Finite Group Databasebeta

← #6463828

Subgroups — #6463828

Order 512.

2022 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic, complemented
2C2×#3076C211yesmaximal, characteristic
3#34214C211yesmaximal, characteristic, complemented
4C4×#536C211yesmaximal, characteristic
5#28216C214yesmaximal, complemented
6#28215C214yesmaximal, complemented
7#28206C212yesmaximal, complemented
8#28205C212yesmaximal, complemented
9#28204C212yesmaximal, complemented
10#28203C212yesmaximal, complemented
11#23662C212yesmaximal, complemented
12#23510C212yesmaximal
13#21066C212yesmaximal, complemented
14#21062C212yesmaximal
15#20755C211yesmaximal, characteristic, complemented
16#20648C211yesmaximal, characteristic
17C2×#1803C211yesmaximal, characteristic, complemented
18C2×#536C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#3078—28no—
23#3076—28no—
24C2×#479C2214yes—
25C2×#479C2212yes—
26C2×#477C2212yes—
27C2×#477C2211yescharacteristic
28C2×#475C2212yes—
29C2×#475C2211yescharacteristic
30C2×#478C2212yes—
31C2×#476C2211yescharacteristic
32C2×#474C2211yescharacteristic
33C2×#428C2212yes—
34C2×#427C2212yes—
35C2×#423C2211yescharacteristic
36C22×C4⋊C8C2211yescharacteristic
37C2×#417C2212yes—
38C2×#419C2212yes—
39C2×#416C2212yes—
40C22×#103C2212yes—
41#2360—22no—
42#2360—22no—
43#2355—21no—
44#2355—21no—
45Q8×C4⋊C4—21no—
46Q8×C4⋊C4—21no—
47#2330C2214yes—
48#2327C2212yes—
49#2326C2212yes—
50#2323—21no—
51#2323—21no—
52#2322—22no—
53#2317C2212yes—
54#2317C2211yescharacteristic
55#2311C2211yescharacteristic
56#2310C2211yescharacteristic
57C4×C4⋊Q8C2212yes—
58C4×C4⋊Q8C2212yes—
59C4×C4⋊Q8C2212yes—
60C4×#124C2212yes—
61C4×C22⋊Q8—21no—
62C4×C22⋊Q8—21no—
63C4×#123—22no—
64C2×#384C2211yescharacteristic
65#2274C2212yes—
66C4×#117C2211yescharacteristic
67#2085C2214yes—
68#2085C2214yes—
69#2084C2214yes—
70#2083C2214yes—
71#2043C2214yes—
72#2043C2214yes—
73#2042C2214yes—
74#2042C2214yes—
75#1928C2214yes—
76#1928C2214yes—
77#1928C2214yes—
78#1927C2214yes—
79#1927C2214yes—
80#1927C2214yes—
81#1926C2214yes—
82#1925C2214yes—
83#1832C2214yes—
84#1831C2212yes—
85#1831C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2214yes—
90#1803C2212yes—
91#1802C2212yes—
92#1802C2212yes—
93#1736C2212yes—
94C2×#340C2212yes—
95#1729C2212yes—
96#1729C2212yes—
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×M16C2312yes—
100C23×C8C2312yes—
101#536—24no—
102#536—24no—
103#533—24no—
104#533D418yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125C2312yes—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C4×#41—24no—
116C4×#41—24no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—22no—
122C2×C4×Q8—22no—
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×#117C2312yes—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2312yes—
143C22×C42—21no—
144C22×C42—21no—
145C22×C42C2311yescharacteristic
146#479—28no—
147#479—216no—
148#478—28no—
149#477—24no—
150#477—28no—
151#476—24no—
152#475—24no—
153#475—28no—
154#474—24no—
155#428—28no—
156#427—28no—
157C2×#107C2312yes—
158C2×#107C2314yes—
159C2×#107C2312yes—
160C2×#106C2312yes—
161C2×#106C2314yes—
162C2×#106C2312yes—
163#423—24no—
164C2×C4⋊C8—24no—
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2312yes—
167C2×C4⋊C8C2311yescharacteristic
168C2×C4⋊C8C2311yescharacteristic
169C2×C4⋊C8C2312yes—
170#419—28no—
171#417—28no—
172#416—28no—
173C2×#103—28no—
174C2×#103C2312yes—
175C2×#103C2314yes—
176C2×#103C2312yes—
177C2×#103C2314yes—
178C2×#103C2314yes—
179C2×#103C2314yes—
180C2×#103C2314yes—
181C2×#102C2312yes—
182C2×#102C2312yes—
183C2×#102C2314yes—
184#391—21no—
185#391—21no—
186#391—22no—
187#391—22no—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#390C2312yes—
192#389—21no—
193#389—21no—
194#389—22no—
195#389—22no—
196#389—22no—
197#389—22no—
198#389—24no—
199#389—24no—
200#389C2314yes—
201#389C2314yes—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—24no—
