Finite Group Databasebeta

← #6630453

Subgroups — #6630453

Order 512.

2632 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3#33149C212yesmaximal, complemented
4#33148C212yesmaximal, complemented
5#33107C212yesmaximal, complemented
6#33106C212yesmaximal, complemented
7#33096C214yesmaximal, complemented
8#32979C212yesmaximal, complemented
9#32978C212yesmaximal, complemented
10#21580C214yesmaximal, complemented
11#21578C214yesmaximal, complemented
12#21395C212yesmaximal, complemented
13#21365C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C2×#525—21no—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42—21no—
20C23×C42C2211yescharacteristic
21C2×#481C2214yes—
22C2×#480C2214yes—
23C2×#432C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2856C2211yescharacteristic
28#2854—21no—
29#2854C2212yes—
30#2850—21no—
31#2850—21no—
32#2847—21no—
33#2847C2212yes—
34C4×#121—21no—
35C4×#121—21no—
36C4×#121C2212yes—
37C4×#121C2212yes—
38#2297—21no—
39#2285C2211yescharacteristic
40C2×C4×C4⋊C4—21no—
41C2×C4×C4⋊C4—21no—
42C2×C4×C4⋊C4—21no—
43C2×C4×C4⋊C4—21no—
44C2×C4×C4⋊C4—21no—
45C2×C4×C4⋊C4—21no—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C22×#95—21no—
53#1987C2214yescomplemented
54#1982C2214yescomplemented
55#1948C2214yescomplemented
56#1947C2214yescomplemented
57#1926C2218yescomplemented
58#1925C2218yescomplemented
59#1921C2218yescomplemented
60#1920C2218yescomplemented
61#1874C2214yes—
62#1874C2214yes—
63#1874C2214yes—
64#1874C2214yes—
65#1853C2214yes—
66#1853C2214yes—
67#1852C2214yes—
68#1852C2214yes—
69#1824C2214yes—
70#1823C2214yes—
71#1817C2212yes—
72#1797C2212yes—
73#1797C2212yes—
74#1779C2212yes—
75#1776C2218yescomplemented
76#1762C2218yescomplemented
77#1749C2212yes—
78#1749C2212yes—
79C22×#41—21no—
80C24×C4—21no—
81C24×C4—21no—
82C24×C4D411yescharacteristic
83C24×C4D411yescharacteristic
84C24×C4C2311yescharacteristic
85#532—28no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#127C2311yescharacteristic
88C2×#124—21no—
89#525—48no—
90C2×#123—22no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×#121—21no—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121—21no—
98C2×#121C2312yes—
99C2×#121C2312yes—
100C2×#121C2312yes—
101C2×#120—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—21no—
105C2×C4×D4C2312yes—
106C2×#117—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—28no—
114C22×C4⋊C4—28no—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C42—28no—
122C22×C42—22no—
123C22×C42—22no—
124C22×C42—22no—
125C22×C42—22no—
126C22×C42—22no—
127C22×C42—22no—
128C22×C42—48no—
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#481—216no—
142#480—216no—
143#432—28no—
144C2×#107C2318yes—
145C2×#106C2318yes—
146C2×C4⋊C8C2314yes—
147C2×#102C2318yes—
148C2×#102C2318yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97C2314yes—
153#391—21no—
154#391—21no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390—21no—
159#390C2312yes—
160#390C2312yes—
161#389—21no—
162#389—21no—
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#386—22no—
174#386—22no—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#382—21no—
178#380—21no—
179#380—21no—
180#380—21no—
181#380C2312yes—
182#380C2312yes—
183#380C2312yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C2×#95—22no—
205C2×#95—22no—
206C2×#95—22no—
207C2×#95—22no—
208C2×#95—22no—
209C2×#95—22no—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#340C2318yes—
214#336C2314yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C2×#41—44no—
218C2×#41—44no—
219C2×#41—44no—
220C22×Q8—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C2412yes—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—28no—
234C23×C4—28no—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—42no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266C23×C4C2411yescharacteristic
267C22×C8C2418yes—
268C4⋊Q8—24no—
269#127—24no—
270#124—44no—
271#123—48no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274#121—28no—
275#121—28no—
276#121—28no—
277#121—44no—
278#121—44no—
