Finite Group Databasebeta

← #6463836

Subgroups — #6463836

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3080C211yesmaximal, characteristic
2C22×#479C211yesmaximal, characteristic, complemented
3#34217C211yesmaximal, characteristic, complemented
4C4×#536C211yesmaximal, characteristic
5#28216C214yesmaximal, complemented
6#28215C214yesmaximal, complemented
7#28209C214yesmaximal, complemented
8#28208C214yesmaximal, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23660C211yesmaximal, characteristic, complemented
11#23512C211yesmaximal, characteristic
12#23512C211yesmaximal, characteristic
13#21068C211yesmaximal, characteristic
14#21067C211yesmaximal, characteristic
15#21064C211yesmaximal, characteristic, complemented
16#21064C211yesmaximal, characteristic, complemented
17#20755C211yesmaximal, characteristic, complemented
18#20647C211yesmaximal, characteristic, complemented
19#20645C211yesmaximal, characteristic
20C2×#536C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24#3080—28no—
25C2×#479—28no—
26C2×#479C2214yes—
27C2×#479C2212yes—
28C2×#479C2214yes—
29C2×#479C2212yes—
30C2×#478C2212yes—
31C2×#478C2212yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C2×#423C2211yescharacteristic
37C22×C4⋊C8C2211yescharacteristic
38C2×#417C2211yescharacteristic
39C2×#417C2211yescharacteristic
40C2×#419C2211yescharacteristic
41C2×#419C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C22×#103C2211yescharacteristic
46#2370—22no—
47#2370—22no—
48#2345—22no—
49#2345—22no—
50#2330C2214yes—
51#2327C2212yes—
52#2326C2212yes—
53#2323—22no—
54#2322—22no—
55#2317C2211yescharacteristic
56#2316C2211yescharacteristic
57#2316C2211yescharacteristic
58#2311C2211yescharacteristic
59#2310C2211yescharacteristic
60C4×C4⋊Q8C2212yes—
61C4×#125C2214yes—
62C4×#124C2212yes—
63C4×C22⋊Q8—22no—
64C4×#123—22no—
65#2285C2211yescharacteristic
66#2274C2211yescharacteristic
67#2274C2211yescharacteristic
68C4×#117C2211yescharacteristic
69#2085C2214yes—
70#2085C2214yes—
71#2084C2214yes—
72#2083C2214yes—
73#2043C2214yes—
74#2043C2214yes—
75#2042C2214yes—
76#2042C2214yes—
77#1928C2214yes—
78#1928C2214yes—
79#1928C2214yes—
80#1927C2214yes—
81#1927C2214yes—
82#1927C2214yes—
83#1926C2214yes—
84#1925C2214yes—
85#1832C2214yes—
86#1831C2214yes—
87#1805C2214yes—
88#1805C2212yes—
89#1804C2212yes—
90#1803C2214yes—
91#1803C2212yes—
92#1802C2212yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#536—24no—
108#536—24no—
109C2×C4⋊Q8C2312yes—
110C2×#125—21no—
111C2×#125—21no—
112C2×#125—24no—
113C2×#125D418yes—
114C2×#125C2312yes—
115C2×#125C2312yes—
116C2×#124C2312yes—
117C2×#123—22no—
118C2×C22⋊Q8—22no—
119C4×#41—24no—
120C4×#41—24no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×#117—21no—
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C42—21no—
146C22×C42—21no—
147C22×C42C2311yescharacteristic
148#479—216no—
149#479—216no—
150#479—28no—
151#479—28no—
152#478—28no—
153#478—28no—
154#428—24no—
155#428—24no—
156C2×#107—24no—
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#107C2314yes—
160C2×#106—24no—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×#106C2314yes—
164#423—24no—
165C2×C4⋊C8—24no—
166C2×C4⋊C8C2312yes—
167C2×C4⋊C8C2312yes—
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2312yes—
170#419—24no—
171#419—24no—
172#417—24no—
173#417—24no—
174#416—24no—
175#416—24no—
176C2×#103—24no—
177C2×#103—24no—
178C2×#103C2314yes—
179C2×#103C2314yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190#391—22no—
191#391C2312yes—
192#391C2312yes—
193#391C2312yes—
194#390C2312yes—
195#389—22no—
196#389—22no—
197#389—22no—
198#389—24no—
199#389—24no—
200#389—24no—
201#389C2314yes—
202#389C2314yes—
203#388—24no—
204#388—24no—
205#387—24no—
206#387—24no—
207#387—24no—
208#387—24no—
209#387—24no—
210#387—24no—
211#386—22no—
212#385—22no—
213#385—24no—
214#384C2312yes—
215#384C2312yes—
216#382C2312yes—
217#382C2312yes—
218#381—22no—
219#380—21no—
220#380—21no—
221#380—22no—
222#380—22no—
223#380—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—24no—
227C4×C4⋊C4—24no—
228C4×C4⋊C4—24no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4D412yes—
232C4×C4⋊C4D412yes—
233C4×C4⋊C4C2314yes—
234C4×C4⋊C4C2314yes—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—24no—
238C4×C22⋊C4D412yes—
239C4×C22⋊C4D412yes—
240#376C2312yes—
241C43C2312yes—
242#340C2312yes—
243#340C2312yes—
244#340C2312yes—
245#340C2312yes—
246#340C2314yes—
247#340C2314yes—
248#336C2314yes—
249#336C2314yes—
250#336C2314yes—
251#336C2314yes—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41#4114yes—
258C22×Q8—21no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—22no—
264C22×Q8#4114yes—
265C22×Q8C2412yes—
266C22×Q8C2411yescharacteristic
267C22×D4—21no—
268C22×D4C2411yescharacteristic
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2411yescharacteristic