211#387—24no—
212#386—22no—
213#385—22no—
214#385—24no—
215#385—22no—
216#385—22no—
217#384—22no—
218#384—21no—
219#384—21no—
220#384C2312yes—
221#384C2312yes—
222#384C2312yes—
223#382—22no—
224#382C2312yes—
225#381—22no—
226#380—21no—
227#380—21no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4D411yescharacteristic
235C4×C4⋊C4D411yescharacteristic
236C4×C4⋊C4D411yescharacteristic
237C4×C4⋊C4D411yescharacteristic
238C4×C4⋊C4C2314yes—
239C4×C4⋊C4C2314yes—
240C4×C22⋊C4—24no—
241C4×C22⋊C4—22no—
242C4×C22⋊C4—22no—
243C4×C22⋊C4D412yes—
244C4×C22⋊C4D412yes—
245#376C2312yes—
246C43C2312yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#336C2314yes—
252#336C2314yes—
253#336C2314yes—
254#336C2314yes—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41#4114yes—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8—22no—
269C22×Q8#4114yes—
270C22×Q8C2412yes—
271C22×Q8C2411yescharacteristic
272C22×D4—21no—
273C22×D4C2411yescharacteristic
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—22no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C2×M16—24no—
284C2×M16—24no—
285C2×M16—24no—
286C22×C8—24no—
287C22×C8—24no—
288C22×C8—24no—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2414yes—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8C2×D414yes—
297C4⋊Q8C2×D414yes—
298#125—24no—
299#125C2×D418yes—
300#124—24no—
301#124—24no—
302#123—216no—
303C22⋊Q8—28no—
304C22⋊Q8—28no—
305C22⋊Q8—28no—
306C22⋊Q8—28no—
307C4×Q8—28no—
308C4×Q8—28no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—44no—
318C4×Q8—44no—
319C4×Q8—84no—
320C4×Q8—84no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—44no—
326C4×D4—44no—
327#117—24no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—44no—
335#117C2×D418yes—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4—82no—
362C2×C4⋊C4—82no—
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2414yes—
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2411yescharacteristic
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D412yes—
383C2×C42—22no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—24no—
390C2×C42—24no—
391C2×C42—44no—
392C2×C42—44no—
393C2×C42C2×D412yes—
394C2×C42C2×D412yes—
395C2×C42C2×D412yes—
396C2×C42C2×D412yes—
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D414yes—
400C2×C42#4114yes—
401C2×C42C2412yes—
402C2×C42C2412yes—
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405#107—24no—
406#107—24no—
407#107—28no—
408#106—24no—
409#106—24no—
410#106—28no—
411C4⋊C8—22no—
412C4⋊C8—22no—
413C4⋊C8—24no—
414C4⋊C8—24no—
415C4⋊C8—24no—
416#103—28no—
417#103—28no—
418#103—28no—
419#103—24no—
420#103—24no—
421#103—28no—
422#103—28no—
423#102—28no—
424#102—24no—
425#102—24no—
426#95—24no—
427#95—24no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—24no—
437#95—24no—
438#95C2×D412yes—
439#95C2×D412yes—
440#95C2×D412yes—
441#95C2×D412yes—
442C24—21no—
443C24C22×D411yescharacteristic
444#41—28no—
445#41—28no—
446#41—44no—
447#41—44no—
448C2×Q8—24no—
449C2×Q8—24no—
450C2×Q8—24no—
451C2×Q8—22no—
452C2×Q8—21no—
453C2×Q8—28no—
454C2×Q8—28no—
455C2×Q8—22no—
456C2×Q8—21no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—44no—
468C2×Q8—44no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8C2×#4114yes—
472C2×Q8C2×#4112yes—
473C2×D4—24no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4C2×#4112yes—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—24no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—24no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—44no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—81no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—81no—
510C22×C4—82no—
511C22×C4—82no—
512C22×C4#12114yes—
513C22×C4#12114yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D412yes—
524C22×C4C2×#4112yes—
525C22×C4C2×#4112yes—
526C22×C4C2×#4112yes—
527C22×C42−1+412yes—
528C22×C42−1+412yes—
