Finite Group Databasebeta

← #6489345

Subgroups — #6489345

Order 512.

2318 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic
3#32959C214yesmaximal, complemented
4#30076C212yesmaximal, complemented
5#30075C212yesmaximal, complemented
6#28722C212yesmaximal, complemented
7#28516C212yesmaximal, complemented
8#28207C214yesmaximal, complemented
9#27953C212yesmaximal, complemented
10#27585C212yesmaximal, complemented
11#27582C212yesmaximal, complemented
12#25860C212yesmaximal
13#22226C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C22×#127C2211yescharacteristic
16C22×C4×D4—21no—
17C22×C4×D4—21no—
18C22×C4×D4C2212yes—
19C2×#490C2214yes—
20C2×#478C2218yescomplemented
21#2864—21no—
22#2864—21no—
23#2859C2211yescharacteristic, complemented
24#2859C2211yescharacteristic, complemented
25#2859C2211yescharacteristic, complemented
26#2859C2211yescharacteristic, complemented
27#2851—21no—
28#2851—21no—
29#2851C2212yes—
30#2851C2212yes—
31C4×#127—21no—
32C4×#127—21no—
33C4×#127—21no—
34C4×#127—21no—
35C4×#127C2212yes—
36C4×#127C2212yes—
37C4×#127C2212yes—
38C4×#127C2212yes—
39C4×#125—21no—
40C4×#125—21no—
41C4×#125C2212yescomplemented
42C4×#125C2212yescomplemented
43#2296—21no—
44#2296—21no—
45C2×#390—21no—
46C2×#390—21no—
47C2×#390C2212yes—
48#2285C2211yescharacteristic
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×C43C2211yescharacteristic
52#2094C2214yescomplemented
53#2093C2214yescomplemented
54#2084C2214yescomplemented
55#2083C2214yescomplemented
56#2071C2214yescomplemented
57#2071C2214yescomplemented
58#2041C2214yes—
59#2041C2214yes—
60#1992C2214yescomplemented
61#1961C2214yes—
62#1926C2218yescomplemented
63#1925C2218yescomplemented
64#1864C2218yescomplemented
65#1862C2218yescomplemented
66#1853C2214yes—
67#1852C2214yes—
68#1847C2214yescomplemented
69#1839C2212yescomplemented
70#1831C2212yescomplemented
71#1817C2212yes—
72#1804C2218yescomplemented
73#1797C2212yes—
74#1786C2214yes—
75#1786C2214yes—
76#1767C2212yescomplemented
77#1749C2212yes—
78#1745C2212yescomplemented
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2312yes—
82C24×C4—22no—
83C2×#127—28no—
84C2×#127—21no—
85C2×#127—21no—
86C2×#127C2312yes—
87C2×#127C2312yes—
88C2×#125—21no—
89C2×#125—21no—
90C2×#125C2312yes—
91C2×#125C2312yes—
92C2×C4×D4—22no—
93C2×C4×D4—22no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—48no—
103C2×C4×D4—48no—
104C2×C4×D4—48no—
105C2×C4×D4C2314yes—
106C2×C4×D4C2314yes—
107C2×#117—21no—
108C2×#117—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—22no—
113C22×C22⋊C4—22no—
114C22×C22⋊C4—22no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#490—216no—
123#478—416no—
124C2×#107C2318yes—
125C2×#106C2318yes—
126C2×C4⋊C8C2314yes—
127C2×#102C2318yes—
128C2×#102C2318yes—
129C2×#102C2318yes—
130C2×#102C2318yes—
131C2×C22⋊C8C2314yes—
132C2×#97C2314yes—
133#390—22no—
134#390—22no—
135#390—22no—
136#390—22no—
137#390—22no—
138#390—22no—
139#390—22no—
140#390—22no—
141#390—22no—
142#390—22no—
143#390C2314yes—
144#390C2314yes—
145#389—21no—
146#389—21no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#384—21no—
174#384—21no—
175#384C2312yes—
176#384C2312yes—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#382—21no—
182#382—21no—
183#382—21no—
184#382—21no—
185#382C2312yes—
186#382C2312yes—
187#380—21no—
188#380—21no—
189#380C2312yes—
190#380C2312yes—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213C25—22no—
214C22×D4—28no—
215C22×D4—28no—
216C22×D4—28no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—28no—
223C22×D4—44no—
224C22×D4—44no—
225C22×D4—44no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4C2414yes—
229C23×C4—22no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—48no—
233C23×C4—48no—
234C23×C4—48no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C22×C8C2418yes—
243#127—28no—
244#127—28no—
245#127—28no—
246#127—24no—
247#127#4118yes—
248#125—28no—
249#125—28no—
250#125—44no—
251#125—44no—
252C4×D4—216no—
253C4×D4—48no—
254C4×D4—48no—
255C4×D4—48no—
256C4×D4—48no—
257C4×D4—48no—
258C4×D4—48no—
259C4×D4—48no—
260C4×D4—48no—
261C4×D4—48no—
262C4×D4—48no—
263C4×D4—48no—
264#117—44no—
