Finite Group Databasebeta

← #6508454

Subgroups — #6508454

Order 512.

1990 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic, complemented
2C2×#2452C211yesmaximal, characteristic, complemented
3#29103C211yesmaximal, characteristic, complemented
4#29095C214yesmaximal, complemented
5#28974C214yesmaximal, complemented
6#28301C211yesmaximal, characteristic
7#26834C214yesmaximal, complemented
8#26490C214yesmaximal, complemented
9#25903C211yesmaximal, characteristic, complemented
10#25811C211yesmaximal, characteristic, complemented
11#24608C212yesmaximal, complemented
12#24506C212yesmaximal, complemented
13#22081C212yesmaximal, complemented
14#21452C212yesmaximal, complemented
15#20534C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C4×D4C2211yescharacteristic
20C2×#498C2212yescomplemented
21C2×#486C2212yescomplemented
22C2×#480C2212yes—
23C2×#447C2212yes—
24C2×#442C2214yes—
25C2×#432C2212yes—
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2854C2212yes—
29#2639—21no—
30#2618—21no—
31#2560—22no—
32#2558—22no—
33#2512C2212yescomplemented
34#2506C2212yescomplemented
35#2452—22no—
36#2452—22no—
37#2452C2214yes—
38#2452C2214yes—
39#2452C2212yescomplemented
40#2445C2212yescomplemented
41C2×#396—21no—
42C2×#393—21no—
43#2325—21no—
44C4×#121C2212yes—
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#385—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2093C2212yes—
52#2087C2214yescomplemented
53#2071C2212yes—
54#2065C2214yes—
55#2065C2214yescomplemented
56#2058C2214yescomplemented
57#2054C2214yes—
58#2023C2212yes—
59#2023C2214yescomplemented
60#2011C2212yes—
61#1990C2212yescomplemented
62#1990C2212yescomplemented
63#1972C2214yes—
64#1970C2214yes—
65#1944C2214yescomplemented
66#1943C2214yescomplemented
67#1936C2214yes—
68#1936C2214yescomplemented
69#1936C2214yescomplemented
70#1934C2214yes—
71#1921C2214yescomplemented
72#1879C2214yes—
73#1874C2212yes—
74#1874C2212yes—
75#1864C2214yes—
76#1864C2214yescomplemented
77#1853C2212yes—
78#1853C2212yes—
79#1823C2212yes—
80#1786C2212yes—
81#1771C2214yescomplemented
82#1771C2214yescomplemented
83#1762C2214yes—
84#1749C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×D8C2314yes—
91#539—48no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127C2311yescharacteristic
99C2×#126—22no—
100C2×#125—22no—
101C2×#123—22no—
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—48no—
108C2×#121—21no—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—44no—
114C2×C4×D4C2314yes—
115C2×C4×D4C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#498—44no—
133#486—44no—
134#480—28no—
135#447—28no—
136#442—48no—
137#432—28no—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×#97C2314yes—
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2312yes—
154#400—22no—
155#398—22no—
156#397—22no—
157#396—22no—
158#396—22no—
159#396—24no—
160#396—24no—
161#394—22no—
162#394—22no—
163#393—22no—
164#393—24no—
165#393—24no—
166#391—21no—
167#390—21no—
168#390—21no—
169#390—22no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2314yes—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#389—21no—
180#389—22no—
181#389—22no—
182#389C2312yes—
183#389C2312yes—
184#389C2314yes—
185#389C2312yes—
186#389C2312yes—
187#389C2314yes—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#387—22no—
193#386—22no—
194#386—22no—
195#386—22no—
196#385—22no—
197#385—24no—
198#385—24no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#380—21no—
202#380C2312yes—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—22no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2312yes—
220#336C2312yes—
221#336C2314yes—
222C25—21no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—21no—
226C22×Q8—44no—
227C22×Q8—44no—
228C22×Q8—82no—
229C22×D4—24no—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—24no—
235C22×D4—42no—
236C22×D4C2412yes—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4—82no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×D8—28no—
254C2×D8—28no—
255C2×D8—44no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265#127—42no—
