Finite Group Databasebeta

← #6449854

Subgroups — #6449854

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic, complemented
2C2×C4×#137C211yesmaximal, characteristic
3#35333C211yesmaximal, characteristic
4#34187C211yesmaximal, characteristic, complemented
5#24691C214yesmaximal, complemented
6#24689C214yesmaximal, complemented
7#24687C214yesmaximal, complemented
8#24685C214yesmaximal, complemented
9#23633C211yesmaximal, characteristic, complemented
10#23633C211yesmaximal, characteristic, complemented
11#23545C211yesmaximal, characteristic
12#23545C211yesmaximal, characteristic
13#21325C211yesmaximal, characteristic, complemented
14#21174C211yesmaximal, characteristic, complemented
15#21173C211yesmaximal, characteristic, complemented
16#21145C211yesmaximal, characteristic, complemented
17#21068C211yesmaximal, characteristic
18#21067C211yesmaximal, characteristic
19#20645C211yesmaximal, characteristic
20C22×#137C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C2×#479—28no—
25C2×#479C2214yes—
26C2×#479C2214yes—
27C4×#137—28no—
28C2×#441C2212yes—
29C2×#440C2212yes—
30C2×C4×Q16C2212yes—
31C2×C4×SD16C2212yes—
32C2×#428C2211yescharacteristic
33C22×#107C2211yescharacteristic
34C22×#106C2211yescharacteristic
35C2×#424C2211yescharacteristic
36C2×#424C2211yescharacteristic
37C22×C4⋊C8C2211yescharacteristic
38C2×#417C2211yescharacteristic
39C2×#419C2211yescharacteristic
40C2×#416C2211yescharacteristic
41C2×#416C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C22×#103C2211yescharacteristic
44C2×C4×M16C2211yescharacteristic
45#2851—22no—
46#2629—22no—
47#2579—22no—
48#2579—22no—
49#2566C2212yes—
50#2566C2212yes—
51#2565C2212yes—
52#2565C2212yes—
53#2356—22no—
54#2356C2214yes—
55#2355—22no—
56#2355C2214yes—
57#2317C2211yescharacteristic
58#2317C2211yescharacteristic
59#2313—22no—
60#2311C2211yescharacteristic
61#2311C2211yescharacteristic
62#2310C2211yescharacteristic
63#2310C2211yescharacteristic
64#2285C2211yescharacteristic
65#2285C2211yescharacteristic
66#2283—22no—
67#2274C2211yescharacteristic
68#2274C2211yescharacteristic
69#2074C2214yes—
70#2074C2214yes—
71#2072C2214yes—
72#2072C2214yes—
73#2056C2214yes—
74#2056C2214yes—
75#2055C2214yes—
76#2055C2214yes—
77#1992C2214yes—
78#1991C2214yes—
79#1949C2214yes—
80#1949C2214yes—
81#1948C2214yes—
82#1947C2214yes—
83#1940C2214yes—
84#1939C2214yes—
85#1928C2214yes—
86#1927C2214yes—
87#1926C2214yes—
88#1925C2214yes—
89#1805C2212yes—
90#1804C2212yes—
91#1803C2212yes—
92#1802C2212yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8C2311yescharacteristic
103C2×#137—24no—
104C2×#137—24no—
105C2×#137—24no—
106C22×Q16C2312yes—
107C22×SD16C2312yes—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C23×C8C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—24no—
115C2×#125—24no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C4×#41—24no—
119C4×#41—24no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—24no—
123C2×C4×Q8D418yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117—21no—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C42—21no—
145C22×C42—21no—
146C22×C42C2311yescharacteristic
147#479—216no—
148#479—216no—
149#441—28no—
150#440—28no—
151C4×Q16—28no—
152C4×SD16—28no—
153#428—24no—
154C2×#107—24no—
155C2×#107C2314yes—
156C2×#107C2312yes—
157C2×#106—24no—
158C2×#106C2314yes—
159C2×#106C2312yes—
160#424—24no—
161#424—24no—
162C2×C4⋊C8—24no—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2314yes—
166C2×C4⋊C8C2314yes—
167#419—24no—
168#417—24no—
169#416—24no—
170#416—24no—
171C2×#103—24no—
172C2×#103—24no—
173C2×#103C2314yes—
174C2×#103C2314yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#102C2312yes—
181C2×#102C2312yes—
182C2×#102C2312yes—
183C4×M16—24no—
184C2×#97C2312yes—
185C2×C4×C8C2312yes—
186#398—24no—
187#398—24no—
188#396—24no—
189#396—24no—
190#391—22no—
191#391—22no—
192#391C2312yes—
193#391C2312yes—
194#391C2314yes—
195#391C2314yes—
196#390C2312yes—
197#390C2312yes—
198#389—22no—
199#389—22no—
200#387—24no—
201#387—24no—
202#387—24no—
203#387—24no—
204#387—24no—
205#387—24no—
206#386—22no—
207#386—22no—
208#385—22no—
209#385—22no—
210#385—24no—
211#385—24no—
212#384D412yes—
213#384C2312yes—
214#384C2312yes—
215#384C2312yes—
216#383—24no—
217#382—24no—
218#382D412yes—
219#382C2312yes—
220#382C2312yes—
221#382C2312yes—
222#381—22no—
223#381—22no—
224#380—21no—
225#380—21no—
226#380—22no—
227#380—22no—
228#380D412yes—
229#380D412yes—
230C4×C4⋊C4—24no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4C2314yes—
236C4×C4⋊C4C2314yes—
237C4×C4⋊C4C2312yes—
238C4×C4⋊C4C2312yes—
239C4×C22⋊C4—22no—
240C4×C22⋊C4—22no—
241#376—24no—
242#340C2314yes—
243#340C2314yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#340C2312yes—
248#340C2314yes—
249#340C2314yes—
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×Q8C2×D414yes—
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
279#137—232no—
280C2×Q16—24no—
281C2×Q16—24no—
282C2×Q16—24no—
283C2×SD16—24no—
284C2×SD16—24no—
