Finite Group Databasebeta

← #6444014

Subgroups — #6444014

Order 512.

2510 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3094C212yesmaximal, complemented
2C2×#2535C211yesmaximal, characteristic
3D4×#117C211yesmaximal, characteristic, complemented
4#25023C214yesmaximal, complemented
5#25022C214yesmaximal, complemented
6#25019C214yesmaximal, complemented
7#25018C214yesmaximal, complemented
8#23642C212yesmaximal, complemented
9#23480C212yesmaximal, complemented
10#21190C211yesmaximal, characteristic, complemented
11#21172C211yesmaximal, characteristic, complemented
12#20846C212yesmaximal, complemented
13#20801C211yesmaximal, characteristic, complemented
14#20646C211yesmaximal, characteristic, complemented
15C4×#418C211yesmaximal, characteristic, complemented
16C22×#121C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2212yes—
20#3094—216no—
21C2×#481C2214yescomplemented
22C2×#482C2214yescomplemented
23C2×#482C2214yescomplemented
24C2×#480C2214yescomplemented
25C2×#428C2212yes—
26C2×#428C2212yes—
27C2×#428C2212yes—
28C2×#423C2211yescharacteristic
29C2×#418C2212yes—
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#407C2212yes—
33C2×#407C2212yes—
34C2×C4×M16C2211yescharacteristic
35#2535—21no—
36#2535—22no—
37#2535—21no—
38#2535C2214yes—
39#2535C2212yes—
40#2535C2212yes—
41C2×#397—21no—
42C2×#392—21no—
43#2354C2212yescomplemented
44#2352C2212yescomplemented
45#2352C2212yescomplemented
46D4×C4⋊C4C2212yescomplemented
47#2344—21no—
48#2342—22no—
49D4×C22⋊C4—21no—
50#2293—21no—
51#2290—22no—
52C2×#386—22no—
53#2285C2211yescharacteristic
54C2×#382C2211yescharacteristic
55#2280C2212yes—
56C2×#380C2212yes—
57C42×D4—21no—
58C4×#117C2211yescharacteristic
59#2077C2214yescomplemented
60#2077C2214yescomplemented
61#2076C2214yescomplemented
62#2075C2214yescomplemented
63#2034C2214yescomplemented
64#2034C2214yescomplemented
65#2033C2214yescomplemented
66#2033C2214yescomplemented
67#1955C2212yescomplemented
68#1955C2212yescomplemented
69#1954C2214yescomplemented
70#1949C2214yescomplemented
71#1949C2214yescomplemented
72#1854C2214yescomplemented
73#1854C2214yescomplemented
74#1853C2214yescomplemented
75#1852C2214yescomplemented
76#1847C2214yescomplemented
77#1846C2212yescomplemented
78#1846C2212yescomplemented
79#1804C2212yescomplemented
80#1804C2212yescomplemented
81#1802C2214yescomplemented
82#1762C2214yescomplemented
83C4×#102C2214yescomplemented
84#1736C2211yescharacteristic
85#1736C2212yes—
86#1736C2212yes—
87#1736C2211yescharacteristic
88C23×D4—22no—
89C23×D4D411yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C24×C4C2312yes—
93C22×M16C2312yes—
94C22×M16C2311yescharacteristic
95C22×M16C2311yescharacteristic
96C2×#121—22no—
97C2×#121—22no—
98C2×#121—28no—
99C2×#121—44no—
100C2×#121C2314yes—
101C2×#121C2314yes—
102C2×C4×D4—22no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C2×C4×D4D411yescharacteristic
110C2×C4×D4D412yes—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117—22no—
118C2×#117—22no—
119C2×#117—28no—
120C2×#117—44no—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4D412yes—
136C22×C42—22no—
137C22×C42C2311yescharacteristic
138#482—216no—
139#482—216no—
140#481—216no—
141#480—216no—
142#428—28no—
143#428—28no—
144#428—44no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×#106C2314yes—
151#423—24no—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154#418—28no—
155#418—24no—
156#418D418yescomplemented
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161#407—28no—
162#407—28no—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×C22⋊C8C2314yes—
167C4×M16—24no—
168C2×#97C2312yes—
169C2×C4×C8C2312yes—
170#397—22no—
171#397—24no—
172#397—22no—
173#392—22no—
174#392—24no—
175#392—22no—
176#390—22no—
177#389—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#386—22no—
185#386—22no—
186#386—24no—
187#386—24no—
188#386—24no—
189#386—24no—
190#385—22no—
191#385—22no—
192#384C2312yes—
193#382—21no—
194#382—21no—
