Finite Group Databasebeta

← #6494847

Subgroups — #6494847

Order 512.

1908 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#33199C214yesmaximal, complemented
4#29982C214yesmaximal, complemented
5#29220C212yesmaximal, complemented
6#29208C211yesmaximal, characteristic, complemented
7#29206C211yesmaximal, characteristic, complemented
8#29139C211yesmaximal, characteristic, complemented
9#29139C211yesmaximal, characteristic, complemented
10#28514C212yesmaximal, complemented
11#27679C214yesmaximal, complemented
12#24304C214yesmaximal, complemented
13#21984C212yesmaximal, complemented
14#21863C212yesmaximal
15#20808C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×#532C2211yescharacteristic
18C2×C4×#41—21no—
19C2×#478C2218yes—
20C2×#431C2212yes—
21#2868C2211yescharacteristic
22#2868C2211yescharacteristic
23#2862C2212yes—
24#2859C2212yes—
25#2859C2212yes—
26#2857—21no—
27#2857—21no—
28#2857—21no—
29#2857—21no—
30C4×C4⋊Q8—21no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×#127C2211yescharacteristic
37C4×#127C2211yescharacteristic
38C4×#124—21no—
39C4×#124—21no—
40C4×#121—21no—
41C4×#121—21no—
42C4×#121C2212yes—
43C4×#121C2212yes—
44#2297—21no—
45#2293—21no—
46#2293C2212yes—
47#2285C2211yescharacteristic
48#2285C2212yes—
49#2285C2212yes—
50C42×D4—21no—
51C42×D4C2212yes—
52C2×C43C2211yescharacteristic
53#2101C2218yes—
54#2092C2218yes—
55#2058C2214yes—
56#2057C2214yes—
57#2023C2214yes—
58#2023C2214yes—
59#1990C2212yes—
60#1990C2212yes—
61#1989C2218yes—
62#1957C2212yes—
63#1957C2212yes—
64#1949C2218yes—
65#1948C2214yes—
66#1947C2214yes—
67#1875C2218yes—
68#1854C2218yes—
69#1845C2212yes—
70#1844C2212yes—
71#1842C2212yes—
72#1834C2212yes—
73#1834C2212yes—
74#1831C2212yes—
75#1804C2218yes—
76#1802C2214yes—
77#1802C2214yes—
78#1797C2212yes—
79#1797C2212yes—
80C4×C4⋊C8C2211yescharacteristic
81C4×C4⋊C8C2211yescharacteristic
82C4×C4⋊C8C2212yes—
83C22×#41—21no—
84#533—48no—
85#532—24no—
86#532Q818yescomplemented
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×#127C2311yescharacteristic
95C2×#125C2314yes—
96C2×#124—21no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×#121—21no—
100C2×#121C2312yes—
101C4×#41—48no—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×#117—21no—
117C2×#117—22no—
118C2×#117C2314yes—
119C22×C42—22no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#478—416no—
128#431—28no—
129C2×#107C2318yes—
130C2×#106C2318yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#102C2318yes—
136C2×#102C2318yes—
137C2×C22⋊C8C2314yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—21no—
145#391—21no—
146#390—21no—
147#390—21no—
148#390—21no—
149#390—21no—
150#390C2312yes—
151#390C2312yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389C2312yes—
163#389C2312yes—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#386—22no—
169#386—22no—
170#384—21no—
171#384—21no—
172#384—21no—
173#384—21no—
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2312yes—
177#384C2312yes—
178#384C2312yes—
179#384C2312yes—
180#382—21no—
181#382C2314yes—
182#382C2314yes—
183#380—22no—
184#380—22no—
185#380—22no—
186#380—22no—
187#380—21no—
188#380C2312yes—
189C4×C4⋊C4—22no—
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⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C43D411yescharacteristic
206C43D411yescharacteristic
207C43D411yescharacteristic
208C43D411yescharacteristic
209C43C2311yescharacteristic
210C43C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213#340C2318yes—
214#340C2318yes—
215#340C2318yes—
216#340C2318yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C2×#41—28no—
221C2×#41—44no—
222C2×#41—44no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C22×C8C2414yes—
241C22×C8C2414yes—
242C4⋊Q8—22no—
243C4⋊Q8—44no—
244C4⋊Q8—44no—
245C4⋊Q8—44no—
246C4⋊Q8—44no—
247C4⋊Q8—44no—
248C4⋊Q8—44no—
249C4⋊Q8C2×Q814yes—
250#127—22no—
251#127C2×Q814yes—
252#125—216no—
253#124—28no—
254C22⋊Q8—48no—
255C22⋊Q8—48no—
256#121—28no—
257#121—24no—
258#121C2×Q818yes—
