Finite Group Databasebeta

← #6468290

Subgroups — #6468290

Order 512.

1964 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#29218C211yesmaximal, characteristic, complemented
4#29208C211yesmaximal, characteristic, complemented
5#29192C212yesmaximal, complemented
6#29177C212yesmaximal, complemented
7#29162C214yesmaximal, complemented
8#29139C211yesmaximal, characteristic, complemented
9#28509C211yesmaximal, characteristic, complemented
10#24843C214yesmaximal, complemented
11#23780C214yesmaximal, complemented
12#21614C214yesmaximal, complemented
13#21410C212yesmaximal
14#20824C212yesmaximal, complemented
15C4×#423C211yesmaximal, characteristic, complemented
16C2×#533—21no—
17C2×#532C2211yescharacteristic
18C2×C4×#41—21no—
19C2×#476C2214yes—
20C2×#423C2212yes—
21C2×C4×M16C2212yes—
22#2868C2211yescharacteristic
23#2868C2211yescharacteristic
24#2862—21no—
25#2859—21no—
26#2859C2212yes—
27#2857—21no—
28#2857—21no—
29#2857C2212yes—
30#2857C2212yes—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2212yes—
35C4×C4⋊Q8C2212yes—
36C4×C4⋊Q8C2211yescharacteristic
37C4×#127C2211yescharacteristic
38C4×#127C2211yescharacteristic
39C4×#124—21no—
40C4×#124—21no—
41C4×#121—21no—
42C4×#121—21no—
43C4×#121C2212yes—
44C4×#121C2212yes—
45#2297—21no—
46#2293—21no—
47#2293C2212yes—
48#2285—21no—
49#2285C2212yes—
50#2285C2211yescharacteristic
51C42×D4—21no—
52C42×D4C2212yes—
53C2×C43C2211yescharacteristic
54#2100C2214yes—
55#2094C2214yes—
56#2088C2218yes—
57#2067C2218yes—
58#2057C2214yes—
59#2023C2214yes—
60#1990C2212yes—
61#1990C2212yes—
62#1987C2214yes—
63#1957C2212yes—
64#1957C2212yes—
65#1947C2214yes—
66#1919C2218yes—
67#1895C2218yes—
68#1862C2218yes—
69#1845C2212yes—
70#1844C2212yes—
71#1834C2212yes—
72#1834C2212yes—
73#1809C2218yes—
74#1802C2214yes—
75#1776C2218yes—
76#1767C2212yes—
77#1767C2212yes—
78C4×C4⋊C8C2212yes—
79C4×C4⋊C8C2211yescharacteristic
80C4×C4⋊C8C2211yescharacteristic
81#1762C2218yes—
82#1728C2212yes—
83C22×#41—21no—
84C22×M16C2314yes—
85#533—48no—
86#532—24no—
87#532Q818yescomplemented
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2312yes—
95C2×#127C2311yescharacteristic
96C2×#125—22no—
97C2×#124—21no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×#121—21no—
101C2×#121C2312yes—
102C4×#41—48no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×#117—21no—
118C2×#117—22no—
119C2×#117—22no—
120C22×C42—22no—
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#476—216no—
129C2×#107C2318yes—
130C2×#106C2318yes—
131#423—28no—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2312yes—
135C2×#102C2318yes—
136C2×#102C2318yes—
137C4×M16—28no—
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—21no—
146#391—21no—
147#390—21no—
148#390—21no—
149#390—21no—
150#390—21no—
151#390C2312yes—
152#390C2312yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389C2312yes—
162#389C2314yes—
163#389C2312yes—
164#389C2314yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#386—22no—
170#386—22no—
171#384—21no—
172#384—21no—
173#384—21no—
174#384—21no—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2312yes—
178#384C2312yes—
179#384C2312yes—
180#384C2312yes—
181#382—22no—
182#382—22no—
183#382—21no—
184#380—22no—
185#380—22no—
186#380—22no—
187#380—21no—
188#380C2312yes—
189#380C2314yes—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C43D411yescharacteristic
207C43D411yescharacteristic
208C43D411yescharacteristic
209C43D411yescharacteristic
210C43C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213C43C2311yescharacteristic
214#340C2318yes—
215#340C2318yes—
216#340C2318yes—
217#340C2318yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C2×#41—28no—
222C2×#41—44no—
223C2×#41—44no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4C2412yes—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×M16—28no—
242C2×M16—28no—
243C2×M16—44no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—22no—
