Finite Group Databasebeta

← #6482125

Subgroups — #6482125

Order 512.

1974 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3238C212yesmaximal, complemented
2#35103C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic, complemented
4#33111C212yesmaximal, complemented
5#33108C211yesmaximal, characteristic, complemented
6#33106C211yesmaximal, characteristic
7#29760C212yesmaximal, complemented
8#29756C211yesmaximal, characteristic, complemented
9#29754C211yesmaximal, characteristic
10#25463C214yesmaximal, complemented
11#25461C214yesmaximal, complemented
12#25286C214yesmaximal, complemented
13#25284C214yesmaximal, complemented
14#21955C211yesmaximal, characteristic, complemented
15#21391C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#529C2212yes—
18C22×C4×Q8—21no—
19#3238—216no—
20C2×#501C2214yes—
21C2×#489C2214yes—
22C2×#483C2212yes—
23C2×#483C2212yes—
24C2×#480C2212yes—
25C2×#480C2212yes—
26C2×#477C2214yes—
27C2×#476C2214yes—
28C2×#432C2212yes—
29#2857C2211yescharacteristic
30#2856C2211yescharacteristic
31#2850C2212yes—
32#2618—21no—
33#2618—21no—
34#2601—22no—
35#2601—22no—
36#2566C2212yes—
37#2565C2212yes—
38#2545C2212yes—
39#2544C2212yes—
40#2365—22no—
41#2365—22no—
42#2365C2214yes—
43#2365C2214yes—
44#2325—21no—
45#2325—21no—
46C2×#391—21no—
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387—21no—
50C2×#387—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55#2085C2214yes—
56#2084C2212yes—
57#2083C2212yes—
58#2068C2214yes—
59#2067C2214yes—
60#2066C2214yes—
61#2065C2214yes—
62#2043C2212yes—
63#2042C2214yes—
64#2041C2212yes—
65#1989C2214yes—
66#1983C2214yes—
67#1983C2214yes—
68#1982C2212yes—
69#1981C2212yes—
70#1949C2214yes—
71#1948C2212yes—
72#1947C2212yes—
73#1940C2214yes—
74#1940C2214yes—
75#1928C2214yes—
76#1927C2214yes—
77#1926C2214yes—
78#1925C2214yes—
79#1874C2212yes—
80#1874C2212yes—
81#1872C2212yes—
82#1872C2212yes—
83#1797C2211yescharacteristic
84#1797C2211yescharacteristic
85#1797C2212yes—
86#1763C2214yes—
87#1762C2214yes—
88C23×Q8—21no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8D418yescomplemented
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2314yes—
94C2×#127C2311yescharacteristic
95#529—28no—
96#529—28no—
97C2×#126—22no—
98C2×#125C2312yes—
99C2×#124C2312yes—
100C2×#124C2312yes—
101C2×#123—22no—
102C2×C22⋊Q8C2312yes—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×Q8—48no—
109C2×C4×Q8C2312yes—
110C2×C4×D4—22no—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C42—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#501—216no—
129#489—216no—
130#483—28no—
131#483—28no—
132#480—28no—
133#480—28no—
134#477—216no—
135#476—216no—
136#432—28no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×#97C2314yes—
156#398—22no—
157#398—22no—
158#396—22no—
159#396—22no—
160#396—22no—
161#396—22no—
162#394—22no—
163#394—22no—
164#391—24no—
165#391—24no—
166#391C2312yes—
167#391C2312yes—
168#390—21no—
169#390—21no—
170#390C2312yes—
171#390C2312yes—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389C2314yes—
178#389C2312yes—
179#389C2312yes—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—22no—
186#387—24no—
187#387—24no—
188#387—24no—
189#387—24no—
190#386—22no—
191#386—22no—
192#386—22no—
193#386—22no—
194#385—22no—
195#385—22no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#382C2312yes—
199#380—21no—
200#380—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95—21no—
218C2×#95D411yescharacteristic
219C2×#95D411yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#336C2312yes—
225#336C2312yes—
226#336C2314yes—
227C22×Q8—28no—
228C22×Q8—24no—
229C22×Q8—44no—
230C22×Q8—44no—
231C22×Q8C2412yes—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—216no—
251C4⋊Q8—28no—
252C4⋊Q8—42no—
253C4⋊Q8C2×D4116yes—
254#127—42no—
255#126—48no—
256#125—24no—
257#125—24no—
258#124—24no—
259#124—24no—
260#124—44no—
261#123—48no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264#121—24no—
265#121—24no—
266#121—44no—
267#121—44no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—416no—
271C4×Q8—416no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—48no—
275#117—48no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42—44no—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D414yes—
341C2×C42#4114yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346#107—28no—
347#107—28no—
348#106—28no—
349#106—28no—
350C4⋊C8—28no—
351C4⋊C8—44no—
352C4⋊C8—42no—
353C4⋊C8—42no—
354#103—28no—
355#103—28no—
356#103—28no—
357#103—44no—
358#102—28no—
359#102—28no—
360#102—28no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364#97—28no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—22no—
383C24C22×D411yescharacteristic
384C2×Q8—216no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—48no—
390C2×Q8—432no—
391C2×Q8—48no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—81no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4—81no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4#12114yes—
445C22×C4#12114yes—
446C22×C4AGL(1,ℤ/8)14yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C4C2511yescharacteristic
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C4⋊C4—24no—
466C4⋊C4—216no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4C22×D414yes—
496C4⋊C4C22×D414yes—
497C4⋊C42−1+414yes—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—41no—
513C42—41no—
514C42—42no—
515C42—48no—
516C42—48no—
517C42—44no—
518C42—44no—
519C42—44no—
520C42#12118yes—
521C42C22×D414yes—
522C42C2×#4114yes—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23C2×#12112yes—
535C23C2×AGL(1,ℤ/8)12yes—
536C23#57512yes—
537C23C23×D411yescharacteristic
538Q8—432no—
539Q8—44no—
540Q8—44no—
541D4—44no—
542D4—44no—
543D4—88no—
544D4—162no—
545D4—162no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—88no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—84no—
602C2×C4C2×#12114yes—
603C2×C4C2×#12112yes—
604C2×C4C2×#12114yes—
605C2×C4C2×AGL(1,ℤ/8)12yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C23×D411yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×2−1+412yes—
610C2×C4#58512yes—
611C8—44no—
612C8—44no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22#303614yes—
627C22C22×#12111yescharacteristic
628C22#348511yescharacteristic
629C22#348611yescharacteristic
630C22C22×AGL(1,ℤ/8)11yescharacteristic
631C22C2×#57511yescharacteristic
632C22#358611yescharacteristic
633C22#358611yescharacteristic
634C4—24no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4#346112yes—
651C4#349712yes—
652C2—24no—
653C2—162no—
654C2#3515311yescharacteristic
655C2C2×#303611yescharacteristic
656C2#6042411yescharacteristic
657C2#6170611yescharacteristic
658C2#6171211yescharacteristic
659C2#6182711yescharacteristic
660C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3238C2semidirect
3. C2×#2365C2semidirect
4. #33111C2semidirect
5. #33108C2semidirect
7. #29760C2semidirect
8. #29756C2semidirect
10. #25463C2semidirect
11. #25461C2semidirect
12. #25286C2semidirect
13. #25284C2semidirect
14. #21955C2semidirect
15. #21391C2semidirect
91. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32382
#351032
C2×#23652
#331112
#331082
#331062
#297602
#297562
#297542
#254632
#254612
#252862
#252842
#219552
#213912