristic
686C2#6190511yescharacteristic
687C2#6253811yescharacteristic
688C2#6258111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2011C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3367C2semidirect
2. #36594C2semidirect
3. #34311C2semidirect
5. #29798C2semidirect
6. #28753C2semidirect
7. #28748C2semidirect
8. #28206C2semidirect
9. #28201C2semidirect
10. #25286C2semidirect
11. #25275C2semidirect
12. #24640C2semidirect
13. #24627C2semidirect
14. #23683C2semidirect
16. #22736C2semidirect
17. #22544C2semidirect
102. D4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33672
#365942
#343112
#298002
#297982
#287532
#287482
#282062
#282012
#252862
#252752
#246402
#246272
#236832
#236762
#227362
#225442