Finite Group Databasebeta

← #6800412

Subgroups — #6800412

Order 512.

1459 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#33147C211yesmaximal, characteristic, complemented
4#32339C214yesmaximal, complemented
5#31483C214yesmaximal, complemented
6#29968C212yesmaximal, complemented
7#28799C212yesmaximal, complemented
8#27969C212yesmaximal, complemented
9#27960C211yesmaximal, characteristic, complemented
10#27736C212yesmaximal, complemented
11#27651C212yesmaximal
12#27644C211yesmaximal, characteristic
13#26509C212yesmaximal, complemented
14#26107C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22081C212yesmaximal
17C4×#481C211yesmaximal, characteristic
18C2×#554—21no—
19C2×#481C2211yescharacteristic
20C2×#480C2212yes—
21#2857C2212yes—
22#2857C2211yescharacteristic
23#2571—21no—
24#2571—21no—
25#2571—22no—
26#2571C2214yes—
27#2571C2212yes—
28#2571C2212yes—
29#2564—22no—
30#2556—22no—
31#2551C2211yescharacteristic
32#2542C2211yescharacteristic
33#2506C2212yes—
34#2492—21no—
35#2452C2212yes—
36#2417—21no—
37C2×#400—22no—
38#2365C2211yescharacteristic
39#2360—21no—
40D4×C4⋊C4C2211yescharacteristic
41C4×#121C2212yes—
42C4×#121C2211yescharacteristic
43C2×#389—21no—
44C2×#389C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382—21no—
47C2×#382C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2088C2214yes—
55#2088C2214yes—
56#2085C2214yes—
57#2083C2212yes—
58#2073C2214yes—
59#2071C2212yes—
60#2058C2214yes—
61#1993C2214yes—
62#1990C2212yes—
63#1988C2212yes—
64#1949C2214yes—
65#1948C2212yes—
66#1947C2212yes—
67#1943C2214yes—
68#1939C2212yes—
69#1936C2214yes—
70#1926C2212yes—
71#1921C2214yes—
72#1919C2212yes—
73#1919C2214yes—
74#1917C2214yes—
75#1874C2211yescharacteristic
76#1864C2214yes—
77#1853C2211yescharacteristic
78#1853C2212yes—
79#1836C2214yes—
80#1834C2212yes—
81#1824C2212yes—
82#1824C2211yescharacteristic
83#1823C2212yes—
84#1823C2212yes—
85#1823C2212yes—
86#1804C2214yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91#1770C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95#554—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#125—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—21no—
105C2×#121C2311yescharacteristic
106C2×C4×Q8—21no—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4C2311yescharacteristic
110C2×C4×D4C2312yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123#481—24no—
124#480—28no—
125C2×#107C2314yes—
126C2×#106C2312yes—
127C2×#106C2312yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×#102C2314yes—
132C2×#102C2312yes—
133C2×#102C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2311yescharacteristic
136C2×C22⋊C8C2311yescharacteristic
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139#400—22no—
140#400—24no—
141#400—24no—
142#400—24no—
143#400—24no—
144#399—22no—
145#398—22no—
146#396—22no—
147#395—22no—
148#393—22no—
149#391—21no—
150#391—21no—
151#391—22no—
152#390C2311yescharacteristic
153#390C2312yes—
154#389—22no—
155#389—22no—
156#389—24no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389C2314yes—
163#389C2314yes—
164#389C2311yescharacteristic
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#388—22no—
170#388—21no—
171#387—22no—
172#387—22no—
173#386—21no—
174#386—22no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384—21no—
179#384C2312yes—
180#384C2312yes—
181#384C2311yescharacteristic
182#384C2312yes—
183#382—22no—
184#382—24no—
185#382—22no—
186#382—22no—
187#382—22no—
188#382C2314yes—
189#382C2314yes—
190#380—21no—
191#380—22no—
192#380—21no—
193#380C2312yes—
194#380C2311yescharacteristic
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C22⋊C4—21no—
