Finite Group Databasebeta

← #6502283

Subgroups — #6502283

Order 512.

2054 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
1C2×#2447C211yesmaximal, characteristic, complemented
2#34206C211yesmaximal, characteristic, complemented
3#29237C212yesmaximal, complemented
4#27586C211yesmaximal, characteristic, complemented
5#27585C211yesmaximal, characteristic, complemented
6#27195C214yesmaximal, complemented
7#26689C214yesmaximal, complemented
8#26585C214yesmaximal, complemented
9#26164C214yesmaximal, complemented
10#24215C212yesmaximal, complemented
11#23931C211yesmaximal, characteristic, complemented
12#23930C211yesmaximal, characteristic
13#22041C212yesmaximal, complemented
14#21912C211yesmaximal, characteristic, complemented
15C4×#447C212yesmaximal, complemented
16C2×#532C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C2×#520—21no—
19C22×C4×D4C2211yescharacteristic
20C2×#463C2214yes—
21C2×#448C2212yes—
22C2×#447C2212yes—
23C2×C4×D8C2214yes—
24#2447—22no—
25#2447—22no—
26#2447C2214yes—
27#2447C2214yes—
28C2×#397—21no—
29C2×#397—21no—
30#2367—22no—
31#2366C2212yescomplemented
32#2365C2212yescomplemented
33#2354C2212yescomplemented
34#2352—22no—
35D4×C4⋊C4C2212yescomplemented
36#2325—21no—
37#2325—21no—
38#2311—21no—
39C4×#121C2212yes—
40C4×#121C2212yes—
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#385—21no—
46#2285C2211yescharacteristic
47C2×#381—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2095C2212yescomplemented
50#2094C2212yes—
51#2087C2214yescomplemented
52#2072C2212yescomplemented
53#2071C2212yes—
54#2065C2214yescomplemented
55#2042C2214yes—
56#2024C2212yescomplemented
57#2023C2212yes—
58#2014C2212yescomplemented
59#2012C2214yes—
60#2011C2212yes—
61#1983C2214yes—
62#1968C2214yescomplemented
63#1933C2214yescomplemented
64#1928C2214yescomplemented
65#1927C2214yescomplemented
66#1926C2214yes—
67#1919C2214yescomplemented
68#1916C2214yescomplemented
69#1916C2214yescomplemented
70#1879C2214yes—
71#1879C2214yes—
72#1877C2214yescomplemented
73#1877C2214yescomplemented
74#1874C2212yes—
75#1874C2212yes—
76#1862C2214yes—
77#1862C2214yescomplemented
78#1853C2212yes—
79#1852C2212yes—
80#1817C2211yescharacteristic
81#1817C2211yescharacteristic
82#1797C2212yes—
83#1787C2212yes—
84#1786C2212yes—
85C4×#102C2214yes—
86C4×C22⋊C8C2212yes—
87C22×#41—21no—
88C23×Q8C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C22×SD16C2314yes—
92C22×D8C2314yes—
93#532—28no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#124—21no—
97C2×#124—22no—
98C2×#124—44no—
99C2×#124C2314yes—
100C2×#121—21no—
101C2×#121C2312yes—
102#520—48no—
103C2×C4×Q8—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4—22no—
109C2×C4×D4—44no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2314yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—21no—
115C2×#117—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#463—48no—
131#448—28no—
132#447—28no—
133C4×D8—48no—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C4×C8C2314yes—
148#397—24no—
149#397—24no—
150#397—24no—
151#397—24no—
152#391—22no—
153#391—22no—
154#391—21no—
155#391—21no—
156#391C2312yes—
157#391C2314yes—
158#391C2312yes—
159#390—22no—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#389—22no—
167#389C2314yes—
168#389C2312yes—
169#389C2312yes—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#387—22no—
175#387—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#385—24no—
181#385—24no—
182#385—22no—
183#385—22no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#382—21no—
187#381—24no—
188#381—24no—
189#380—21no—
190#380—22no—
191#380—22no—
192#380C2312yes—
193C4×C4⋊C4—21no—
194C4×C4⋊C4D412yes—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25—21no—
221C2×#41—28no—
222C2×#41—44no—
223C2×#41—44no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—24no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8C2412yes—
230C22×D4—22no—
231C22×D4—24no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—24no—
236C22×D4—42no—
237C22×D4C2412yes—
238C22×D4C2412yes—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—24no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×SD16—28no—
257C2×SD16—44no—
258C2×SD16—44no—
259C2×D8—28no—
260C2×D8—28no—
261C2×D8—44no—
262C22×C8C2414yes—
263C22×C8C2414yes—
264C4⋊Q8—24no—
265#127—24no—
266#124—28no—
267#124—28no—
268#124—44no—
269#124—48no—
270#121—28no—
271#121—24no—
