Finite Group Databasebeta

← #6577883

Subgroups — #6577883

Order 512.

2289 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3112C212yesmaximal, complemented
2#36127C211yesmaximal, characteristic
3C4×D4×Q8C211yesmaximal, characteristic, complemented
4#33198C211yesmaximal, characteristic, complemented
5#32957C212yesmaximal, complemented
6#28511C211yesmaximal, characteristic, complemented
7#28252C211yesmaximal, characteristic, complemented
8#26566C214yesmaximal, complemented
9#25010C212yesmaximal, complemented
10#24910C214yesmaximal, complemented
11#24711C212yesmaximal, complemented
12#24239C212yesmaximal, complemented
13#24156C212yesmaximal, complemented
14#23002C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic, complemented
16#22671C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#547—21no—
20C2×#532C2212yes—
21C22×C4×Q8C2212yes—
22#3112—216no—
23C2×#480C2214yes—
24C2×#477C2212yes—
25C2×#477C2212yes—
26C2×#477C2212yes—
27C2×#460C2212yes—
28C2×#460C2211yescharacteristic
29C2×#461C2212yes—
30C2×#448C2214yes—
31C2×#440C2214yes—
32C2×#440C2214yes—
33C2×C4×SD16C2212yes—
34C2×#432C2212yes—
35#2868C2211yescharacteristic
36#2857C2212yes—
37#2547—21no—
38#2537—21no—
39#2537C2212yes—
40#2432—22no—
41#2422—22no—
42#2422C2214yes—
43#2365—21no—
44#2365C2211yescharacteristic
45#2360—22no—
46#2360C2212yes—
47Q8×C4⋊C4C2211yescharacteristic
48D4×C4⋊C4—21no—
49D4×C4⋊C4—21no—
50D4×C4⋊C4C2212yes—
51D4×C4⋊C4C2211yescharacteristic
52Q8×C22⋊C4C2212yes—
53C4×C4⋊Q8—21no—
54C4×C4⋊Q8C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C4×C22⋊Q8—22no—
57C4×C22⋊Q8C2212yes—
58C42×Q8C2211yescharacteristic
59C42×D4—21no—
60C42×D4C2211yescharacteristic
61#2095C2214yes—
62#2075C2214yes—
63#2055C2214yes—
64#2035C2214yes—
65#2033C2214yes—
66#2000C2214yes—
67#1987C2212yes—
68#1982C2211yescharacteristic
69#1956C2212yes—
70#1954C2212yes—
71#1948C2211yescharacteristic
72#1947C2212yes—
73#1947C2212yes—
74#1944C2214yes—
75#1935C2214yes—
76#1926C2212yes—
77#1924C2214yes—
78#1880C2212yes—
79#1877C2214yes—
80#1863C2214yes—
81#1863C2214yes—
82#1862C2214yes—
83#1852C2214yes—
84#1845C2212yes—
85#1842C2211yescharacteristic
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1803C2211yescharacteristic
90#1802C2212yes—
91#1797C2212yes—
92#1787C2214yes—
93#1770C2212yes—
94C4×C4⋊C8C2211yescharacteristic
95C4×#102C2212yes—
96C22×#41—22no—
97C23×Q8C2312yes—
98C22×SD16C2314yes—
99C22×SD16C2312yes—
100D4×Q8—24no—
101D4×Q8—24no—
102#547—48no—
103#532—28no—
104#532—44no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×#127C2311yescharacteristic
111C2×#124—22no—
112C2×#123—24no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8C2312yes—
116C2×#121—22no—
117C2×#121—22no—
118C2×#121C2314yes—
119C2×C4×Q8—22no—
120C2×C4×Q8—22no—
121C2×C4×Q8—28no—
122C2×C4×Q8—44no—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—22no—
134C2×C4×D4D411yescharacteristic
135C2×C4×D4D411yescharacteristic
136C2×C4×D4C2312yes—
137C2×C4×D4C2314yes—
138C2×C4×D4C2311yescharacteristic
139C22×C4⋊C4—22no—
140C22×C4⋊C4—22no—
141C22×C4⋊C4C2312yes—
142C22×C42—22no—
143C22×C42C2312yes—
144#480—216no—
145#477—28no—
146#477—28no—
147#477—44no—
148#461—28no—
149#460—28no—
150#460D418yescomplemented
151#448—216no—
152#440—216no—
153#440—216no—
154C4×SD16—28no—
155#432—28no—
156C2×#107C2314yes—
157C2×#107C2314yes—
158C2×#107C2314yes—
159C2×#106C2312yes—
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2311yescharacteristic
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2314yes—
170C2×#102C2314yes—
171C2×#102C2312yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×C22⋊C8C2314yes—
175C2×C22⋊C8C2314yes—
176C2×#97C2314yes—
177C2×C4×C8C2312yes—
178#397—24no—
179#394—24no—
180#394—24no—
181#391—22no—
182#391—22no—
183#391—22no—
184#391C2312yes—
185#391C2312yes—
186#390—21no—
187#390—21no—
