Finite Group Databasebeta

← #6481777

Subgroups — #6481777

Order 512.

1879 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic, complemented
2C2×#3108C211yesmaximal, characteristic
3#35241C211yesmaximal, characteristic
4C2×#2355C211yesmaximal, characteristic, complemented
5#33111C212yesmaximal
6#33108C212yesmaximal, complemented
7#29719C212yesmaximal
8#29715C212yesmaximal, complemented
9#24691C214yesmaximal, complemented
10#24683C212yesmaximal, complemented
11#24678C212yesmaximal, complemented
12#24641C214yesmaximal, complemented
13#24632C212yesmaximal, complemented
14#24629C212yesmaximal, complemented
15#21953C211yesmaximal, characteristic, complemented
16#21867C211yesmaximal, characteristic
17#21860C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3113—28no—
22#3108—28no—
23C2×#483C2211yescharacteristic
24C2×#483C2212yes—
25C2×#480C2211yescharacteristic
26C2×#479C2212yes—
27C2×#475C2212yes—
28C2×#464C2212yes—
29C2×#462C2212yes—
30C2×#451C2211yescharacteristic
31C2×#451C2212yes—
32C2×#448C2211yescharacteristic
33C2×#441C2214yes—
34C2×C4×Q16C2212yes—
35C2×C4×SD16C2212yes—
36C2×#432C2211yescharacteristic
37C2×#431C2211yescharacteristic
38#2857C2212yes—
39#2850C2212yes—
40#2664—22no—
41#2568C2211yescharacteristic
42#2565C2212yes—
43#2547C2211yescharacteristic
44#2546C2211yescharacteristic
45#2545C2212yes—
46#2542C2211yescharacteristic
47#2498—22no—
48#2417—21no—
49#2417—21no—
50#2355—21no—
51#2355—21no—
52#2355—22no—
53#2355C2212yes—
54#2355C2212yes—
55#2355C2214yes—
56#2323—21no—
57#2323—21no—
58C2×#391C2212yes—
59C2×#387—22no—
60C2×#384C2211yescharacteristic
61C2×#382—21no—
62C2×C4×C4⋊C4—21no—
63C2×C4×C4⋊C4C2212yes—
64C2×C4×C4⋊C4C2211yescharacteristic
65#2074C2214yes—
66#2072C2214yes—
67#2056C2214yes—
68#2056C2214yes—
69#2056C2214yes—
70#2055C2214yes—
71#2016C2214yes—
72#2015C2214yes—
73#1992C2212yes—
74#1990C2212yes—
75#1983C2214yes—
76#1981C2214yes—
77#1959C2212yes—
78#1956C2212yes—
79#1949C2214yes—
80#1948C2214yes—
81#1940C2214yes—
82#1939C2212yes—
83#1939C2212yes—
84#1928C2214yes—
85#1927C2214yes—
86#1927C2214yes—
87#1926C2214yes—
88#1874C2212yes—
89#1872C2212yes—
90#1872C2212yes—
91#1872C2212yes—
92#1803C2212yes—
93#1803C2214yes—
94#1802C2212yes—
95#1797C2212yes—
96#1797C2212yes—
97C23×Q8C2311yescharacteristic
98C22×Q16C2312yes—
99C22×Q16C2314yes—
100C22×SD16C2312yes—
101#533—24no—
102#533—24no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106#529—24no—
107#529—24no—
108C2×#126—21no—
109C2×#125—21no—
110C2×#125C2311yescharacteristic
111C2×#124C2311yescharacteristic
112C2×#124C2312yes—
113C2×#123—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2311yescharacteristic
119C2×#121C2311yescharacteristic
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—24no—
124C2×C4×Q8D418yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—21no—
130C2×C4×D4C2311yescharacteristic
131C2×#117—21no—
132C2×#117—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4D411yescharacteristic
137C22×C4⋊C4D411yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C42—21no—
141C22×C42—22no—
142C22×C42C2312yes—
143C22×C42C2311yescharacteristic
144#483—24no—
145#483—28no—
146#480—24no—
147#479—28no—
148#475—28no—
149#464—28no—
150#462—28no—
151#451—24no—
152#451—28no—
153#448—24no—
154#441—216no—
155C4×Q16—28no—
156C4×SD16—28no—
157#432—24no—
158#431—24no—
159C2×#107C2312yes—
160C2×#106C2314yes—
161C2×#106C2312yes—
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2314yes—