247C4⋊Q8—28no—
248C4⋊Q8—44no—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253C4⋊Q8C2×Q814yes—
254#127—22no—
255#127C2×Q814yes—
256#125—48no—
257#124—28no—
258C22⋊Q8—48no—
259C22⋊Q8—48no—
260#121—28no—
261#121—24no—
262#121C2×Q818yes—
263C4×Q8—44no—
264C4×Q8—48no—
265C4×Q8—48no—
266C4×Q8—44no—
267C4×D4—28no—
268C4×D4—24no—
269C4×D4—28no—
270C4×D4—44no—
271C4×D4—44no—
272C4×D4—48no—
273C4×D4—48no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4C2×Q818yes—
278#117—48no—
279#117—44no—
280#117—48no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—48no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×Q814yes—
349C2×C42#4114yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—48no—
357#106—216no—
358C4⋊C8—28no—
359C4⋊C8—24no—
360C4⋊C8#4118yescomplemented
361#102—216no—
362#102—48no—
363#97—28no—
364C4×C8—24no—
365C4×C8—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376C24—22no—
377C24C22×D411yescharacteristic
378#41—432no—
379C2×Q8—24no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×D4—28no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4C22×Q814yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4C22⋊Q814yes—
434C22×C4C22⋊Q814yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×Q812yes—
449C22×C4C2×#4112yes—
450C22×C4C2511yescharacteristic
451M16—416no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8C2×#4118yes—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4C22×Q814yes—
483C22⋊C4—28no—
484C22⋊C4—28no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—44no—
514C42—44no—
515C42—48no—
516C42—44no—
517C42C22⋊Q814yes—
518C42C22⋊Q814yes—
519C42AGL(1,ℤ/8)14yes—
520C42AGL(1,ℤ/8)14yes—
521C42#13714yes—
522C42C22×Q812yes—
523C42C22×Q812yes—
524C42C2×#4112yes—
525C42C2×#4112yes—
526C42C2×#4114yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—24no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C2×C22⋊Q812yes—
541C23C23×D411yescharacteristic
542Q8—48no—
543Q8—88no—
544Q8—88no—
545D4—28no—
546D4—416no—
547D4—48no—
548D4—48no—
549D4—44no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—88no—
597C2×C4—88no—
598C2×C4C2×C22⋊Q812yes—
599C2×C4C2×C22⋊Q812yes—
600C2×C4C2×C22⋊Q812yes—
601C2×C4C2×C22⋊Q812yes—
602C2×C4C2×C22⋊Q812yes—
603C2×C4#53212yes—
604C2×C4#53212yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4C2×AGL(1,ℤ/8)12yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C2×#13712yes—
612C2×C4C2×#13712yes—
613C2×C4C2×#13712yes—
614C2×C4#57512yes—
615C2×C4C23×Q811yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4112yes—
618C2×C4#58512yes—
619C8—28no—
620C8—44no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22C22×C22⋊Q811yescharacteristic
636C22C2×#53211yescharacteristic
637C22#344911yescharacteristic
638C22C22×AGL(1,ℤ/8)11yescharacteristic
639C22C22×AGL(1,ℤ/8)11yescharacteristic
640C22C22×#13711yescharacteristic
641C22C2×#57511yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—44no—
653C4—82no—
654C4—84no—
655C4—82no—
656C4—84no—
657C4#307714yes—
658C4#307914yes—
659C4#318214yes—
660C4#347512yes—
661C4Q8×#4112yes—
662C2—42no—
663C2—84no—
664C2#3494711yescharacteristic
665C2C2×#307711yescharacteristic
666C2C2×#307911yescharacteristic
667C2C2×#318211yescharacteristic
668C2#6070011yescharacteristic
669C2#6096611yescharacteristic
670C2#6096811yescharacteristic
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. #36645C2semidirect
3. #29218C2semidirect
4. #29208C2semidirect
5. #29192C2semidirect
6. #29177C2semidirect
7. #29162C2semidirect
8. #29139C2semidirect
9. #28509C2semidirect
10. #24843C2semidirect
11. #23780C2semidirect
12. #21614C2semidirect
14. #20824C2semidirect
15. C4×#423C2semidirect
87. #532Q8semidirect
360. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#366452
C4×#5322
#292182
#292082
#291922
#291772
#291622
#291392
#285092
#248432
#237802
#216142
#214102
#208242
C4×#4232