Finite Group Databasebeta

← #6464346

Subgroups — #6464346

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#29414C214yesmaximal, complemented
4#29393C212yesmaximal, complemented
5#29369C212yesmaximal, complemented
6#29272C211yesmaximal, characteristic, complemented
7#29254C211yesmaximal, characteristic, complemented
8#28205C211yesmaximal, characteristic, complemented
9#28160C211yesmaximal, characteristic, complemented
10#27629C212yesmaximal, complemented
11#24154C212yesmaximal
12#23761C212yesmaximal, complemented
13#23574C212yesmaximal, complemented
14#21615C212yesmaximal, complemented
15#21505C212yesmaximal, complemented
16#21121C214yesmaximal, complemented
17#20778C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×C4×#41—21no—
21C2×#475C2212yes—
22C2×#460C2212yes—
23C2×#440C2214yes—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×C4⋊Q8C2211yescharacteristic
30C4×#127C2211yescharacteristic
31C4×#127C2211yescharacteristic
32C4×#125—21no—
33C4×#125—21no—
34C4×#124—21no—
35C4×#124—21no—
36C4×C22⋊Q8—21no—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8C2212yes—
39C4×C22⋊Q8C2212yes—
40C4×#121—21no—
41C4×#121—21no—
42C4×#121C2212yes—
43C4×#121C2212yes—
44#2297—21no—
45#2296—21no—
46#2295—21no—
47#2295C2212yes—
48#2293—21no—
49#2293C2212yes—
50#2285C2211yescharacteristic
51#2285C2211yescharacteristic
52C42×Q8—21no—
53C42×Q8C2212yes—
54C42×D4—21no—
55C42×D4C2212yes—
56C4×#117—21no—
57C2×C43C2211yescharacteristic
58#2099C2212yes—
59#2093C2212yes—
60#2084C2212yes—
61#2076C2214yes—
62#2071C2212yes—
63#2057C2212yes—
64#2043C2212yes—
65#2036C2214yes—
66#2034C2214yes—
67#2023C2212yes—
68#2013C2212yes—
69#2000C2214yes—
70#1988C2212yes—
71#1982C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1928C2214yes—
75#1927C2214yes—
76#1925C2214yes—
77#1921C2214yes—
78#1920C2214yes—
79#1918C2214yes—
80#1880C2214yes—
81#1877C2214yes—
82#1877C2214yes—
83#1864C2214yes—
84#1863C2214yes—
85#1862C2214yes—
86#1839C2212yes—
87#1831C2211yescharacteristic
88#1831C2211yescharacteristic
89#1817C2212yes—
90#1803C2212yes—
91#1803C2212yes—
92#1797C2212yes—
93#1776C2214yes—
94#1767C2212yes—
95#1763C2214yes—
96#1762C2214yes—
97#1749C2212yes—
98#1745C2212yes—
99C22×#41—21no—
100C22×SD16C2314yes—
101#533—28no—
102#532—24no—
103#532—24no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#125—21no—
109C2×#124—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2312yes—
112C2×#121—21no—
113C2×#121C2312yes—
114C4×#41—48no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117—22no—
137C22×C42—22no—
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42D411yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#475—28no—
146#460—28no—
147#440—216no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×C22⋊C8C2314yes—
160C2×#97C2314yes—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391C2312yes—
166#391C2312yes—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390C2312yes—
172#390C2312yes—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#386—22no—
194#386—22no—
195#385—22no—
196#385—22no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#382—21no—
202#382—21no—
203#380—21no—
204#380—21no—
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237#376—22no—
238C43—22no—
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#336C2314yes—
254#336C2312yes—
255#336C2312yes—
256#336C2312yes—
257#336C2314yes—
258#336C2312yes—
259C2×#41—28no—
260C2×#41—44no—
261C2×#41—44no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8C2412yes—
265C22×D4—22no—
266C22×D4—21no—
267C22×D4—21no—
268C22×D4C2412yes—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2411yescharacteristic
279C2×SD16—28no—
280C2×SD16—28no—
281C2×SD16—44no—
282C22×C8C2414yes—
283C22×C8C2414yes—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—22no—
287C4⋊Q8#4114yes—
288#127—22no—
289#127#4114yes—
290#125—28no—
291#124—28no—
292C22⋊Q8—28no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295#121—28no—
296#121—24no—
297#121—24no—
298C4×Q8—28no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—42no—
303C4×Q8—42no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×Q8#4118yescomplemented
