Finite Group Databasebeta

← #6514559

Subgroups — #6514559

Order 512.

1734 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#29288C212yesmaximal, complemented
4#29274C212yesmaximal, complemented
5#29256C212yesmaximal, complemented
6#28286C212yesmaximal, complemented
7#26420C214yesmaximal, complemented
8#26348C214yesmaximal, complemented
9#26107C212yesmaximal, complemented
10#26087C212yesmaximal, complemented
11#26041C212yesmaximal, complemented
12#26021C212yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21972C211yesmaximal, characteristic, complemented
15#21884C211yesmaximal, characteristic
16#21386C211yesmaximal, characteristic
17C2×#1948C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520—21no—
20C2×#478C2214yes—
21C2×#431C2211yescharacteristic
22C22×#106C2212yes—
23C22×C4⋊C8C2211yescharacteristic
24#2571C2211yescharacteristic
25#2571C2211yescharacteristic
26#2570—21no—
27#2570—21no—
28#2569—22no—
29#2552C2211yescharacteristic
30#2552C2211yescharacteristic
31#2442C2212yes—
32#2442C2212yes—
33#2402—22no—
34#2361—22no—
35#2361C2214yes—
36#2354—21no—
37#2354C2212yes—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40#2326—22no—
41#2311—21no—
42C4×C4⋊Q8C2212yes—
43C4×#127C2212yes—
44C4×#125C2212yes—
45C4×#125C2212yes—
46#2296—21no—
47#2285—21no—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C4×#117C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2101C2214yes—
53#2092C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2067C2214yes—
57#2065C2214yes—
58#2059C2214yes—
59#2025C2214yes—
60#1989C2214yes—
61#1988C2211yescharacteristic
62#1988C2212yes—
63#1988C2211yescharacteristic
64#1982C2211yescharacteristic
65#1982C2212yes—
66#1982C2211yescharacteristic
67#1949C2214yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1939C2212yes—
76#1921C2214yes—
77#1919C2214yes—
78#1893C2214yes—
79#1874C2212yes—
80#1874C2212yes—
81#1864C2214yes—
82#1862C2214yes—
83#1852C2214yes—
84#1850C2214yes—
85#1845C2212yes—
86#1842C2212yes—
87#1834C2212yes—
88#1831C2212yes—
89#1804C2214yes—
90#1802C2212yes—
91#1802C2212yes—
92C4×#102C2212yes—
93C4×#102C2212yes—
94C2×#340C2212yes—
95C22×#41—21no—
96C23×C8C2312yes—
97#532—24no—
98#532Q818yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#125C2312yes—
102C2×#125C2312yes—
103C2×#124—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106#520—48no—
107C2×C4×Q8—22no—
108C2×C4×D4—24no—
109C2×C4×D4—21no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2314yes—
114C2×#117—21no—
115C2×#117—22no—
116C2×#117D411yescharacteristic
117C2×#117D411yescharacteristic
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126#478—48no—
127#431—24no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106—44no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8—24no—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#400—22no—
150#400—22no—
151#400—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#398—22no—
156#391—22no—
157#390—22no—
158#390—22no—
159#390C2314yes—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#388—24no—
171#388—24no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#386—24no—
177#385—22no—
178#385—22no—
179#385—24no—
180#385—24no—
181#384—21no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#382—21no—
190#382—21no—
191#382—21no—
192#382C2312yes—
193#382C2312yes—
194#380—21no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C2×#95—21no—
222C43C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235C2×#41—28no—
236C2×#41—44no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C22×C8—24no—
252C22×C8—42no—
253C22×C8—42no—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—22no—
258C4⋊Q8C2×Q814yes—
259#127—22no—
260#127C2×Q814yes—
261#125—24no—
262#125—24no—
263#125—42no—
264#125—42no—
265#124—28no—
266#121—28no—
267#121—24no—
268#121C2×Q818yes—
269C4×Q8—48no—
270C4×D4—28no—
271C4×D4—24no—