265#117—44no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—48no—
305C2×C22⋊C4—48no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—48no—
344#106—48no—
345C4⋊C8—28no—
346#102—216no—
347#102—216no—
348#102—48no—
349#102—48no—
350C22⋊C8—28no—
351#97—28no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364C24—28no—
365C24—24no—
366C24—44no—
367C24—44no—
368C24—44no—
369C24—44no—
370C24—44no—
371C24—44no—
372C24C22×D411yescharacteristic
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—24no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—432no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—432no—
387C2×D4—432no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4C2×#4118yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
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—41no—
418C22×C4—41no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—832no—
429C22×C4AGL(1,ℤ/8)14yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
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×C42−1+414yes—
444C22×C4C2511yescharacteristic
445C2×C8—28no—
446C2×C8—28no—
447C2×C8—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—28no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
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—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—44no—
508C42—44no—
509C42AGL(1,ℤ/8)14yes—
510C42C2×#4114yes—
511C42H(5,𝔽2)14yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—28no—
517C23—416no—
518C23—416no—
519C23—416no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—88no—
527C23—88no—
528C23—88no—
529C23—88no—
530C23#53612yes—
531C23C2×AGL(1,ℤ/8)12yes—
532C23#57512yes—
533C23C23×D411yescharacteristic
534D4—216no—
535D4—416no—
536D4—416no—
537D4—416no—
538D4—416no—
539D4—416no—
540D4—48no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—28no—
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—21no—
567C2×C4—28no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—816no—
592C2×C4—816no—
593C2×C4—88no—
594C2×C4#52512yes—
595C2×C4#52512yes—
596C2×C4#53512yes—
597C2×C4#53512yes—
598C2×C4#53612yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×AGL(1,ℤ/8)12yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4#57512yes—
605C2×C4C22×#4112yes—
606C2×C4C2×H(5,𝔽2)12yes—
607C2×C4C2×2−1+412yes—
608C8—48no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—48no—
616C22—48no—
617C22—48no—
618C22—832no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22#323414yes—
624C22C2×#52511yescharacteristic
625C22C2×#53511yescharacteristic
626C22C2×#53611yescharacteristic
627C22C22×AGL(1,ℤ/8)11yescharacteristic
628C22C22×AGL(1,ℤ/8)11yescharacteristic
629C22C2×#57511yescharacteristic
630C22C2×#57511yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—41no—
643C4—41no—
644C4—82no—
645C4—82no—
646C4—84no—
647C4—82no—
648C4—82no—
649C4#352014yes—
650C2—24no—
651C2—88no—
652C2#3524011yescharacteristic
653C2#6055811yescharacteristic
654C2#6055811yescharacteristic
655C2#6079011yescharacteristic
656C2#6079811yescharacteristic
657C2C2×#323411yescharacteristic
658C2#6086411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. #36640C2semidirect
3. #32959C2semidirect
4. #30076C2semidirect
5. #30075C2semidirect
6. #28722C2semidirect
7. #28516C2semidirect
8. #28207C2semidirect
9. #27953C2semidirect
10. #27585C2semidirect
11. #27582C2semidirect
13. #22226C2semidirect
14. #20778C2semidirect
20. C2×#478C22semidirect
23. #2859C22semidirect
24. #2859C22semidirect
25. #2859C22semidirect
26. #2859C22semidirect
41. C4×#125C22semidirect
42. C4×#125C22semidirect
52. #2094C22semidirect
53. #2093C22semidirect
54. #2084C22semidirect
55. #2083C22semidirect
56. #2071C22semidirect
57. #2071C22semidirect
60. #1992C22semidirect
62. #1926C22semidirect
63. #1925C22semidirect
64. #1864C22semidirect
65. #1862C22semidirect
68. #1847C22semidirect
69. #1839C22semidirect
70. #1831C22semidirect
72. #1804C22semidirect
76. #1767C22semidirect
78. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#366402
C2×C4×#1272
#329592
#300762
#300752
#287222
#285162
#282072
#279532
#275852
#275822
#258602
#222262
#207782