Finite Group Databasebeta

← #6502287

Subgroups — #6502287

Order 512.

2302 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2445C211yesmaximal, characteristic, complemented
2#34206C211yesmaximal, characteristic, complemented
3#29235C212yesmaximal, complemented
4#27585C211yesmaximal, characteristic, complemented
5#27585C211yesmaximal, characteristic, complemented
6#27211C214yesmaximal, complemented
7#26705C214yesmaximal, complemented
8#26601C214yesmaximal, complemented
9#26174C214yesmaximal, complemented
10#24208C212yesmaximal, complemented
11#23930C211yesmaximal, characteristic, complemented
12#23930C211yesmaximal, characteristic, complemented
13#22046C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21452C212yesmaximal, complemented
16C2×#532C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#123—21no—
19C2×#520—21no—
20C22×C4×D4C2211yescharacteristic
21C2×#459C2214yes—
22C2×#447C2212yes—
23C2×#447C2212yes—
24C2×#442C2214yescomplemented
25#2445—22no—
26#2445—22no—
27#2445C2214yes—
28#2445C2214yes—
29C2×#394—21no—
30C2×#394—21no—
31#2367—22no—
32#2366C2212yescomplemented
33#2365C2212yescomplemented
34#2354C2212yescomplemented
35#2352—22no—
36D4×C4⋊C4C2212yescomplemented
37#2325—21no—
38#2325—21no—
39#2311—21no—
40C4×#121C2212yes—
41C4×#121C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#386—21no—
47#2285C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2094C2212yescomplemented
50#2094C2212yescomplemented
51#2088C2214yescomplemented
52#2071C2212yescomplemented
53#2071C2212yescomplemented
54#2065C2214yescomplemented
55#2041C2214yes—
56#2023C2212yescomplemented
57#2023C2212yescomplemented
58#2011C2214yes—
59#2011C2212yescomplemented
60#2011C2212yescomplemented
61#1982C2214yes—
62#1972C2214yescomplemented
63#1936C2214yescomplemented
64#1926C2214yescomplemented
65#1926C2214yescomplemented
66#1926C2214yescomplemented
67#1920C2214yescomplemented
68#1917C2214yescomplemented
69#1917C2214yescomplemented
70#1879C2214yescomplemented
71#1879C2214yescomplemented
72#1879C2214yescomplemented
73#1879C2214yescomplemented
74#1874C2212yes—
75#1874C2212yes—
76#1864C2214yescomplemented
77#1864C2214yescomplemented
78#1853C2212yes—
79#1853C2212yes—
80#1817C2211yescharacteristic
81#1817C2211yescharacteristic
82#1797C2212yes—
83#1786C2212yes—
84#1786C2212yes—
85#1762C2214yescomplemented
86#1749C2212yes—
87C22×#41—21no—
88C23×D4—21no—
89C23×D4—21no—
90C23×D4C2311yescharacteristic
91C23×D4C2311yescharacteristic
92C24×C4—21no—
93C22×D8C2314yes—
94C22×D8C2314yes—
95#532—28no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127—22no—
98C2×#127—44no—
99C2×#127C2311yescharacteristic
100C2×#127C2314yes—
101C2×#124—21no—
102C2×#123—24no—
103C2×#123—24no—
104C2×#123—48no—
105C2×#121—21no—
106C2×#121C2312yes—
107#520—48no—
108C2×C4×Q8—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4—44no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2314yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×#117—21no—
120C2×#117—22no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#459—48no—
138#447—28no—
139#447—28no—
140#442—48no—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×#97C2314yes—
155#394—24no—
156#394—24no—
157#394—24no—
158#394—24no—
159#391—22no—
160#390—22no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—22no—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#390C2314yes—
171#390C2312yes—
172#390C2312yes—
173#389—22no—
174#389C2314yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—22no—
182#387—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—24no—
188#386—24no—
189#385—22no—
190#385—22no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#382—21no—
194#380—21no—
195#380—22no—
196#380—22no—
197#380C2312yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25—21no—
222C25—21no—
223C2×#41—28no—
224C2×#41—44no—
225C2×#41—44no—
226C22×Q8—21no—
227C22×D4—22no—
228C22×D4—24no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—24no—
234C22×D4—24no—
235C22×D4—24no—
236C22×D4—24no—
237C22×D4—42no—
238C22×D4—42no—
239C22×D4—42no—
240C22×D4—44no—
241C22×D4—44no—
242C22×D4—44no—
243C22×D4—44no—
244C22×D4—82no—
245C22×D4—82no—
246C22×D4C2412yes—
247C22×D4C2412yes—
248C22×D4C2412yes—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—24no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—82no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×D8—28no—
266C2×D8—28no—
267C2×D8—28no—
268C2×D8—44no—
269C2×D8—44no—
270C2×D8—44no—
271C22×C8C2414yes—
272C22×C8C2414yes—
273C4⋊Q8—24no—
274#127—24no—
275#127—28no—
276#127—44no—
277#127—48no—
278#124—28no—
279#123—416no—
280#123—416no—
281#121—28no—
282#121—24no—
283#121#4118yes—
284C4×Q8—48no—
285C4×D4—28no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—28no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4—48no—