272#121#4118yes—
273C4×Q8—48no—
274C4×D4—28no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—28no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—48no—
282C4×D4—48no—
283C4×D4—48no—
284C4×D4—48no—
285#117—48no—
286#117—44no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—48no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
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—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—82no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347#107—44no—
348#106—44no—
349C4⋊C8—44no—
350#103—28no—
351#103—44no—
352#103—44no—
353#102—28no—
354#102—28no—
355#102—44no—
356#102—44no—
357#102—44no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C4×C8—44no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24C22×D411yescharacteristic
388#41—432no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—88no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—28no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—416no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4C2×#4114yes—
420C2×D4H(5,𝔽2)14yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—28no—
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—44no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—84no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—816no—
467C22×C4#13514yes—
468C22×C4AGL(1,ℤ/8)14yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C2×#4114yes—
479C22×C4C2511yescharacteristic
480SD16—88no—
481D8—416no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—84no—
507C4⋊C4C2×#4114yes—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—42no—
531C42—42no—
532C42—44no—
533C42—44no—
534C42—44no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—24no—
540C23—24no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23#52512yes—
561C23C2×#13512yes—
562C23C2×AGL(1,ℤ/8)12yes—
563C23C23×D411yescharacteristic
564Q8—48no—
565Q8—44no—
566Q8—44no—
567Q8—84no—
568D4—28no—
569D4—28no—
570D4—48no—
571D4—48no—
572D4—416no—
573D4—44no—
574D4—44no—
575D4—84no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—28no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—44no—
603C2×C4—416no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
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—816no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—164no—
630C2×C4—164no—
631C2×C4#52512yes—
632C2×C4C2×#13512yes—
633C2×C4C2×#13512yes—
634C2×C4C2×AGL(1,ℤ/8)14yes—
635C2×C4C22×#4112yes—
636C2×C4C2×H(5,𝔽2)12yes—
637C2×C4#58512yes—
638C8—44no—
639C8—44no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—816no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22#310514yes—
662C22C2×#52511yescharacteristic
663C22#348411yescharacteristic
664C22#348711yescharacteristic
665C22C22×#13511yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22#358511yescharacteristic
668C22#358611yescharacteristic
669C4—22no—
670C4—22no—
671C4—22no—
672C4—24no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—41no—
678C4—41no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4#351314yes—
685C2—24no—
686C2—84no—
687C2—84no—
688C2#3535611yescharacteristic
689C2C2×#310511yescharacteristic
690C2#6056711yescharacteristic
691C2#6156511yescharacteristic
692C2#6156911yescharacteristic
693C2#6188511yescharacteristic
694C2#6188811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23930C2
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. C2×#2447C2semidirect
2. #34206C2semidirect
3. #29237C2semidirect
4. #27586C2semidirect
5. #27585C2semidirect
6. #27195C2semidirect
7. #26689C2semidirect
8. #26585C2semidirect
9. #26164C2semidirect
10. #24215C2semidirect
11. #23931C2semidirect
13. #22041C2semidirect
14. #21912C2semidirect
15. C4×#447C2semidirect
31. #2366C22semidirect
32. #2365C22semidirect
33. #2354C22semidirect
35. D4×C4⋊C4C22semidirect
49. #2095C22semidirect
51. #2087C22semidirect
52. #2072C22semidirect
54. #2065C22semidirect
56. #2024C22semidirect
58. #2014C22semidirect
62. #1968C22semidirect
63. #1933C22semidirect
64. #1928C22semidirect
65. #1927C22semidirect
67. #1919C22semidirect
68. #1916C22semidirect
69. #1916C22semidirect
72. #1877C22semidirect
73. #1877C22semidirect
77. #1862C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24472
#342062
#292372
#275862
#275852
#271952
#266892
#265852
#261642
#242152
#239312
#239302
#220412
#219122
C4×#4472