Finite Group Databasebeta

← #6471596

Subgroups — #6471596

Order 512.

2218 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#33002C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic, complemented
5#32902C211yesmaximal, characteristic
6#29968C211yesmaximal, characteristic, complemented
7#29907C212yesmaximal, complemented
8#29754C211yesmaximal, characteristic, complemented
9#27571C214yesmaximal, complemented
10#27315C214yesmaximal, complemented
11#26392C214yesmaximal, complemented
12#26174C214yesmaximal, complemented
13#22081C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21452C212yesmaximal, complemented
16C2×#539—21no—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4—21no—
21C2×#480C2212yes—
22C2×#476C2212yes—
23C2×#459C2212yes—
24C2×#447C2212yes—
25C2×#442C2214yes—
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2854C2212yes—
29#2639—21no—
30#2618—21no—
31#2560—22no—
32#2558—22no—
33#2512C2212yescomplemented
34#2506C2212yescomplemented
35#2452C2212yescomplemented
36#2445C2212yescomplemented
37D4×C4⋊C4—22no—
38D4×C4⋊C4—22no—
39D4×C4⋊C4C2214yes—
40D4×C4⋊C4C2214yes—
41#2325—21no—
42C4×#121C2212yes—
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#385—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#380—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2100C2212yescomplemented
51#2100C2214yescomplemented
52#2094C2214yescomplemented
53#2088C2214yescomplemented
54#2084C2214yescomplemented
55#2083C2212yescomplemented
56#2071C2214yescomplemented
57#2067C2214yescomplemented
58#2058C2212yescomplemented
59#2041C2212yescomplemented
60#1943C2214yescomplemented
61#1936C2214yescomplemented
62#1921C2214yescomplemented
63#1917C2214yescomplemented
64#1917C2214yescomplemented
65#1879C2214yes—
66#1879C2214yes—
67#1879C2214yes—
68#1879C2214yes—
69#1874C2212yes—
70#1874C2212yes—
71#1864C2214yes—
72#1864C2214yes—
73#1864C2214yescomplemented
74#1862C2214yes—
75#1862C2214yes—
76#1853C2212yes—
77#1853C2212yes—
78#1823C2212yes—
79#1817C2212yes—
80#1802C2212yes—
81#1802C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1786C2212yes—
85#1762C2214yes—
86#1749C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C24×C4—21no—
90C24×C4—21no—
91C22×D8C2314yes—
92#539—48no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#126—22no—
96C2×#125—22no—
97C2×#123—22no—
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—22no—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×D4—24no—
105C2×C4×D4—22no—
106C2×C4×D4—24no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—44no—
112C2×C4×D4—44no—
113C2×C4×D4C2314yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2314yes—
117C2×C4×D4C2312yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4—48no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#480—28no—
137#476—44no—
138#459—44no—
139#447—28no—
140#442—48no—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×#97C2314yes—
155#400—22no—
156#398—22no—
157#397—22no—
158#396—22no—
159#396—22no—
160#394—22no—
161#394—22no—
162#393—22no—
163#391—21no—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2314yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#385—22no—
190#385—24no—
191#385—24no—
192#385—24no—
193#385—24no—
194#384—22no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2314yes—
198#380—21no—
199#380—24no—
200#380—24no—
201#380C2312yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—22no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25—21no—
224C22×Q8—21no—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4#4114yes—
233C22×D4C2412yes—
234C22×D4C2412yes—
235C23×C4—28no—
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—24no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4—82no—
251C23×C4—82no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×D8—28no—
260C2×D8—28no—
261C2×D8—28no—
262C22×C8C2414yes—
263C22×C8C2414yes—
264C4⋊Q8—42no—
265#127—42no—
266#126—48no—
267#125—48no—
268#123—48no—
269#123—48no—
270C22⋊Q8—48no—
271C22⋊Q8—44no—
272#121—24no—
273#121—24no—
274#121—44no—
275#121—44no—
276C4×D4—216no—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—28no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—44no—
286C4×D4—48no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—48no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—416no—