195#382C2311yescharacteristic
196#382C2311yescharacteristic
197#382C2312yes—
198#382C2312yes—
199#381—22no—
200#380—21no—
201#380—21no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207#380C2314yes—
208#380C2312yes—
209#380C2312yes—
210#379—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222#376—21no—
223C2×#95—21no—
224C2×#95—22no—
225C2×#95—21no—
226C43—21no—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233C25—21no—
234C25C2×D412yes—
235C22×D4—24no—
236C22×D4—24no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—44no—
245C22×D4—48no—
246C22×D4—48no—
247C22×D4—84no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2×D414yes—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2412yes—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—24no—
256C23×C4—24no—
257C23×C4—24no—
258C23×C4—24no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—44no—
266C23×C4—84no—
267C23×C4C2×D412yes—
268C23×C4C2×D412yes—
269C23×C4C2×D412yes—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2412yes—
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—24no—
280C2×M16—24no—
281C2×M16—24no—
282C2×M16C2×D414yes—
283C2×M16#4114yes—
284C22×C8C2414yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287#121—28no—
288#121—28no—
289#121—44no—
290#121—44no—
291#121—48no—
292#121—48no—
293C4×D4—24no—
294C4×D4—28no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—24no—
301C4×D4—22no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—48no—
307C4×D4—44no—
308C4×D4—44no—
309#117—22no—
310#117—24no—
311#117—22no—
312#117—24no—
313#117—22no—
314#117—24no—
315#117—24no—
316#117—48no—
317#117—48no—
318#117—42no—
319#117—42no—
320#117C2×D414yes—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—28no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—48no—
362C2×C22⋊C4—48no—
363C2×C22⋊C4—44no—
364C2×C22⋊C4—84no—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D412yes—
368C2×C22⋊C4C2×D412yes—
369C2×C42—21no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—48no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42#4114yes—
380C2×C42#4114yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—28no—
384#107—28no—
385#107—44no—
386#106—28no—
387#106—28no—
388#106—44no—
389C4⋊C8—24no—
390C4⋊C8—24no—
391#102—28no—
392#102—28no—
393#102—28no—
394#102C2×D4116yescomplemented
395C22⋊C8—28no—
396C22⋊C8—28no—
397C22⋊C8—28no—
398C22⋊C8—28no—
399#97—24no—
400C4×C8—24no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—21no—
410#95—21no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—21no—
416#95—21no—
417C24—24no—
418C24—24no—
419C24—22no—
420C24—24no—
421C24—24no—
422C24—24no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24—44no—
427C24—44no—
428C24—42no—
429C24—42no—
430C24C22×D412yes—
431C24C22×D412yes—
432C24C22×D411yescharacteristic
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—28no—
439C2×D4—24no—
440C2×D4—28no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—416no—
449C2×D4—48no—
450C2×D4—48no—
451C2×D4—48no—
452C2×D4—48no—
453C2×D4—42no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4—84no—
458C2×D4—84no—
459C2×D4—832no—
460C2×D4C22×D414yes—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—42no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—84no—
506C22×C4—84no—
507C22×C4—82no—
508C22×C4—82no—
509C22×C4—816no—
510C22×C4#12314yes—
511C22×C4#12314yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D412yes—
520C22×C4C2×#4112yes—
521C22×C4C2×#4112yes—
522C22×C4C2×#4112yes—
523C22×C4H(5,𝔽2)12yes—
524C22×C42−1+412yes—
525C22×C4C2511yescharacteristic
526M16—28no—
527M16—28no—
528M16—48no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—24no—
534C2×C8—24no—
535C2×C8—24no—
536C2×C8C22×D414yes—
537C2×C8C2×#4114yes—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—24no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—82no—
558C4⋊C4C22×D414yes—