272C4×D4—48no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—416no—
276C4×D4—84no—
277C4×D4C2×Q818yes—
278#117—24no—
279#117—24no—
280#117—48no—
281#117—44no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—41no—
306C2×C4⋊C4—41no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—44no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×Q814yes—
346C2×C42#4114yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—44no—
352#107—44no—
353#106—28no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—24no—
357C4⋊C8—22no—
358C4⋊C8—22no—
359C4⋊C8—44no—
360C4⋊C8—44no—
361#102—24no—
362#102—24no—
363#102—42no—
364#102—42no—
365#102—44no—
366#102—44no—
367C22⋊C8—24no—
368C4×C8—24no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—22no—
390C24C22×D411yescharacteristic
391#41—432no—
392C2×Q8—24no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4C22×Q814yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—28no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4C22⋊Q814yes—
440C22×C4C22⋊Q814yes—
441C22×C4#13514yes—
442C22×C4AGL(1,ℤ/8)14yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×Q812yes—
453C22×C4C2×#4112yes—
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—
464C2×C8—48no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4C22×Q814yes—
504C4⋊C4H(5,𝔽2)12yes—
505C4⋊C42−1+412yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—42no—
521C42—42no—
522C42—48no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42C22⋊Q814yes—
529C42C22⋊Q814yes—
530C42C22×Q812yes—
531C42C22×Q812yes—
532C42C2×#4112yes—
533C42C2×#4112yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×C22⋊Q812yes—
546C23C2×#13512yes—
547C23C2×AGL(1,ℤ/8)12yes—
548C23C23×D411yescharacteristic
549Q8—48no—
550D4—24no—
551D4—24no—
552D4—48no—
553D4—416no—
554D4—48no—
555D4—42no—
556D4—42no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
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—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—416no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—168no—
608C2×C4—164no—
609C2×C4#478116yescomplemented
610C2×C4C2×C22⋊Q812yes—
611C2×C4C2×C22⋊Q812yes—
612C2×C4C2×C22⋊Q812yes—
613C2×C4C2×C22⋊Q812yes—
614C2×C4C2×C22⋊Q812yes—
615C2×C4C2×#13512yes—
616C2×C4C2×#13512yes—
617C2×C4C2×AGL(1,ℤ/8)12yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C23×Q811yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4#58511yescharacteristic
624C2×C4#58511yescharacteristic
625C8—42no—
626C8—44no—
627C8—42no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22C2×#47814yes—
642C22C22×C22⋊Q811yescharacteristic
643C22#348811yescharacteristic
644C22#348911yescharacteristic
645C22C22×#13511yescharacteristic
646C22C22×AGL(1,ℤ/8)11yescharacteristic
647C22#358511yescharacteristic
648C22#358611yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—84no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4C2×#47814yes—
665C4C2×#47814yes—
666C4#350311yescharacteristic
667C4#350411yescharacteristic
668C4#350611yescharacteristic
669C4#350711yescharacteristic
670C2—24no—
671C2—84no—
672C2#3551911yescharacteristic
673C2C22×#47811yescharacteristic
674C2#6049211yescharacteristic
675C2#6195011yescharacteristic
676C2#6196311yescharacteristic
677C2#6209811yescharacteristic
678C2#6211111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34948C2semidirect
3. #29288C2semidirect
4. #29274C2semidirect
5. #29256C2semidirect
6. #28286C2semidirect
7. #26420C2semidirect
8. #26348C2semidirect
9. #26107C2semidirect
10. #26087C2semidirect
11. #26041C2semidirect
12. #26021C2semidirect
13. #21982C2semidirect
14. #21972C2semidirect
17. C2×#1948C2semidirect
609. C2×C4#478semidirect

Maximal subgroups

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

subgroupindex
#349482
#342072
#292882
#292742
#292562
#282862
#264202
#263482
#261072
#260872
#260412
#260212
#219822
#219722
#218842
#213862
C2×#19482