208C4×C22⋊C4—22no—
209C2×#95—21no—
210C2×#95—22no—
211C2×#95—22no—
212C2×#95—22no—
213C2×#95D412yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#336C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×D4—21no—
234C22×D4C2411yescharacteristic
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2411yescharacteristic
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C4⋊Q8—24no—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251#125—48no—
252C22⋊Q8—44no—
253C22⋊Q8—44no—
254C22⋊Q8—48no—
255C22⋊Q8—48no—
256C22⋊Q8—44no—
257#121—22no—
258#121#4114yes—
259C4×Q8—44no—
260C4×D4—24no—
261C4×D4—44no—
262C4×D4—48no—
263C4×D4—44no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—84no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2412yes—
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—28no—
334#106—24no—
335#106—42no—
336#106—44no—
337C4⋊C8—24no—
338C4⋊C8—44no—
339#102—28no—
340#102—24no—
341#102—42no—
342C22⋊C8—24no—
343C22⋊C8—22no—
344C22⋊C8—22no—
345#97—44no—
346C4×C8—24no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95—22no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24—21no—
369C24C22×D411yescharacteristic
370C2×Q8—44no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—82no—
375C2×Q8—82no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4C2×#4112yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4AGL(1,ℤ/8)14yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4112yes—
430C22×C42−1+412yes—
431C22×C4C2511yescharacteristic
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4C2×#4112yes—
479C22⋊C4—24no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—81no—
487C22⋊C4—82no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—24no—
492C42—24no—
493C42—42no—
494C42—42no—
495C42—44no—
496C42—44no—
497C42—42no—
498C42—44no—
499C422−1+414yes—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23#54812yes—
512C23C2×AGL(1,ℤ/8)12yes—
513C23#57712yes—
514C23C23×D411yescharacteristic
515Q8—84no—
516Q8—164no—
517D4—24no—
518D4—48no—
519D4—42no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—88no—
574C2×C4—84no—
575C2×C4—164no—
576C2×C4#53612yes—
577C2×C4#53612yes—
578C2×C4#54812yes—
579C2×C4C2×AGL(1,ℤ/8)12yes—
580C2×C4C2×AGL(1,ℤ/8)12yes—
581C2×C4#57512yes—
582C2×C4C22×#4111yescharacteristic
583C2×C4C2×2−1+412yes—
584C2×C4C2×2−1+411yescharacteristic
585C2×C4#58512yes—
586C8—42no—
587C8—44no—
588C8—42no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22#337314yes—
603C22C2×#53611yescharacteristic
604C22C2×#54811yescharacteristic
605C22#348511yescharacteristic
606C22C22×AGL(1,ℤ/8)11yescharacteristic
607C22C2×#57511yescharacteristic
608C22C2×#57711yescharacteristic
609C22#358611yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—41no—
619C4—82no—
620C4—81no—
621C4—82no—
622C4—84no—
623C4—82no—
624C4—81no—
625C4—82no—
626C4#323414yes—
627C4#351512yes—
628C4#356312yes—
629C2—24no—
630C2—82no—
631C2#3912211yescharacteristic
632C2C2×#323411yescharacteristic
633C2#6087211yescharacteristic
634C2C2×#337311yescharacteristic
635C2#6132311yescharacteristic
636C2#6172211yescharacteristic
637C2#6173111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2571C2semidirect
3. #33147C2semidirect
4. #32339C2semidirect
5. #31483C2semidirect
6. #29968C2semidirect
7. #28799C2semidirect
8. #27969C2semidirect
9. #27960C2semidirect
10. #27736C2semidirect
13. #26509C2semidirect
14. #26107C2semidirect
15. #22629C2semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#25712
#331472
#323392
#314832
#299682
#287992
#279692
#279602
#277362
#276512
#276442
#265092
#261072
#226292
#220812
C4×#4812