Finite Group Databasebeta

← #6792923

Subgroups — #6792923

Order 512.

1623 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36117C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33106C211yesmaximal, characteristic
4#31588C214yesmaximal, complemented
5#31470C214yesmaximal, complemented
6#29907C212yesmaximal, complemented
7#28800C212yesmaximal, complemented
8#27973C211yesmaximal, characteristic
9#27758C212yesmaximal, complemented
10#27680C211yesmaximal, characteristic, complemented
11#26530C212yesmaximal, complemented
12#26112C212yesmaximal, complemented
13#24608C212yesmaximal, complemented
14#24066C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16#21867C212yesmaximal, complemented
17C4×#483C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C22×#125—21no—
20C2×#483C2211yescharacteristic
21C2×#480C2212yes—
22C2×#432C2212yes—
23#2856C2211yescharacteristic
24#2854C2212yes—
25#2552—21no—
26#2552—22no—
27#2552—21no—
28#2552C2212yes—
29#2552C2214yes—
30#2552C2212yes—
31#2552C2211yescharacteristic
32#2548C2211yescharacteristic
33#2512—21no—
34#2501—21no—
35#2479—22no—
36#2454C2212yes—
37#2448C2212yes—
38#2425—22no—
39C2×#398—22no—
40#2366C2211yescharacteristic
41#2361—21no—
42#2354—21no—
43Q8×C4⋊C4C2211yescharacteristic
44C4×C22⋊Q8C2211yescharacteristic
45C2×#389—21no—
46C2×#389C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2100C2214yes—
52#2099C2212yes—
53#2096C2212yes—
54#2094C2214yes—
55#2087C2214yes—
56#2084C2214yes—
57#2071C2214yes—
58#2071C2214yes—
59#2065C2214yes—
60#2059C2214yes—
61#2042C2214yes—
62#1992C2214yes—
63#1982C2212yes—
64#1959C2214yes—
65#1957C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1944C2214yes—
69#1939C2212yes—
70#1936C2214yes—
71#1927C2212yes—
72#1925C2214yes—
73#1920C2212yes—
74#1917C2214yes—
75#1874C2212yes—
76#1874C2212yes—
77#1874C2212yes—
78#1872C2212yes—
79#1872C2211yescharacteristic
80#1864C2214yes—
81#1863C2214yes—
82#1853C2212yes—
83#1853C2211yescharacteristic
84#1834C2212yes—
85#1823C2211yescharacteristic
86#1817C2212yes—
87#1803C2212yes—
88#1803C2214yes—
89#1802C2214yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92#1770C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×#127C2311yescharacteristic
96#529—28no—
97#529—44no—
98C2×#126—22no—
99C2×#125—22no—
100C2×#125—22no—
101C2×#125—24no—
102C2×#125—48no—
103C2×#125C2312yes—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×#123—22no—
107C2×C22⋊Q8C2311yescharacteristic
108C2×#121—21no—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×#117—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#483—24no—
131#480—44no—
132#432—44no—
133C2×#107C2314yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×#106C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2311yescharacteristic
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#399—22no—
152#399—22no—
153#398—24no—
154#398—24no—
155#398—24no—
156#398—24no—
157#398—22no—
158#397—22no—
159#396—22no—
160#394—22no—
161#391C2311yescharacteristic
162#390—21no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389—24no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2314yes—
179#389C2312yes—
180#389C2312yes—
181#389C2311yescharacteristic
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—21no—
187#386—22no—
188#386—22no—
189#385—21no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384—22no—
194#384—22no—
195#384—21no—
196#384—21no—
197#384C2314yes—
198#384C2312yes—
199#384C2312yes—
200#384C2314yes—
201#384C2311yescharacteristic
202#380—22no—
203#380C2311yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C2×#95—21no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8C2411yescharacteristic
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2412yes—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C22×C8C2412yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251#127—24no—
252#126—48no—
253#125—24no—
254#125—42no—
255#125—416no—
256#125—48no—
257#125—48no—
258#124—24no—
259#124—42no—
260#123—48no—
261#123—84no—
262C22⋊Q8—22no—
263C22⋊Q8#4114yes—
264#121—24no—
265#121—24no—
266#121—44no—
267#121—82no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—42no—
271C4×D4—24no—
272C4×D4—42no—
273C4×D4—48no—
274C4×D4—44no—
275#117—48no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
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—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4—84no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—44no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—44no—
353#106—24no—
354#106—44no—
355#106—42no—
356C4⋊C8—24no—
357C4⋊C8—44no—
358#103—24no—
359#103—44no—
360#103—42no—
361#102—44no—
362#102—44no—
363#102—44no—
364#102—44no—
365C22⋊C8—22no—
366C22⋊C8—22no—
367C22⋊C8—24no—
368#97—44no—
369C4×C8—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—22no—
386C24C22×D411yescharacteristic
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8C2×#4112yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—81no—
403C2×D4—81no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—42no—
424C22×C4—41no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—82no—
437C22×C4AGL(1,ℤ/8)14yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4C2511yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—82no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4#13714yes—
499C4⋊C4C2×#4112yes—
500C22⋊C4—24no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C42—22no—
513C42—22no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—44no—
522C42—42no—
523C42—44no—
524C42—44no—
525C42—42no—
526C422−1+414yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23#54612yes—
539C23C2×AGL(1,ℤ/8)12yes—
540C23#57712yes—
541C23C23×D411yescharacteristic
542Q8—24no—
543Q8—42no—
544D4—44no—
545D4—44no—
546D4—88no—
547D4—84no—
548D4—162no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—41no—
577C2×C4—48no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—88no—
600C2×C4—84no—
601C2×C4—88no—
602C2×C4—84no—
603C2×C4#53612yes—
604C2×C4#53612yes—
605C2×C4#54612yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×#13714yes—
609C2×C4C2×#13712yes—
610C2×C4C22×#4111yescharacteristic
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4C2×2−1+412yes—
613C2×C4#58512yes—
614C8—42no—
615C8—42no—
616C8—44no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22#333214yes—
631C22C2×#53611yescharacteristic
632C22C2×#54611yescharacteristic
633C22#348711yescharacteristic
634C22C22×AGL(1,ℤ/8)11yescharacteristic
635C22C22×#13711yescharacteristic
636C22C2×#57711yescharacteristic
637C22#358511yescharacteristic
638C4—22no—
639C4—22no—
640C4—24no—
641C4—22no—
642C4—41no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—82no—
649C4—84no—
650C4—81no—
651C4—81no—
652C4—82no—
653C4—82no—
654C4#323814yes—
655C4#351512yes—
656C4#355912yes—
657C2—24no—
658C2—162no—
659C2#3890711yescharacteristic
660C2C2×#323811yescharacteristic
661C2#6087311yescharacteristic
662C2C2×#333211yescharacteristic
663C2#6116211yescharacteristic
664C2#6189011yescharacteristic
665C2#6190611yescharacteristic
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
2. C2×#2552C2semidirect
4. #31588C2semidirect
5. #31470C2semidirect
6. #29907C2semidirect
7. #28800C2semidirect
9. #27758C2semidirect
10. #27680C2semidirect
11. #26530C2semidirect
12. #26112C2semidirect
13. #24608C2semidirect
14. #24066C2semidirect
15. #22603C2semidirect
16. #21867C2semidirect
17. C4×#483C2semidirect

Maximal subgroups

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

subgroupindex
#361172
C2×#25522
#331062
#315882
#314702
#299072
#288002
#279732
#277582
#276802
#265302
#261122
#246082
#240662
#226032
#218672
C4×#4832