Finite Group Databasebeta

← #6468960

Subgroups — #6468960

Order 512.

2562 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic
3#29366C214yesmaximal, complemented
4#29258C212yesmaximal, complemented
5#28160C212yesmaximal, complemented
6#27680C212yesmaximal, complemented
7#27585C212yesmaximal, complemented
8#24211C212yesmaximal
9#23933C212yesmaximal, complemented
10#23815C212yesmaximal, complemented
11#23799C212yesmaximal, complemented
12C4×#460C214yesmaximal, complemented
13C4×#450C214yesmaximal, complemented
14#20777C211yesmaximal, characteristic, complemented
15C2×#533C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C22×C4×D4—21no—
18C22×C4×D4—21no—
19C22×C4×D4C2212yes—
20C2×#460C2214yes—
21C2×#450C2214yes—
22C2×C4×SD16C2218yescomplemented
23C4×C4⋊Q8C2211yescharacteristic, complemented
24C4×C4⋊Q8C2211yescharacteristic, complemented
25C4×C4⋊Q8C2211yescharacteristic, complemented
26C4×C4⋊Q8C2211yescharacteristic, complemented
27C4×#127—21no—
28C4×#127—21no—
29C4×#127—21no—
30C4×#127—21no—
31C4×#127C2212yes—
32C4×#127C2212yes—
33C4×#127C2212yes—
34C4×#127C2212yes—
35C4×#125—21no—
36C4×#125—21no—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41#2296—21no—
42#2295—21no—
43#2295C2212yescomplemented
44C2×#390—21no—
45C2×#390—21no—
46C2×#390C2212yes—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C42×Q8—21no—
50C42×Q8C2212yescomplemented
51C4×#117—21no—
52C2×C43C2211yescharacteristic
53#2099C2214yescomplemented
54#2096C2214yescomplemented
55#2094C2214yescomplemented
56#2071C2214yescomplemented
57#2041C2214yes—
58#2023C2214yescomplemented
59#2014C2214yes—
60#2013C2214yescomplemented
61#1982C2214yes—
62#1948C2214yescomplemented
63#1926C2218yescomplemented
64#1880C2218yescomplemented
65#1877C2218yescomplemented
66#1872C2214yes—
67#1864C2218yescomplemented
68#1853C2214yes—
69#1839C2212yescomplemented
70#1831C2212yescomplemented
71#1817C2212yes—
72#1803C2214yescomplemented
73#1797C2212yes—
74#1786C2214yes—
75C4×C4⋊C8C2212yescomplemented
76C4×#102C2218yescomplemented
77C4×C22⋊C8C2212yes—
78#1744C2212yescomplemented
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2312yes—
82C24×C4—22no—
83C22×SD16C2318yes—
84#533—28no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×#127—28no—
88C2×#127—21no—
89C2×#127—21no—
90C2×#127C2312yes—
91C2×#127C2312yes—
92C2×#125—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8C2312yes—
95C2×C4×Q8—21no—
96C2×C4×Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2312yes—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—48no—
112C2×C4×D4—48no—
113C2×C4×D4—48no—
114C2×C4×D4C2314yes—
115C2×C4×D4C2314yes—
116C2×#117—21no—
117C2×#117—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4—22no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#460—216no—
132#450—216no—
133C4×SD16—416no—
134C2×#106C2318yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2318yes—
137C2×#102C2318yes—
138C2×#102C2318yes—
139C2×#102C2318yes—
140C2×C22⋊C8C2314yes—
141C2×C4×C8C2314yes—
142#391—21no—
143#391—21no—
144#391C2312yes—
145#391C2312yes—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390C2314yes—
157#390C2314yes—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389C2312yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#385—22no—
171#385—22no—
172#384—21no—
173#384—21no—
174#384C2312yes—
175#384C2312yes—
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#382—21no—
179#380—21no—
180#380C2312yes—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199#376—22no—
200C43—22no—
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209#340C2318yes—
210#340C2318yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2314yes—
215C25—22no—
216C22×Q8—21no—
217C22×Q8C2412yes—
218C22×D4—28no—
219C22×D4—28no—
220C22×D4—28no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—28no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4—44no—
232C22×D4C2414yes—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—48no—
237C23×C4—48no—
238C23×C4—48no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×SD16—216no—
247C2×SD16—216no—
248C2×SD16—48no—
249C22×C8C2418yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252#127—28no—
253#127—28no—
254#127—28no—
255#127—24no—
256#127#4118yes—
257#125—28no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8#4118yes—
261C4×Q8—28no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—28no—
265C4×Q8—28no—
266C4×Q8—44no—
267C4×Q8—44no—
268C4×D4—216no—
269C4×D4—48no—
270C4×D4—48no—
271C4×D4—48no—
272C4×D4—48no—
273C4×D4—48no—
274C4×D4—48no—
275C4×D4—48no—
276C4×D4—48no—
277C4×D4—48no—
278C4×D4—48no—
279C4×D4—48no—