259C4×Q8—44no—
260C4×Q8—48no—
261C4×Q8—48no—
262C4×Q8—44no—
263C4×D4—28no—
264C4×D4—24no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—44no—
268C4×D4—44no—
269C4×D4—48no—
270C4×D4—48no—
271C4×D4—44no—
272C4×D4—44no—
273C4×D4C2×Q818yes—
274#117—48no—
275#117—48no—
276#117—44no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—48no—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
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×Q814yes—
345C2×C42#4114yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—48no—
353#106—48no—
354C4⋊C8—28no—
355C4⋊C8—28no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358#102—48no—
359#102—48no—
360C22⋊C8—28no—
361C4×C8—24no—
362C4×C8—24no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373C24—22no—
374C24C22×D411yescharacteristic
375#41—432no—
376C2×Q8—24no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×D4—28no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4C22×Q814yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4C22⋊Q814yes—
431C22×C4C22⋊Q814yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
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×Q812yes—
446C22×C4C2×#4112yes—
447C22×C4C2511yescharacteristic
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—42no—
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—44no—
477C4⋊C4—44no—
478C4⋊C4C22×Q814yes—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
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—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—44no—
510C42—44no—
511C42—48no—
512C42—44no—
513C42C22⋊Q814yes—
514C42C22⋊Q814yes—
515C42#13514yes—
516C42AGL(1,ℤ/8)14yes—
517C42C22×Q812yes—
518C42C22×Q812yes—
519C42C2×#4112yes—
520C42C2×#4112yes—
521C42H(5,𝔽2)14yes—
522C422−1+414yes—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—24no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C2×C22⋊Q812yes—
537C23C23×D411yescharacteristic
538Q8—48no—
539Q8—88no—
540Q8—88no—
541D4—28no—
542D4—416no—
543D4—48no—
544D4—48no—
545D4—44no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
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—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—416no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—88no—
593C2×C4—88no—
594C2×C4C2×C22⋊Q812yes—
595C2×C4C2×C22⋊Q812yes—
596C2×C4C2×C22⋊Q812yes—
597C2×C4C2×C22⋊Q812yes—
598C2×C4C2×C22⋊Q812yes—
599C2×C4#53512yes—
600C2×C4#53512yes—
601C2×C4#53612yes—
602C2×C4#53612yes—
603C2×C4C2×#13512yes—
604C2×C4C2×#13512yes—
605C2×C4C2×#13512yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4#57512yes—
610C2×C4#57512yes—
611C2×C4C23×Q811yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C2×H(5,𝔽2)12yes—
614C2×C4C2×2−1+412yes—
615C8—44no—
616C8—44no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22C22×C22⋊Q811yescharacteristic
632C22C2×#53511yescharacteristic
633C22C2×#53611yescharacteristic
634C22C22×#13511yescharacteristic
635C22C22×AGL(1,ℤ/8)11yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57511yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—44no—
647C4—41no—
648C4—41no—
649C4—84no—
650C4—82no—
651C4—84no—
652C4—82no—
653C4#307614yes—
654C4#307714yes—
655C4#322814yes—
656C4#350312yes—
657C4#351112yes—
658C2—42no—
659C2—84no—
660C2#3527911yescharacteristic
661C2C2×#307611yescharacteristic
662C2C2×#307711yescharacteristic
663C2#6078211yescharacteristic
664C2#6080311yescharacteristic
665C2C2×#322811yescharacteristic
666C2#6086211yescharacteristic
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. #33199C2semidirect
4. #29982C2semidirect
5. #29220C2semidirect
6. #29208C2semidirect
7. #29206C2semidirect
8. #29139C2semidirect
9. #29139C2semidirect
10. #28514C2semidirect
11. #27679C2semidirect
12. #24304C2semidirect
13. #21984C2semidirect
15. #20808C2semidirect
86. #532Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
C4×#5322
#331992
#299822
#292202
#292082
#292062
#291392
#285142
#276792
#243042
#219842
#218632
#208082