Finite Group Databasebeta

← #6849000

Subgroups — #6849000

Order 512.

1535 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#31749C214yesmaximal, complemented
5#31127C214yesmaximal, complemented
6#29692C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#28792C212yesmaximal, complemented
9#28284C211yesmaximal, characteristic, complemented
10#27748C212yesmaximal, complemented
11#27649C212yesmaximal, complemented
12#26233C212yesmaximal, complemented
13#24968C212yesmaximal
14#24702C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16#21862C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C2×#485C2214yes—
20C2×#477C2212yes—
21C2×#476C2211yescharacteristic
22C2×#474C2212yes—
23C2×#431C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2571C2212yes—
27#2563—21no—
28#2553—22no—
29#2552C2211yescharacteristic
30#2547C2211yescharacteristic
31#2542C2211yescharacteristic
32#2538—21no—
33#2537C2211yescharacteristic
34#2497—22no—
35#2496—21no—
36#2492—22no—
37#2489C2212yes—
38#2439C2214yes—
39#2427C2212yes—
40#2422—21no—
41#2395—22no—
42#2369—21no—
43#2355—21no—
44#2354C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C4×#121—21no—
49#2285—21no—
50#2285C2212yes—
51C42×D4C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2085C2214yes—
54#2076C2214yes—
55#2074C2214yes—
56#2068C2214yes—
57#2059C2214yes—
58#2042C2214yes—
59#2036C2214yes—
60#2035C2214yes—
61#2033C2214yes—
62#2024C2214yes—
63#1991C2214yes—
64#1988C2212yes—
65#1987C2211yescharacteristic
66#1981C2212yes—
67#1958C2214yes—
68#1956C2212yes—
69#1954C2212yes—
70#1948C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1942C2214yes—
75#1933C2214yes—
76#1923C2214yes—
77#1920C2212yes—
78#1875C2214yes—
79#1873C2214yes—
80#1864C2212yes—
81#1842C2211yescharacteristic
82#1834C2212yes—
83#1825C2214yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1797C2212yes—
91C4×#106C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100#529—28no—
101#529—44no—
102C2×#126—22no—
103C2×#125C2312yes—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×C4×Q8—21no—
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C2×#117C2314yes—
122C22×C42—21no—
123C22×C42C2312yes—
124#485—48no—
125#477—44no—
126#476—24no—
127#474—44no—
128#431—44no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2314yes—
149#400—24no—
150#400—22no—
151#399—22no—
152#398—22no—
153#397—22no—
154#397—22no—
155#397—22no—
156#396—24no—
157#394—22no—
158#394—22no—
159#393—24no—
160#391—21no—
161#391—22no—
162#391—24no—
163#391C2312yes—
164#391C2312yes—
165#390—22no—
166#390—21no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2314yes—
175#389C2312yes—
176#388—24no—
177#388—22no—
178#387—22no—
179#387—24no—
180#386—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#382—22no—
191#382—22no—
192#382C2314yes—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—22no—
200#380C2312yes—
201#380C2314yes—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C43C2311yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2412yes—
238C22×D4—22no—
239C22×D4C2×D411yescharacteristic
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—42no—
255C4⋊Q8—42no—
256C4⋊Q8—44no—
257C4⋊Q8—82no—
258#126—48no—
259#125—24no—
260#125—42no—
261#124—24no—
262#124—42no—
263#123—48no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266C22⋊Q8—42no—
267C22⋊Q8—42no—
268C22⋊Q8—84no—
269#121—24no—
270#121—42no—
271#121—44no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×D4—22no—
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—42no—
283C4×D4—41no—
284C4×D4—48no—
285C4×D4#4114yes—
286#117—48no—
287#117—44no—
288#117—44no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—21no—
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×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42—44no—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—28no—
353#107—44no—
354#106—24no—
355#106—24no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360C4⋊C8—42no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#102—24no—
366#102—42no—
367#102—44no—
368C22⋊C8—44no—
369C22⋊C8—44no—
370C4×C8—24no—
371C4×C8—44no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—21no—
391C24C22×D412yes—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—82no—
415C2×D4C2×#4112yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—81no—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—84no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4C2×#4112yes—
500C4⋊C4H(5,𝔽2)12yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—21no—
523C42—41no—
524C42—44no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—44no—
529C42—84no—
530C42#13514yes—
531C42C2×#4112yes—
532C42H(5,𝔽2)12yes—
533C422−1+412yes—
534C422−1+412yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C23×D411yescharacteristic
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—84no—
555Q8—164no—
556D4—22no—
557D4—22no—
558D4—44no—
559D4—41no—
560D4—41no—
561D4—88no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4#53612yes—
614C2×C4#53612yes—
615C2×C4#54612yes—
616C2×C4#54612yes—
617C2×C4C2×#13512yes—
618C2×C4C2×#13512yes—
619C2×C4C2×#13512yes—
620C2×C4#57512yes—
621C2×C4#57812yes—
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C2×C4C2×2−1+411yescharacteristic
627C2×C4#58511yescharacteristic
628C8—42no—
629C8—42no—
630C8—44no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22C2×#53611yescharacteristic
646C22C2×#54611yescharacteristic
647C22#348811yescharacteristic
648C22C22×#13511yescharacteristic
649C22C2×#57511yescharacteristic
650C22C2×#57811yescharacteristic
651C22#358511yescharacteristic
652C4—22no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—82no—
662C4—82no—
663C4—81no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4#323314yes—
668C4#330214yes—
669C4#350711yescharacteristic
670C4#351111yescharacteristic
671C4#354711yescharacteristic
672C4#354811yescharacteristic
673C2—82no—
674C2—82no—
675C2#3947011yescharacteristic
676C2C2×#323311yescharacteristic
677C2#6089311yescharacteristic
678C2C2×#330211yescharacteristic
679C2#6116611yescharacteristic
680C2#6194911yescharacteristic
681C2#6199811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #33195C2semidirect
4. #31749C2semidirect
5. #31127C2semidirect
6. #29692C2semidirect
7. #29141C2semidirect
8. #28792C2semidirect
9. #28284C2semidirect
10. #27748C2semidirect
11. #27649C2semidirect
12. #26233C2semidirect
14. #24702C2semidirect
15. #22795C2semidirect
16. #21862C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349322
#331952
#317492
#311272
#296922
#291412
#287922
#282842
#277482
#276492
#262332
#249682
#247022
#227952
#218622
C4×#4762