529C22×C4C2511yescharacteristic
530M16—48no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—24no—
534C2×C8—24no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—24no—
540C2×C8—48no—
541C4⋊C4—28no—
542C4⋊C4—28no—
543C4⋊C4—28no—
544C4⋊C4—28no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—24no—
557C4⋊C4—24no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—44no—
562C4⋊C4—42no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—44no—
568C4⋊C4—44no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C4⋊C4—82no—
572C4⋊C4—82no—
573C4⋊C4—84no—
574C4⋊C4—84no—
575C4⋊C4C22×D414yes—
576C4⋊C4C22×D414yes—
577C22⋊C4—28no—
578C22⋊C4—28no—
579C22⋊C4—28no—
580C22⋊C4—24no—
581C22⋊C4—24no—
582C22⋊C4—42no—
583C22⋊C4—42no—
584C22⋊C4—42no—
585C22⋊C4—82no—
586C22⋊C4—82no—
587C42—28no—
588C42—24no—
589C42—24no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—42no—
595C42—42no—
596C42—42no—
597C42—42no—
598C42—42no—
599C42—44no—
600C42—44no—
601C42—84no—
602C42—84no—
603C42#12114yes—
604C42#12114yes—
605C42C22×D412yes—
606C42C22×D412yes—
607C42C2×#4112yes—
608C42C2×#4112yes—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—22no—
614C23—41no—
615C23—41no—
616C23—41no—
617C23—41no—
618C23—41no—
619C23—41no—
620C23C2×#12112yes—
621C23#52512yes—
622C23#53612yes—
623C23C23×D411yescharacteristic
624Q8—24no—
625Q8—22no—
626Q8—24no—
627Q8—22no—
628Q8—48no—
629Q8—42no—
630Q8—41no—
631Q8—42no—
632Q8—41no—
633Q8—48no—
634Q8—48no—
635D4—22no—
636D4—22no—
637D4—48no—
638D4—41no—
639D4—41no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—28no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—24no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—48no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—48no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—84no—
687C2×C4—84no—
688C2×C4—84no—
689C2×C4—164no—
690C2×C4C2×#12112yes—
691C2×C4C2×#12112yes—
692C2×C4C2×#12112yes—
693C2×C4C2×#12112yes—
694C2×C4C2×#12112yes—
695C2×C4#52512yes—
696C2×C4#53612yes—
697C2×C4#54814yes—
698C2×C4#54814yes—
699C2×C4#55214yes—
700C2×C4#55214yes—
701C2×C4#57712yes—
702C2×C4#57812yes—
703C2×C4C23×D411yescharacteristic
704C2×C4C22×#4111yescharacteristic
705C2×C4C22×#4111yescharacteristic
706C2×C4C22×#4111yescharacteristic
707C2×C4C2×2−1+411yescharacteristic
708C2×C4C2×2−1+411yescharacteristic
709C8—42no—
710C8—42no—
711C8—44no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—22no—
716C22—22no—
717C22—22no—
718C22—42no—
719C22—42no—
720C22—42no—
721C22—81no—
722C22—81no—
723C22—81no—
724C22—81no—
725C22#253614yes—
726C22C22×#12111yescharacteristic
727C22C2×#52511yescharacteristic
728C22C2×#53611yescharacteristic
729C22C2×#57711yescharacteristic
730C22C2×#57811yescharacteristic
731C22#358511yescharacteristic
732C22#358611yescharacteristic
733C4—22no—
734C4—22no—
735C4—22no—
736C4—44no—
737C4—44no—
738C4—42no—
739C4—44no—
740C4—42no—
741C4—42no—
742C4—42no—
743C4—42no—
744C4—81no—
745C4—82no—
746C4—81no—
747C4—84no—
748C4#253614yes—
749C4#304614yes—
750C4C2×#54811yescharacteristic
751C4C2×#54811yescharacteristic
752C4C2×#55211yescharacteristic
753C4C2×#55211yescharacteristic
754C2—24no—
755C2—82no—
756C2C2×#253611yescharacteristic
757C2C2×#304611yescharacteristic
758C2#6042511yescharacteristic
759C2#6059311yescharacteristic
760C2#6059411yescharacteristic
761C2#6088911yescharacteristic
762C2#6089311yescharacteristic
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
1. C2×#3078C2semidirect
3. #34214C2semidirect
5. #28216C2semidirect
6. #28215C2semidirect
7. #28206C2semidirect
8. #28205C2semidirect
9. #28204C2semidirect
10. #28203C2semidirect
11. #23662C2semidirect
13. #21066C2semidirect
15. #20755C2semidirect
17. C2×#1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×#30762
#342142
C4×#5362
#282162
#282152
#282062
#282052
#282042
#282032
#236622
#235102
#210662
#210622
#207552
#206482
C2×#18032