266#126—48no—
267#125—48no—
268#123—48no—
269#123—48no—
270C22⋊Q8—416no—
271C22⋊Q8—416no—
272C22⋊Q8—48no—
273C22⋊Q8—44no—
274#121—24no—
275#121—24no—
276#121—44no—
277#121—44no—
278C4×D4—28no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—48no—
282C4×D4—48no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
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—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—82no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352#107—28no—
353#106—28no—
354#106—44no—
355C4⋊C8—28no—
356#102—28no—
357#102—28no—
358#102—28no—
359#102—28no—
360#102—44no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8#4118yes—
364#97—28no—
365#97—44no—
366C4×C8—42no—
367C4×C8—42no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24C22×D411yescharacteristic
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—81no—
398C2×Q8—81no—
399C2×Q8—816no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—416no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—81no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D42−1+414yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—84no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—816no—
461C22×C4AGL(1,ℤ/8)14yes—
462C22×C4AGL(1,ℤ/8)14yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4114yes—
473C22×C4C2511yescharacteristic
474D8—416no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8C2×#4118yes—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4H(5,𝔽2)14yes—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—84no—
527C22⋊C4—82no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—42no—
535C42—41no—
536C42—41no—
537C42—44no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—24no—
543C23—48no—
544C23—48no—
545C23—48no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23#53212yes—
564C23C2×AGL(1,ℤ/8)12yes—
565C23C2×AGL(1,ℤ/8)12yes—
566C23C23×D411yescharacteristic
567Q8—88no—
568Q8—88no—
569Q8—162no—
570D4—28no—
571D4—48no—
572D4—44no—
573D4—44no—
574D4—44no—
575D4—88no—
576D4—162no—
577D4—162no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—816no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—164no—
630C2×C4#53212yes—
631C2×C4C2×AGL(1,ℤ/8)12yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C22×#4112yes—
636C2×C4C2×H(5,𝔽2)12yes—
637C2×C4C2×2−1+412yes—
638C8—28no—
639C8—44no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—816no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22#318414yes—
662C22C2×#53211yescharacteristic
663C22#348411yescharacteristic
664C22#348611yescharacteristic
665C22C22×AGL(1,ℤ/8)11yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22#358511yescharacteristic
668C22#358611yescharacteristic
669C4—22no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4#352014yes—
685C2—24no—
686C2—84no—
687C2—162no—
688C2#3539711yescharacteristic
689C2C2×#318411yescharacteristic
690C2#6070311yescharacteristic
691C2#6157111yescharacteristic
692C2#6157211yescharacteristic
693C2#6182511yescharacteristic
694C2#6182711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28301C2
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
1. #35006C2semidirect
2. C2×#2452C2semidirect
3. #29103C2semidirect
4. #29095C2semidirect
5. #28974C2semidirect
7. #26834C2semidirect
8. #26490C2semidirect
9. #25903C2semidirect
10. #25811C2semidirect
11. #24608C2semidirect
12. #24506C2semidirect
13. #22081C2semidirect
14. #21452C2semidirect
15. #20534C2semidirect
20. C2×#498C22semidirect
21. C2×#486C22semidirect
33. #2512C22semidirect
34. #2506C22semidirect
39. #2452C22semidirect
40. #2445C22semidirect
52. #2087C22semidirect
55. #2065C22semidirect
56. #2058C22semidirect
59. #2023C22semidirect
61. #1990C22semidirect
62. #1990C22semidirect
65. #1944C22semidirect
66. #1943C22semidirect
68. #1936C22semidirect
69. #1936C22semidirect
71. #1921C22semidirect
76. #1864C22semidirect
81. #1771C22semidirect
82. #1771C22semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#24522
#291032
#290952
#289742
#283012
#268342
#264902
#259032
#258112
#246082
#245062
#220812
#214522
#205342