313C2×C4⋊C4—416no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4#4114yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—42no—
351C2×C42—42no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360#107—28no—
361#106—44no—
362#106—44no—
363C4⋊C8—42no—
364C4⋊C8—42no—
365#102—28no—
366#102—28no—
367#102—28no—
368#102—28no—
369#102—44no—
370#102—44no—
371C22⋊C8—24no—
372C22⋊C8—24no—
373#97—44no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—24no—
388C24—24no—
389C24—24no—
390C24—22no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C22×D411yescharacteristic
397C2×Q8—42no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—216no—
403C2×D4—216no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4C2×#4114yes—
422C2×D4H(5,𝔽2)14yes—
423C2×D42−1+414yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—216no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—48no—
451C22×C4—432no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—48no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—84no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—816no—
474C22×C4—816no—
475C22×C4AGL(1,ℤ/8)14yes—
476C22×C4AGL(1,ℤ/8)14yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2511yescharacteristic
489D8—416no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—81no—
523C4⋊C4—84no—
524C4⋊C4—81no—
525C4⋊C4C2×#4114yes—
526C22⋊C4—216no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—24no—
545C42—22no—
546C42—22no—
547C42—41no—
548C42—41no—
549C42—42no—
550C42—44no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—216no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—48no—
560C23—48no—
561C23—48no—
562C23—48no—
563C23—48no—
564C23—42no—
565C23—42no—
566C23—48no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23#52512yes—
577C23C2×AGL(1,ℤ/8)12yes—
578C23C2×AGL(1,ℤ/8)12yes—
579C23C23×D411yescharacteristic
580Q8—162no—
581D4—216no—
582D4—24no—
583D4—24no—
584D4—24no—
585D4—44no—
586D4—44no—
587D4—88no—
588D4—162no—
589D4—162no—
590D4#53918yescomplemented
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—44no—
619C2×C4—432no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—88no—
641C2×C4—88no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—816no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—164no—
648C2×C4#52512yes—
649C2×C4C2×AGL(1,ℤ/8)14yes—
650C2×C4C2×AGL(1,ℤ/8)12yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)12yes—
655C2×C4C2×2−1+412yes—
656C8—44no—
657C8—44no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—28no—
665C22—432no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22#310914yes—
680C22C2×#52511yescharacteristic
681C22#348411yescharacteristic
682C22#348611yescharacteristic
683C22C22×AGL(1,ℤ/8)11yescharacteristic
684C22C22×AGL(1,ℤ/8)11yescharacteristic
685C22#358511yescharacteristic
686C22#358611yescharacteristic
687C4—24no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—41no—
695C4—41no—
696C4—42no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4C2×#53912yes—
703C4#349212yes—
704C2—24no—
705C2—48no—
706C2—162no—
707C2#3498911yescharacteristic
708C2C2×#310911yescharacteristic
709C2#6056511yescharacteristic
710C2#6157111yescharacteristic
711C2#6158511yescharacteristic
712C2#6182511yescharacteristic
713C2#6182711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32902C2
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×D4×C4⋊C4C2semidirect
3. #33002C2semidirect
4. #32956C2semidirect
6. #29968C2semidirect
7. #29907C2semidirect
8. #29754C2semidirect
9. #27571C2semidirect
10. #27315C2semidirect
11. #26392C2semidirect
12. #26174C2semidirect
13. #22081C2semidirect
14. #21916C2semidirect
15. #21452C2semidirect
33. #2512C22semidirect
34. #2506C22semidirect
35. #2452C22semidirect
36. #2445C22semidirect
50. #2100C22semidirect
51. #2100C22semidirect
52. #2094C22semidirect
53. #2088C22semidirect
54. #2084C22semidirect
55. #2083C22semidirect
56. #2071C22semidirect
57. #2067C22semidirect
58. #2058C22semidirect
59. #2041C22semidirect
60. #1943C22semidirect
61. #1936C22semidirect
62. #1921C22semidirect
63. #1917C22semidirect
64. #1917C22semidirect
73. #1864C22semidirect
590. D4#539semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×D4×C4⋊C42
#330022
#329562
#329022
#299682
#299072
#297542
#275712
#273152
#263922
#261742
#220812
#219162
#214522