559C22⋊C4—24no—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C22⋊C4—24no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—48no—
567C22⋊C4—48no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—42no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—84no—
577C22⋊C4—84no—
578C42—24no—
579C42—44no—
580C42—44no—
581C42—44no—
582C42—44no—
583C42—42no—
584C42—82no—
585C42#12314yes—
586C42#12314yes—
587C42C2×#4112yes—
588C42C2×#4112yes—
589C42C2×#4112yes—
590C42C2×#4112yes—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—24no—
604C23—22no—
605C23—22no—
606C23—22no—
607C23—48no—
608C23—48no—
609C23—48no—
610C23—416no—
611C23—44no—
612C23—42no—
613C23—48no—
614C23—48no—
615C23—42no—
616C23—44no—
617C23—84no—
618C23—84no—
619C23—84no—
620C23—84no—
621C23—88no—
622C23—84no—
623C23C2×#12312yes—
624C23D4212yes—
625C23#54712yes—
626C23#57514yes—
627C23C23×D411yescharacteristic
628D4—28no—
629D4—24no—
630D4—24no—
631D4—48no—
632D4—44no—
633D4—48no—
634D4—42no—
635D4—42no—
636D4—816no—
637D4—816no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—48no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—48no—
657C2×C4—48no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—88no—
679C2×C4—88no—
680C2×C4—88no—
681C2×C4—88no—
682C2×C4—84no—
683C2×C4—88no—
684C2×C4C2×#12312yes—
685C2×C4C2×#12312yes—
686C2×C4C2×#12312yes—
687C2×C4C2×#12312yes—
688C2×C4C2×#12312yes—
689C2×C4#53214yes—
690C2×C4#54214yes—
691C2×C4D4212yes—
692C2×C4#54712yes—
693C2×C4#54714yes—
694C2×C4#55314yes—
695C2×C4C23×D411yescharacteristic
696C2×C4C22×#4111yescharacteristic
697C2×C4C22×#4111yescharacteristic
698C2×C4C22×#4111yescharacteristic
699C2×C4C2×H(5,𝔽2)11yescharacteristic
700C2×C4C2×2−1+411yescharacteristic
701C8—24no—
702C8—24no—
703C8—44no—
704C22—24no—
705C22—22no—
706C22—22no—
707C22—24no—
708C22—24no—
709C22—24no—
710C22—24no—
711C22—42no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—42no—
720C22—44no—
721C22—42no—
722C22—48no—
723C22—42no—
724C22—816no—
725C22—816no—
726C22—84no—
727C22—82no—
728C22—82no—
729C22#242214yes—
730C22C22×#12311yescharacteristic
731C22C2×D4211yescharacteristic
732C22C2×#54711yescharacteristic
733C22C2×#57512yes—
734C22#358512yes—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—42no—
740C4—42no—
741C4—44no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—84no—
747C4#242214yes—
748C4C2×#53211yescharacteristic
749C4C2×#54211yescharacteristic
750C4C2×#54711yescharacteristic
751C4C2×#55311yescharacteristic
752C2—24no—
753C2—44no—
754C2—82no—
755C2—84no—
756C2C2×#242211yescharacteristic
757C2#6051111yescharacteristic
758C2#6051811yescharacteristic
759C2#6106612yes—
760C2#6125512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3094C2semidirect
3. D4×#117C2semidirect
4. #25023C2semidirect
5. #25022C2semidirect
6. #25019C2semidirect
7. #25018C2semidirect
8. #23642C2semidirect
9. #23480C2semidirect
10. #21190C2semidirect
11. #21172C2semidirect
12. #20846C2semidirect
13. #20801C2semidirect
14. #20646C2semidirect
15. C4×#418C2semidirect
21. C2×#481C22semidirect
22. C2×#482C22semidirect
23. C2×#482C22semidirect
24. C2×#480C22semidirect
43. #2354C22semidirect
44. #2352C22semidirect
45. #2352C22semidirect
46. D4×C4⋊C4C22semidirect
59. #2077C22semidirect
60. #2077C22semidirect
61. #2076C22semidirect
62. #2075C22semidirect
63. #2034C22semidirect
64. #2034C22semidirect
65. #2033C22semidirect
66. #2033C22semidirect
67. #1955C22semidirect
68. #1955C22semidirect
69. #1954C22semidirect
70. #1949C22semidirect
71. #1949C22semidirect
72. #1854C22semidirect
73. #1854C22semidirect
74. #1853C22semidirect
75. #1852C22semidirect
76. #1847C22semidirect
77. #1846C22semidirect
78. #1846C22semidirect
79. #1804C22semidirect
80. #1804C22semidirect
81. #1802C22semidirect
82. #1762C22semidirect
83. C4×#102C22semidirect
156. #418D4semidirect
394. #102C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30942
C2×#25352
D4×#1172
#250232
#250222
#250192
#250182
#236422
#234802
#211902
#211722
#208462
#208012
#206462
C4×#4182