280#117—28no—
281#117—44no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—48no—
317C2×C22⋊C4—48no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42SD1618yescomplemented
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#106—48no—
360C4⋊C8—28no—
361#103—48no—
362#102—216no—
363#102—216no—
364#102—48no—
365C22⋊C8—28no—
366C4×C8—28no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377C24—28no—
378C24—24no—
379C24—44no—
380C24—44no—
381C24—44no—
382C24—44no—
383C24—44no—
384C24—44no—
385C24C22×D411yescharacteristic
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8C2×#4114yes—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—28no—
401C2×D4—432no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—432no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—432no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4C2×#4118yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
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—24no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—832no—
447C22×C4C2×SD1614yes—
448C22×C4C2×SD1614yes—
449C22×C4C2×SD1614yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2×#4114yes—
464C22×C4C2511yescharacteristic
465SD16—432no—
466C2×C8—28no—
467C2×C8—28no—
468C2×C8—28no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4C2×#4114yes—
492C22⋊C4—28no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C42—28no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—44no—
523C42—44no—
524C42—44no—
525C42—44no—
526C42—44no—
527C42C2×SD1614yes—
528C42C2×SD1614yes—
529C42C2×SD1614yes—
530C42C2×SD1614yes—
531C42AGL(1,ℤ/8)14yes—
532C42C2×#4114yes—
533C42H(5,𝔽2)14yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—28no—
539C23—416no—
540C23—416no—
541C23—416no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—88no—
549C23—88no—
550C23—88no—
551C23—88no—
552C23#52512yes—
553C23C22×SD1612yes—
554C23#57512yes—
555C23C23×D411yescharacteristic
556Q8—28no—
557Q8—48no—
558Q8—44no—
559D4—216no—
560D4—416no—
561D4—416no—
562D4—416no—
563D4—416no—
564D4—416no—
565D4—48no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
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—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—21no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—816no—
613C2×C4—88no—
614C2×C4—816no—
615C2×C4#52512yes—
616C2×C4#52512yes—
617C2×C4#52512yes—
618C2×C4#53512yes—
619C2×C4#53512yes—
620C2×C4C22×SD1612yes—
621C2×C4C22×SD1612yes—
622C2×C4C22×SD1612yes—
623C2×C4C22×SD1612yes—
624C2×C4C22×SD1612yes—
625C2×C4C22×SD1612yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4#57512yes—
630C2×C4C22×#4112yes—
631C2×C4C22×#4112yes—
632C2×C4C2×H(5,𝔽2)12yes—
633C8—48no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—48no—
641C22—48no—
642C22—48no—
643C22—832no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22#310314yes—
649C22C2×#52511yescharacteristic
650C22C2×#52511yescharacteristic
651C22C2×#53511yescharacteristic
652C22C23×SD1611yescharacteristic
653C22C22×AGL(1,ℤ/8)11yescharacteristic
654C22C2×#57511yescharacteristic
655C22C2×#57511yescharacteristic
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—41no—
667C4—44no—
668C4—44no—
669C4—84no—
670C4—82no—
671C4—82no—
672C4#310114yes—
673C4#320814yes—
674C4#348114yes—
675C2—24no—
676C2—88no—
677C2#3496311yescharacteristic
678C2C2×#310111yescharacteristic
679C2C2×#310311yescharacteristic
680C2#6055811yescharacteristic
681C2#6056111yescharacteristic
682C2C2×#320811yescharacteristic
683C2#6079011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C4×#533C2semidirect
3. #29366C2semidirect
4. #29258C2semidirect
5. #28160C2semidirect
6. #27680C2semidirect
7. #27585C2semidirect
9. #23933C2semidirect
10. #23815C2semidirect
11. #23799C2semidirect
12. C4×#460C2semidirect
13. C4×#450C2semidirect
14. #20777C2semidirect
22. C2×C4×SD16C22semidirect
23. C4×C4⋊Q8C22semidirect
24. C4×C4⋊Q8C22semidirect
25. C4×C4⋊Q8C22semidirect
26. C4×C4⋊Q8C22semidirect
43. #2295C22semidirect
50. C42×Q8C22semidirect
53. #2099C22semidirect
54. #2096C22semidirect
55. #2094C22semidirect
56. #2071C22semidirect
58. #2023C22semidirect
60. #2013C22semidirect
62. #1948C22semidirect
63. #1926C22semidirect
64. #1880C22semidirect
65. #1877C22semidirect
67. #1864C22semidirect
69. #1839C22semidirect
70. #1831C22semidirect
72. #1803C22semidirect
75. C4×C4⋊C8C22semidirect
76. C4×#102C22semidirect
78. #1744C22semidirect
330. C2×C42SD16semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
C2×C4×#1272
#293662
#292582
#281602
#276802
#275852
#242112
#239332
#238152
#237992
C4×#4602
C4×#4502
#207772