309C4×D4—28no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—28no—
313C4×D4—44no—
314C4×D4—44no—
315C4×D4—44no—
316C4×D4—44no—
317C4×D4—48no—
318C4×D4—48no—
319C4×D4—44no—
320#117—28no—
321#117—48no—
322#117—44no—
323#117—44no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
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⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C42—24no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—42no—
376C2×C42—42no—
377C2×C42—42no—
378C2×C42—48no—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407#107—28no—
408#107—44no—
409#106—28no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412#103—28no—
413#103—28no—
414#103—44no—
415#102—28no—
416#102—28no—
417#102—44no—
418C22⋊C8—44no—
419#97—28no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436C24—22no—
437C24C22×D411yescharacteristic
438#41—432no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8C2×#4114yes—
451C2×D4—28no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—24no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4C2×#4114yes—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—28no—
470C22×C4—22no—
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—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4#13714yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C2511yescharacteristic
516SD16—416no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—28no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—28no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4C2×#4114yes—
552C4⋊C4C2×#4114yes—
553C22⋊C4—28no—
554C22⋊C4—28no—
555C22⋊C4—28no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—28no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—21no—
581C42—21no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—41no—
586C42—41no—
587C42—41no—
588C42—41no—
589C42—41no—
590C42—41no—
591C42—44no—
592C42—44no—
593C42—44no—
594C42—44no—
595C42—44no—
596C42—44no—
597C42—44no—
598C42—44no—
599C42AGL(1,ℤ/8)14yes—
600C42C2×#4114yes—
601C42C2×#4112yes—
602C42C2×#4112yes—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—24no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23C2×#13712yes—
617C23C23×D411yescharacteristic
618Q8—24no—
619Q8—24no—
620Q8—48no—
621Q8—48no—
622Q8—42no—
623Q8—42no—
624D4—28no—
625D4—48no—
626D4—416no—
627D4—48no—
628D4—44no—
629C2×C4—28no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—28no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—416no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—88no—
688C2×C4—88no—
689C2×C4#52512yes—
690C2×C4#52512yes—
691C2×C4#52512yes—
692C2×C4#52512yes—
693C2×C4C2×AGL(1,ℤ/8)12yes—
694C2×C4C2×AGL(1,ℤ/8)12yes—
695C2×C4C2×AGL(1,ℤ/8)12yes—
696C2×C4C2×#13712yes—
697C2×C4C2×#13712yes—
698C2×C4#57512yes—
699C2×C4#57512yes—
700C2×C4C22×#4111yescharacteristic
701C2×C4C22×#4112yes—
702C2×C4C22×#4111yescharacteristic
703C2×C4#58512yes—
704C8—44no—
705C8—44no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22C2×#52511yescharacteristic
721C22C2×#52511yescharacteristic
722C22#344911yescharacteristic
723C22C22×AGL(1,ℤ/8)11yescharacteristic
724C22C22×#13711yescharacteristic
725C22C2×#57511yescharacteristic
726C22C2×#57511yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—44no—
739C4—44no—
740C4—41no—
741C4—44no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4#311014yes—
747C4#347512yes—
748C4#348012yes—
749C2—42no—
750C2—84no—
751C2#3490911yescharacteristic
752C2C2×#311011yescharacteristic
753C2#6055911yescharacteristic
754C2#6056011yescharacteristic
755C2#6056111yescharacteristic
756C2#6096311yescharacteristic
757C2#6096511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×#533C2semidirect
3. #29414C2semidirect
4. #29393C2semidirect
5. #29369C2semidirect
6. #29272C2semidirect
7. #29254C2semidirect
8. #28205C2semidirect
9. #28160C2semidirect
10. #27629C2semidirect
12. #23761C2semidirect
13. #23574C2semidirect
14. #21615C2semidirect
15. #21505C2semidirect
16. #21121C2semidirect
17. #20778C2semidirect
308. C4×Q8#41semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
C4×#5322
#294142
#293932
#293692
#292722
#292542
#282052
#281602
#276292
#241542
#237612
#235742
#216152
#215052
#211212
#207782