188#390C2312yes—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389C2314yes—
197#389C2312yes—
198#387—22no—
199#387—22no—
200#387—22no—
201#385—22no—
202#385—22no—
203#385—22no—
204#385—24no—
205#385—24no—
206#384—21no—
207#384—21no—
208#384—21no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2312yes—
212#384C2312yes—
213#380—22no—
214#380—22no—
215#380—22no—
216#380—22no—
217#380C2312yes—
218#380C2314yes—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D411yescharacteristic
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2311yescharacteristic
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4D412yes—
237C4×C22⋊C4D412yes—
238C4×C22⋊C4C2312yes—
239C43—21no—
240C43C2311yescharacteristic
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2314yes—
245#340C2312yes—
246#336C2314yes—
247#336C2312yes—
248C2×#41—48no—
249C2×#41—48no—
250C2×#41—84no—
251C22×Q8—24no—
252C22×Q8—24no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—24no—
256C22×Q8—24no—
257C22×Q8—42no—
258C22×Q8—42no—
259C22×Q8C2412yes—
260C22×Q8C2411yescharacteristic
261C22×D4—24no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4C2×D411yescharacteristic
265C22×D4C2412yes—
266C23×C4—22no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—24no—
270C23×C4C2×D412yes—
271C23×C4C2×D412yes—
272C23×C4C2412yes—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—28no—
276C2×SD16—28no—
277C2×SD16—44no—
278C2×SD16C2×D418yes—
279C22×C8C2414yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C4⋊Q8—28no—
283C4⋊Q8—24no—
284C4⋊Q8—22no—
285C4⋊Q8—24no—
286C4⋊Q8—44no—
287C4⋊Q8—82no—
288C4⋊Q8#4114yescomplemented
289#127—22no—
290#127C2×D414yes—
291#124—48no—
292#123—416no—
293C22⋊Q8—216no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—48no—
297#121—28no—
298#121—44no—
299#121—48no—
300#121—48no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—24no—
304C4×Q8—22no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—48no—
309C4×Q8—42no—
310C4×Q8—42no—
311C4×Q8—41no—
312C4×Q8—48no—
313C4×Q8C2×D414yes—
314C4×Q8#4114yescomplemented
315C4×D4—28no—
316C4×D4—24no—
317C4×D4—28no—
318C4×D4—24no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4—44no—
324C4×D4—48no—
325C4×D4—44no—
326C4×D4—44no—
327C4×D4—44no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—48no—
352C2×C4⋊C4—48no—
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4C2×D412yes—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42—21no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—48no—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D412yes—
387C2×C42C2×D412yes—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2412yes—
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395#107—28no—
396#107—28no—
397#107—44no—
398#106—24no—
399C4⋊C8—24no—
400C4⋊C8—24no—
401C4⋊C8—22no—
402C4⋊C8—42no—
403C4⋊C8C2×D414yes—
404#103—28no—
405#103—24no—
406#103—28no—
407#103—28no—
408#103—44no—
409#102—28no—
410#102—24no—
411#102—42no—
412#102C2×D418yes—
413C22⋊C8—28no—
414C22⋊C8—44no—
415#97—28no—
416C4×C8—24no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—24no—
427#95—24no—
428#95C2×D412yes—
429#95C2×D412yes—
430#95C2×D412yes—
431C24—22no—
432C24C22×D412yes—
433#41—832no—
434C2×Q8—28no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—48no—
448C2×Q8—48no—
449C2×Q8—48no—
450C2×Q8—48no—
451C2×Q8—44no—
452C2×Q8C22×D412yes—
453C2×Q8C2×#4114yes—
454C2×Q8C2×#4112yes—
455C2×Q82−1+412yes—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—48no—
468C2×D4—48no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—84no—
472C2×D4AGL(1,ℤ/8)14yes—
473C2×D4C22×D414yes—
474C22×C4—21no—
475C22×C4—28no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—42no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—84no—
505C22×C4—82no—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C2511yescharacteristic