165C2×C4⋊C8C2314yes—
166C2×#103C2314yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2314yes—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#102C2312yes—
175C2×#102C2314yes—
176C2×C22⋊C8C2311yescharacteristic
177C2×C22⋊C8C2311yescharacteristic
178C2×C22⋊C8C2312yes—
179C2×#97C2312yes—
180C2×C4×C8C2312yes—
181#400—22no—
182#399—22no—
183#398—22no—
184#398—22no—
185#398—22no—
186#397—22no—
187#396—22no—
188#396—22no—
189#391—21no—
190#391—21no—
191#391—22no—
192#391—22no—
193#391C2312yes—
194#391C2312yes—
195#391C2314yes—
196#391C2312yes—
197#391C2314yes—
198#390C2312yes—
199#389—21no—
200#389—21no—
201#389—22no—
202#389—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#387—22no—
208#387—22no—
209#387—24no—
210#387—24no—
211#387—24no—
212#387—24no—
213#386—22no—
214#386—22no—
215#384—21no—
216#384—21no—
217#384C2312yes—
218#384C2312yes—
219#384C2312yes—
220#382—22no—
221#382—22no—
222#382—24no—
223#382C2312yes—
224#380—22no—
225#380—22no—
226#380—21no—
227#380—21no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—24no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4D411yescharacteristic
234C4×C4⋊C4D411yescharacteristic
235C4×C4⋊C4D412yes—
236C4×C4⋊C4C2314yes—
237C4×C4⋊C4C2314yes—
238C4×C4⋊C4C2312yes—
239C4×C4⋊C4C2311yescharacteristic
240C4×C4⋊C4C2311yescharacteristic
241C2×#95—22no—
242C2×#95—22no—
243C2×#95—22no—
244C2×#95D412yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2314yes—
250#336C2314yes—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—22no—
254C22×Q8—42no—
255C22×Q8—42no—
256C22×Q8C2×D414yes—
257C22×Q8C2411yescharacteristic
258C22×Q8C2412yes—
259C22×Q8C2412yes—
260C22×D4—21no—
261C22×D4C2411yescharacteristic
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—22no—
265C23×C4—21no—
266C23×C4C2×D412yes—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×Q16—24no—
272C2×Q16—28no—
273C2×Q16—44no—
274C2×Q16—42no—
275C2×Q16—42no—
276C2×Q16—44no—
277C2×SD16—24no—
278C2×SD16—42no—
279C2×SD16—42no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2414yes—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—44no—
288#126—44no—
289#125—22no—
290#125—22no—
291#125—44no—
292#124—22no—
293#124—22no—
294#124—44no—
295#123—44no—
296C22⋊Q8—22no—
297C22⋊Q8—24no—
298C22⋊Q8—22no—
299C22⋊Q8—24no—
300C22⋊Q8—44no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303#121—22no—
304#121—22no—
305C4×Q8—28no—
306C4×Q8—28no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—22no—
310C4×Q8—24no—
311C4×Q8—22no—
312C4×Q8—24no—
313C4×Q8—44no—
314C4×Q8C2×D418yes—
315C4×Q8C2×D418yes—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—44no—
319#117—44no—
320#117—44no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—82no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C42—28no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—24no—
377C2×C42—24no—
378C2×C42—44no—
379C2×C42—84no—
380C2×C42C2×D412yes—
381C2×C42C2×D412yes—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D414yes—
385C2×C42#4114yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#107—24no—
391#106—24no—
392#106—28no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—44no—
396C4⋊C8—44no—
397#103—24no—
398#103—24no—
399#103—28no—
400#103—24no—
401#103—28no—
402#103—44no—
403#103—44no—
404#103—44no—
405#102—24no—
406#102—44no—
407C22⋊C8—22no—
408C22⋊C8—22no—
409C22⋊C8—24no—
410#97—24no—