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C4⋊Q8—24no—
296C4⋊Q8—24no—
297#125—28no—
298#125—28no—
299#125—24no—
300#125—24no—
301#125C2×D418yes—
302#125C2×D418yes—
303#124—24no—
304#124—24no—
305#123—216no—
306C22⋊Q8—216no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—44no—
316C4×Q8—44no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—44no—
322C4×D4—44no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—44no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2×D414yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—24no—
386C2×C42—24no—
387C2×C42—44no—
388C2×C42—44no—
389C2×C42C2×D414yes—
390C2×C42C2×D414yes—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D414yes—
394C2×C42#4114yes—
395C2×C42C2412yes—
396C2×C42C2412yes—
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399#107—28no—
400#107—24no—
401#107—24no—
402#106—28no—
403#106—24no—
404#106—24no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407C4⋊C8—24no—
408C4⋊C8—24no—
409#103—24no—
410#103—24no—
411#103—28no—
412#103—28no—
413#103—28no—
414#103—28no—
415#103—24no—
416#103—24no—
417#102—24no—
418#102—24no—
419#102—24no—
420#102—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—22no—
426#95—22no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95—24no—
431#95C2×D414yes—
432#95C2×D414yes—
433C24—21no—
434C24C22×D411yescharacteristic
435#41—28no—
436#41—28no—
437#41—44no—
438#41—44no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—28no—
443C2×Q8—28no—
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—44no—
455C2×Q8—44no—
456C2×Q8C2×#4114yes—
457C2×Q8C2×#4112yes—
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4C2×#4112yes—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4#12114yes—
495C22×C4#12114yes—
496C22×C4C22×D414yes—
497C22×C4C22×D414yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C2×#4112yes—
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C22×C42−1+412yes—
508C22×C42−1+412yes—
509C22×C4C2511yescharacteristic
510M16—44no—
511M16—44no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—44no—
525C2×C8—44no—
526C4⋊C4—216no—
527C4⋊C4—28no—
528C4⋊C4—28no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—84no—
555C4⋊C4—84no—
556C4⋊C4—84no—
557C4⋊C4C22×D414yes—
558C4⋊C4C22×D414yes—
559C22⋊C4—216no—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C22⋊C4—84no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—44no—
573C42—44no—
574C42—42no—
575C42—42no—
576C42—42no—
577C42—44no—
578C42—44no—
579C42—42no—
580C42—42no—
581C42—84no—
582C42—84no—
583C42#12114yes—
584C42#12114yes—
585C42C22×D412yes—
586C42C22×D412yes—
587C42C2×#4112yes—
588C42C2×#4112yes—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—42no—
593C23—42no—
594C23—41no—
595C23—41no—
596C23C2×#12112yes—
597C23#52512yes—
598C23#53612yes—
599C23C23×D411yescharacteristic
600Q8—24no—
601Q8—22no—
602Q8—24no—
603Q8—22no—
604Q8—42no—
605Q8—41no—
606Q8—42no—
607Q8—41no—
608Q8—48no—
609D4—22no—
610D4—22no—
611D4—48no—
612D4—41no—
613D4—41no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—48no—
637C2×C4—48no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4C2×#12112yes—
656C2×C4C2×#12112yes—
657C2×C4C2×#12112yes—
658C2×C4C2×#12112yes—
659C2×C4C2×#12112yes—
660C2×C4#52512yes—
661C2×C4#53612yes—
662C2×C4#54814yes—
663C2×C4#54814yes—
664C2×C4#55214yes—
665C2×C4#55214yes—
666C2×C4C23×D411yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C2×2−1+411yescharacteristic
671C2×C4C2×2−1+411yescharacteristic
672C8—42no—
673C8—42no—
674C8—42no—
675C8—42no—
676C22—22no—
677C22—22no—
678C22—24no—
679C22—24no—
680C22—42no—
681C22—44no—
682C22—81no—
683C22—81no—
684C22—82no—
685C22#253614yes—
686C22C22×#12111yescharacteristic
687C22C2×#52511yescharacteristic
688C22C2×#53611yescharacteristic
689C22#358412yes—
690C22#358612yes—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—44no—
696C4—44no—
697C4—42no—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—82no—
702C4—81no—
703C4—81no—
704C4—82no—
705C4—82no—
706C4#253614yes—
707C4C2×#54811yescharacteristic
708C4C2×#54811yescharacteristic
709C4C2×#55211yescharacteristic
710C4C2×#55211yescharacteristic
711C2—24no—
712C2—82no—
713C2C2×#253611yescharacteristic
714C2#6042211yescharacteristic
715C2#6042711yescharacteristic
716C2#6059812yes—
717C2#6089112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#479C2semidirect
3. #34217C2semidirect
5. #28216C2semidirect
6. #28215C2semidirect
7. #28209C2semidirect
8. #28208C2semidirect
9. #23661C2semidirect
10. #23660C2semidirect
15. #21064C2semidirect
16. #21064C2semidirect
17. #20755C2semidirect
18. #20647C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30802
C22×#4792
#342172
C4×#5362
#282162
#282152
#282092
#282082
#236612
#236602
#235122
#210682
#210672
#210642
#207552
#206472
#206452