516SD16—216no—
517SD16—416no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—24no—
521C2×C8—22no—
522C2×C8—24no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—24no—
526C2×C8C22×D414yes—
527C4⋊C4—28no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—48no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—81no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4C22×D412yes—
562C4⋊C4C2×#4112yes—
563C22⋊C4—28no—
564C22⋊C4—24no—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—48no—
568C22⋊C4—48no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—48no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C42—24no—
575C42—24no—
576C42—24no—
577C42—24no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—21no—
583C42—21no—
584C42—41no—
585C42—44no—
586C42—44no—
587C42—42no—
588C42—44no—
589C42—42no—
590C42—42no—
591C42—81no—
592C42#13714yes—
593C42C22×D412yes—
594C42C2×#4112yes—
595C42C2×#4112yes—
596C42H(5,𝔽2)12yes—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—24no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23C2×AGL(1,ℤ/8)14yes—
611C23C23×D411yescharacteristic
612Q8—22no—
613Q8—22no—
614Q8—22no—
615Q8—24no—
616Q8—24no—
617Q8—416no—
618Q8—42no—
619Q8—42no—
620Q8—88no—
621Q8#54714yescomplemented
622D4—28no—
623D4—24no—
624D4—24no—
625D4—48no—
626D4—42no—
627D4—42no—
628D4—48no—
629D4—816no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—21no—
648C2×C4—22no—
649C2×C4—21no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—21no—
653C2×C4—22no—
654C2×C4—48no—
655C2×C4—416no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—48no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—41no—
679C2×C4—88no—
680C2×C4—816no—
681C2×C4—88no—
682C2×C4#52512yes—
683C2×C4#52512yes—
684C2×C4D4212yes—
685C2×C4D4212yes—
686C2×C4C2×AGL(1,ℤ/8)12yes—
687C2×C4C2×#13712yes—
688C2×C4C2×#13712yes—
689C2×C4C2×#13712yes—
690C2×C4#57812yes—
691C2×C4C23×D411yescharacteristic
692C2×C4C22×#4111yescharacteristic
693C2×C4C22×#4111yescharacteristic
694C2×C4C2×H(5,𝔽2)11yescharacteristic
695C2×C4C2×2−1+411yescharacteristic
696C2×C4#58511yescharacteristic
697C8—24no—
698C8—44no—
699C8—42no—
700C22—24no—
701C22—24no—
702C22—24no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22C2×#52511yescharacteristic
715C22C2×D4211yescharacteristic
716C22#348911yescharacteristic
717C22C22×AGL(1,ℤ/8)11yescharacteristic
718C22C22×#13711yescharacteristic
719C22C2×#57811yescharacteristic
720C22#358511yescharacteristic
721C4—24no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—42no—
726C4—44no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—41no—
731C4—42no—
732C4—82no—
733C4—82no—
734C4—84no—
735C4—81no—
736C4#311214yescomplemented
737C4#327614yes—
738C4C2×#54711yescharacteristic
739C4D4×#4111yescharacteristic
740C4#350611yescharacteristic
741C4#352011yescharacteristic
742C2—44no—
743C2—84no—
744C2#3610011yescharacteristic
745C2C2×#311211yescharacteristic
746C2#6060011yescharacteristic
747C2C2×#327611yescharacteristic
748C2D4×AGL(1,ℤ/8)11yescharacteristic
749C2#6211411yescharacteristic
750C2#6213211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3112C2semidirect
3. C4×D4×Q8C2semidirect
4. #33198C2semidirect
5. #32957C2semidirect
6. #28511C2semidirect
7. #28252C2semidirect
8. #26566C2semidirect
9. #25010C2semidirect
10. #24910C2semidirect
11. #24711C2semidirect
12. #24239C2semidirect
13. #24156C2semidirect
14. #23002C2semidirect
15. #22799C2semidirect
16. #22671C2semidirect
17. C4×#460C2semidirect
150. #460D4semidirect
288. C4⋊Q8#41semidirect
314. C4×Q8#41semidirect
621. Q8#547semidirect
736. C4#3112semidirect

Maximal subgroups

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

subgroupindex
C2×#31122
#361272
C4×D4×Q82
#331982
#329572
#285112
#282522
#265662
#250102
#249102
#247112
#242392
#241562
#230022
#227992
#226712
C4×#4602