411C4×C8—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95C2×D412yes—
427#95C2×D412yes—
428#95C2×D412yes—
429#95C2×D412yes—
430C24—21no—
431C24C22×D411yescharacteristic
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—48no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—48no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8C22×D414yes—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4#12114yes—
494C22×C4#12114yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C2×#4112yes—
506C22×C4C2511yescharacteristic
507Q16—48no—
508Q16—416no—
509SD16—48no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—24no—
519C4⋊C4—28no—
520C4⋊C4—28no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4#13714yes—
555C4⋊C4C22×D414yes—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—44no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C42—28no—
567C42—28no—
568C42—24no—
569C42—24no—
570C42—42no—
571C42—42no—
572C42—42no—
573C42—44no—
574C42—84no—
575C42—84no—
576C42#12114yes—
577C42#12114yes—
578C42C22×D412yes—
579C42C22×D412yes—
580C42C2×#4112yes—
581C42C2×#4112yes—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23C2×#12112yes—
594C23#57712yes—
595C23#57812yes—
596C23C23×D411yescharacteristic
597Q8—24no—
598Q8—24no—
599Q8—48no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—44no—
604Q8—42no—
605Q8—44no—
606Q8—84no—
607D4—42no—
608D4—42no—
609D4—84no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—28no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—48no—
636C2×C4—48no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—48no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—44no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—41no—
657C2×C4—44no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—84no—
661C2×C4—88no—
662C2×C4—84no—
663C2×C4C2×#12112yes—
664C2×C4C2×#12112yes—
665C2×C4C2×#12112yes—
666C2×C4C2×#12112yes—
667C2×C4C2×#12112yes—
668C2×C4C2×#13714yes—
669C2×C4C2×#13712yes—
670C2×C4#57512yes—
671C2×C4C23×D411yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4#58512yes—
674C2×C4#58512yes—
675C8—42no—
676C8—42no—
677C8—44no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22#304614yes—
692C22C22×#12111yescharacteristic
693C22#348511yescharacteristic
694C22#348711yescharacteristic
695C22C22×#13711yescharacteristic
696C22C2×#57511yescharacteristic
697C22C2×#57711yescharacteristic
698C22C2×#57811yescharacteristic
699C4—22no—
700C4—22no—
701C4—24no—
702C4—22no—
703C4—22no—
704C4—44no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—44no—
709C4—81no—
710C4—81no—
711C4—82no—
712C4—84no—
713C4—81no—
714C4—82no—
715C4#303714yes—
716C4#344912yes—
717C4#349312yes—
718C2—24no—
719C2—161no—
720C2#3515011yescharacteristic
721C2C2×#303711yescharacteristic
722C2C2×#304611yescharacteristic
723C2#6173011yescharacteristic
724C2#6173511yescharacteristic
725C2#6190011yescharacteristic
726C2#6190411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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. C2×#3113C2semidirect
4. C2×#2355C2semidirect
6. #33108C2semidirect
8. #29715C2semidirect
9. #24691C2semidirect
10. #24683C2semidirect
11. #24678C2semidirect
12. #24641C2semidirect
13. #24632C2semidirect
14. #24629C2semidirect
15. #21953C2semidirect
17. #21860C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
C2×#31082
#352412
C2×#23552
#331112
#331082
#297192
#297152
#246912
#246832
#246782
#246412
#246322
#246292
#219532
#218672
#218602