293C4×D4—48no—
294C4×D4—48no—
295C4×D4—48no—
296#117—48no—
297#117—44no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—82no—
348C2×C22⋊C4—82no—
349C2×C22⋊C4—82no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2412yes—
363C2×C42C2411yescharacteristic
364#106—44no—
365#106—44no—
366C4⋊C8—44no—
367#102—28no—
368#102—28no—
369#102—28no—
370#102—44no—
371#102—44no—
372#102—44no—
373#102—44no—
374#102—44no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377#97—44no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—24no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24C22×D411yescharacteristic
407#41—432no—
408C2×Q8—24no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—416no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—48no—
439C2×D4—88no—
440C2×D4—81no—
441C2×D4—81no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—816no—
445C2×D4—816no—
446C2×D4C2×#4114yes—
447C2×D4H(5,𝔽2)14yes—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—84no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4—816no—
490C22×C4AGL(1,ℤ/8)14yes—
491C22×C4AGL(1,ℤ/8)14yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C2×#4114yes—
502C22×C4C2511yescharacteristic
503D8—416no—
504D8—88no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—81no—
529C4⋊C4—81no—
530C4⋊C4C2×#4114yes—
531C22⋊C4—28no—
532C22⋊C4—28no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C22⋊C4—84no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—24no—
556C42—24no—
557C42—42no—
558C42—42no—
559C42—44no—
560C42—44no—
561C42—44no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—24no—
567C23—24no—
568C23—48no—
569C23—48no—
570C23—48no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23—84no—
597C23—84no—
598C23—84no—
599C23—84no—
600C23—84no—
601C23#52512yes—
602C23C2×AGL(1,ℤ/8)12yes—
603C23C2×AGL(1,ℤ/8)12yes—
604C23C23×D411yescharacteristic
605Q8—48no—
606D4—28no—
607D4—28no—
608D4—48no—
609D4—48no—
610D4—416no—
611D4—44no—
612D4—44no—
613D4—44no—
614D4—44no—
615D4—816no—
616D4—84no—
617D4—84no—
618D4—816no—
619C2×C4—22no—
620C2×C4—28no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—28no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—24no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—44no—
646C2×C4—416no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—82no—
664C2×C4—82no—
665C2×C4—816no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4—164no—
669C2×C4—164no—
670C2×C4#52512yes—
671C2×C4C2×AGL(1,ℤ/8)14yes—
672C2×C4C2×AGL(1,ℤ/8)12yes—
673C2×C4C2×AGL(1,ℤ/8)12yes—
674C2×C4C22×#4112yes—
675C2×C4C2×H(5,𝔽2)12yes—
676C2×C4#58512yes—
677C8—44no—
678C8—44no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—816no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—168no—
708C22#310914yes—
709C22C2×#52511yescharacteristic
710C22#348411yescharacteristic
711C22#348711yescharacteristic
712C22C22×AGL(1,ℤ/8)11yescharacteristic
713C22C22×AGL(1,ℤ/8)11yescharacteristic
714C22#358511yescharacteristic
715C22#358511yescharacteristic
716C4—22no—
717C4—22no—
718C4—22no—
719C4—24no—
720C4—44no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—41no—
725C4—41no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4#351314yes—
731C2—24no—
732C2—84no—
733C2—84no—
734C2—162no—
735C2#3535611yescharacteristic
736C2C2×#310911yescharacteristic
737C2#6056811yescharacteristic
738C2#6156911yescharacteristic
739C2#6158211yescharacteristic
740C2#6189011yescharacteristic
741C2#6189111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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×#2445C2semidirect
2. #34206C2semidirect
3. #29235C2semidirect
4. #27585C2semidirect
5. #27585C2semidirect
6. #27211C2semidirect
7. #26705C2semidirect
8. #26601C2semidirect
9. #26174C2semidirect
10. #24208C2semidirect
11. #23930C2semidirect
12. #23930C2semidirect
13. #22046C2semidirect
14. #21916C2semidirect
15. #21452C2semidirect
24. C2×#442C22semidirect
32. #2366C22semidirect
33. #2365C22semidirect
34. #2354C22semidirect
36. D4×C4⋊C4C22semidirect
49. #2094C22semidirect
50. #2094C22semidirect
51. #2088C22semidirect
52. #2071C22semidirect
53. #2071C22semidirect
54. #2065C22semidirect
56. #2023C22semidirect
57. #2023C22semidirect
59. #2011C22semidirect
60. #2011C22semidirect
62. #1972C22semidirect
63. #1936C22semidirect
64. #1926C22semidirect
65. #1926C22semidirect
66. #1926C22semidirect
67. #1920C22semidirect
68. #1917C22semidirect
69. #1917C22semidirect
70. #1879C22semidirect
71. #1879C22semidirect
72. #1879C22semidirect
73. #1879C22semidirect
76. #1864C22semidirect
77. #1864C22semidirect
85. #1762C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24452
#342062
#292352
#275852
#272112
#267052
#266012
#261742
#242082
#239302
#220462
#219162
#214522