285C2×SD16—24no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16#4114yes—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302#125—24no—
303#125—24no—
304#125—24no—
305#125—24no—
306#125—44no—
307#125—44no—
308C4×Q8—28no—
309C4×Q8—28no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8C2×D418yes—
317C4×Q8C2×D418yes—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—44no—
323C4×D4—44no—
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—22no—
334#117—44no—
335C2×C4⋊C4—28no—
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—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
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—44no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2×D414yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2411yescharacteristic
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
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⋊C4—24no—
377C2×C22⋊C4C2×D412yes—
378C2×C22⋊C4C2×D412yes—
379C2×C42—28no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—24no—
388C2×C42—24no—
389C2×C42—44no—
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D414yes—
393C2×C42#4114yes—
394C2×C42C2412yes—
395C2×C42C2412yes—
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398#107—28no—
399#107—24no—
400#106—28no—
401#106—24no—
402C4⋊C8—28no—
403C4⋊C8—28no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406#103—28no—
407#103—28no—
408#103—24no—
409#103—24no—
410#103—24no—
411#103—24no—
412#103—24no—
413#102—24no—
414#102—24no—
415#102—24no—
416#97—24no—
417C4×C8—24no—
418#95—22no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—22no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95C2×D414yes—
431#95C2×D414yes—
432C24—21no—
433C24C22×D411yescharacteristic
434#41—28no—
435#41—28no—
436#41—44no—
437#41—44no—
438C2×Q8—24no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—28no—
442C2×Q8—28no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8C22×D414yes—
456C2×Q8C2×#4112yes—
457C2×D4—24no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4C2×#4112yes—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4#12114yes—
494C22×C4#12114yes—
495C22×C4C22×D414yes—
496C22×C4C22×D414yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4C2×#4112yes—
506C22×C42−1+412yes—
507C22×C42−1+412yes—
508C22×C4C2511yescharacteristic
509Q16—216no—
510SD16—216no—
511M16—28no—
512M16—44no—
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—
526C2×C8C2×#4114yes—
527C4⋊C4—28no—
528C4⋊C4—28no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—82no—
557C4⋊C4—84no—
558C4⋊C4—84no—
559C4⋊C4C22×D414yes—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C22⋊C4—84no—
568C22⋊C4—84no—
569C42—28no—
570C42—28no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—42no—
576C42—42no—
577C42—42no—
578C42—42no—
579C42—42no—
580C42—84no—
581C42—84no—
582C42#12114yes—
583C42#12114yes—
584C42C22×D412yes—
585C42C22×D412yes—
586C42C2×#4112yes—
587C42C2×#4112yes—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—42no—
592C23—42no—
593C23—41no—
594C23—41no—
595C23C2×#12112yes—
596C23#54712yes—
597C23#54812yes—
598C23C23×D411yescharacteristic
599Q8—24no—
600Q8—22no—
601Q8—24no—
602Q8—22no—
603Q8—48no—
604Q8—41no—
605Q8—42no—
606Q8—42no—
607Q8—41no—
608D4—22no—
609D4—22no—
610D4—48no—
611D4—41no—
612D4—41no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—48no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—48no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4C2×#12112yes—
655C2×C4C2×#12112yes—
656C2×C4C2×#12112yes—
657C2×C4C2×#12112yes—
658C2×C4C2×#12112yes—
659C2×C4#52514yes—
660C2×C4#53614yes—
661C2×C4#54712yes—
662C2×C4#54812yes—
663C2×C4#55214yes—
664C2×C4#55614yes—
665C2×C4C23×D411yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C2×C4C2×2−1+411yescharacteristic
671C8—24no—
672C8—42no—
673C8—42no—
674C8—42no—
675C22—22no—
676C22—22no—
677C22—24no—
678C22—24no—
679C22—42no—
680C22—44no—
681C22—81no—
682C22—81no—
683C22—82no—
684C22#241514yes—
685C22C22×#12111yescharacteristic
686C22C2×#54711yescharacteristic
687C22C2×#54811yescharacteristic
688C22#358412yes—
689C22#358612yes—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—44no—
695C4—44no—
696C4—44no—
697C4—42no—
698C4—44no—
699C4—82no—
700C4—81no—
701C4—82no—
702C4—81no—
703C4—82no—
704C4—82no—
705C4#241514yes—
706C4C2×#52511yescharacteristic
707C4C2×#53611yescharacteristic
708C4C2×#55211yescharacteristic
709C4C2×#55611yescharacteristic
710C2—24no—
711C2—82no—
712C2C2×#241511yescharacteristic
713C2#6042211yescharacteristic
714C2#6042411yescharacteristic
715C2#6128012yes—
716C2#6135012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#479C2semidirect
4. #34187C2semidirect
5. #24691C2semidirect
6. #24689C2semidirect
7. #24687C2semidirect
8. #24685C2semidirect
9. #23633C2semidirect
10. #23633C2semidirect
13. #21325C2semidirect
14. #21174C2semidirect
15. #21173C2semidirect
16. #21145C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4792
C2×C4×#1372
#353332
#341872
#246912
#246892
#246872
#246852
#236332
#235452
#213252
#211742
#211732
#211452
#210682
#210672
#206452