Finite Group Databasebeta

← #6849003

Subgroups — #6849003

Order 512.

1573 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34934C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#31751C214yesmaximal, complemented
5#31129C214yesmaximal, complemented
6#29689C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic, complemented
8#28792C212yesmaximal, complemented
9#28279C211yesmaximal, characteristic, complemented
10#27750C212yesmaximal, complemented
11#27646C212yesmaximal, complemented
12#26235C212yesmaximal
13#24970C212yesmaximal, complemented
14#24700C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16#21862C212yesmaximal, complemented
17C4×#475C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C2×#482C2214yes—
20C2×#477C2212yes—
21C2×#475C2211yescharacteristic
22C2×#474C2212yes—
23C2×#431C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2571C2212yes—
27#2563—21no—
28#2554—22no—
29#2552C2211yescharacteristic
30#2548C2211yescharacteristic
31#2541C2211yescharacteristic
32#2538—21no—
33#2538C2211yescharacteristic
34#2512—21no—
35#2497—22no—
36#2492—22no—
37#2490C2212yes—
38#2445—21no—
39#2439C2214yes—
40#2426C2212yes—
41#2392—22no—
42#2369—21no—
43#2355—21no—
44#2354C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127—21no—
47C4×#127C2211yescharacteristic
48C4×#121C2212yes—
49#2293—21no—
50#2285C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2085C2214yes—
54#2075C2214yes—
55#2073C2214yes—
56#2067C2214yes—
57#2059C2214yes—
58#2042C2214yes—
59#2036C2214yes—
60#2034C2214yes—
61#2033C2214yes—
62#2023C2214yes—
63#1991C2214yes—
64#1988C2211yescharacteristic
65#1987C2212yes—
66#1982C2212yes—
67#1958C2214yes—
68#1957C2212yes—
69#1954C2212yes—
70#1948C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1942C2214yes—
75#1934C2214yes—
76#1923C2214yes—
77#1919C2212yes—
78#1875C2214yes—
79#1873C2214yes—
80#1865C2212yes—
81#1842C2211yescharacteristic
82#1834C2212yes—
83#1825C2214yes—
84#1803C2211yescharacteristic
85#1803C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1797C2212yes—
91C4×#106C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#127—21no—
100C2×#127C2311yescharacteristic
101#529—28no—
102#529—44no—
103C2×#126—22no—
104C2×#125C2312yes—
105C2×#124C2312yes—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×C4×Q8—21no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117C2314yes—
125C22×C42C2312yes—
126#482—48no—
127#477—44no—
128#475—24no—
129#474—44no—
130#431—44no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2314yes—
151#400—24no—
152#398—22no—
153#397—22no—
154#396—22no—
155#396—24no—
156#396—22no—
157#394—22no—
158#394—22no—
159#394—22no—
160#393—24no—
161#392—22no—
162#391—21no—
163#391—22no—
164#391—24no—
165#390—21no—
166#390—22no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—22no—
171#390—22no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#389—22no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389C2312yes—
182#389C2314yes—
183#389C2312yes—
184#388—24no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—24no—
189#386—22no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#382—22no—
198#382C2314yes—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2314yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222C43—21no—
223C43C2311yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2411yescharacteristic
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—42no—
254C4⋊Q8—42no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257#127—22no—
258#127—22no—
259#127—44no—
260#126—48no—
261#125—24no—
262#125—42no—
263#124—24no—
264#124—42no—
265#123—48no—
266C22⋊Q8—24no—
267C22⋊Q8—42no—
268C22⋊Q8—84no—
269#121—24no—
270#121—24no—
271#121—42no—
272#121—42no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—41no—
278C4×Q8—44no—
279C4×Q8#4114yes—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—42no—
286C4×D4—48no—
287C4×D4—44no—
288C4×D4—82no—
289C4×D4—84no—
290#117—48no—
291#117—44no—
292#117—44no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—21no—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—28no—
359#107—44no—
360#106—24no—
361#106—24no—
362C4⋊C8—22no—
363C4⋊C8—22no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—42no—
367#103—24no—
368#103—42no—
369#103—44no—
370#102—44no—
371#102—44no—
372#102—44no—
373#102—44no—
374C22⋊C8—44no—
375C22⋊C8—44no—
376C4×C8—24no—
377C4×C8—44no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—22no—
393C24C22×D412yes—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8C2×#4112yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—81no—
424C2×D4—81no—
425C2×D4—82no—
426C2×D4—82no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4C2×#4112yes—
504C4⋊C4H(5,𝔽2)12yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C22⋊C4—82no—
520C22⋊C4—84no—
521C42—24no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—21no—
529C42—21no—
530C42—41no—
531C42—42no—
532C42—42no—
533C42—44no—
534C42—42no—
535C42—42no—
536C42—81no—
537C42—84no—
538C42#13514yes—
539C42C2×#4112yes—
540C42H(5,𝔽2)12yes—
541C422−1+412yes—
542C422−1+412yes—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23C23×D411yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562Q8—84no—
563Q8—164no—
564D4—44no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—84no—
570D4—88no—
571D4—88no—
572D4—84no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—48no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—84no—
621C2×C4—88no—
622C2×C4#53612yes—
623C2×C4#53612yes—
624C2×C4#54612yes—
625C2×C4#54612yes—
626C2×C4C2×#13512yes—
627C2×C4C2×#13512yes—
628C2×C4C2×#13512yes—
629C2×C4#57512yes—
630C2×C4#57612yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C2×C4#58511yescharacteristic
637C8—42no—
638C8—42no—
639C8—44no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22C2×#53611yescharacteristic
655C22C2×#54611yescharacteristic
656C22#348811yescharacteristic
657C22C22×#13511yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57611yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—41no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4—82no—
675C4—81no—
676C4#323314yes—
677C4#330514yes—
678C4#350711yescharacteristic
679C4#351111yescharacteristic
680C4#354711yescharacteristic
681C4#354811yescharacteristic
682C2—82no—
683C2—162no—
684C2#3947011yescharacteristic
685C2C2×#323311yescharacteristic
686C2#6088811yescharacteristic
687C2C2×#330511yescharacteristic
688C2#6116511yescharacteristic
689C2#6195411yescharacteristic
690C2#6199511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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. #33198C2semidirect
4. #31751C2semidirect
5. #31129C2semidirect
6. #29689C2semidirect
7. #29139C2semidirect
8. #28792C2semidirect
9. #28279C2semidirect
10. #27750C2semidirect
11. #27646C2semidirect
13. #24970C2semidirect
14. #24700C2semidirect
15. #22795C2semidirect
16. #21862C2semidirect
17. C4×#475C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349342
#331982
#317512
#311292
#296892
#291392
#287922
#282792
#277502
#276462
#262352
#249702
#247002
#227952
#218622
C4×#4752