279#121—44no—
280#121—44no—
281#121—44no—
282#120—48no—
283C4×D4—28no—
284C4×D4—48no—
285C4×D4—48no—
286C4×D4—44no—
287#117—44no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4#4118yescomplemented
314C2×C4⋊C4#4118yescomplemented
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2414yes—
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—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—48no—
348C2×C42—48no—
349C2×C42—48no—
350C2×C42—48no—
351C2×C42—48no—
352C2×C42—48no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42#4118yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—216no—
371#106—216no—
372C4⋊C8—28no—
373#102—216no—
374#102—216no—
375C22⋊C8—28no—
376C22⋊C8—28no—
377C22⋊C8—28no—
378#97—28no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24AGL(1,ℤ/8)14yes—
397C24C22×D411yescharacteristic
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400#41—816no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C22×C4—232no—
423C22×C4—232no—
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—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—22no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—416no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—432no—
490C22×C4—432no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4#12314yes—
501C22×C4#12314yes—
502C22×C4AGL(1,ℤ/8)14yes—
503C22×C4#13714yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C2×#4114yes—
516C22×C4C2×#4114yes—
517C22×C4C2×#4114yes—
518C22×C4C2511yescharacteristic
519C2×C8—28no—
520C2×C8—28no—
521C2×C8—28no—
522C4⋊C4—28no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—28no—
526C4⋊C4—28no—
527C4⋊C4—28no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4C2×#4118yes—
541C4⋊C4C2×#4118yes—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—48no—
566C42—48no—
567C42—44no—
568C42—48no—
569C42—48no—
570C42#12314yes—
571C42#12314yes—
572C42C2×#4114yes—
573C42C2×#4114yes—
574C23—28no—
575C23—28no—
576C23—28no—
577C23—28no—
578C23—28no—
579C23—28no—
580C23—28no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23C2×#12312yes—
601C23#52512yes—
602C23#52512yes—
603C23C2×AGL(1,ℤ/8)12yes—
604C23C2×AGL(1,ℤ/8)12yes—
605C23C2×AGL(1,ℤ/8)12yes—
606C23C2×AGL(1,ℤ/8)12yes—
607C23C2×#13712yes—
608C23#57512yes—
609C23C23×D411yescharacteristic
610Q8—84no—
611D4—48no—
612D4—84no—
613D4—88no—
614D4—88no—
615D4—88no—
616D4—84no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—28no—
641C2×C4—28no—
642C2×C4—28no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—28no—
649C2×C4—28no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—416no—
659C2×C4—416no—
660C2×C4—416no—
661C2×C4—416no—
662C2×C4—416no—
663C2×C4—44no—
664C2×C4—416no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—88no—
679C2×C4C2×#12312yes—
680C2×C4C2×#12312yes—
681C2×C4C2×#12312yes—
682C2×C4C2×#12312yes—
683C2×C4C2×#12312yes—
684C2×C4#52512yes—
685C2×C4#52512yes—
686C2×C4C2×AGL(1,ℤ/8)12yes—
687C2×C4C2×AGL(1,ℤ/8)12yes—
688C2×C4C2×#13712yes—
689C2×C4C2×#13712yes—
690C2×C4C22×#4112yes—
691C2×C4C22×#4112yes—
692C2×C4C22×#4112yes—
693C8—48no—
694C22—232no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22#309414yes—
721C22#310914yes—
722C22#311214yes—
723C22C22×#12311yescharacteristic
724C22C2×#52511yescharacteristic
725C22C2×#52511yescharacteristic
726C22C22×AGL(1,ℤ/8)11yescharacteristic
727C22C22×AGL(1,ℤ/8)11yescharacteristic
728C22C22×#13711yescharacteristic
729C22C2×#57511yescharacteristic
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—41no—
739C4—41no—
740C4—48no—
741C4—48no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4#309214yes—
746C4#344414yes—
747C2—24no—
748C2—24no—
749C2—24no—
750C2—162no—
751C2#3662611yescharacteristic
752C2C2×#309211yescharacteristic
753C2C2×#309411yescharacteristic
754C2C2×#310911yescharacteristic
755C2C2×#311211yescharacteristic
756C2#6055811yescharacteristic
757C2#6056011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C4⋊C4C2semidirect
3. #33149C2semidirect
4. #33148C2semidirect
5. #33107C2semidirect
6. #33106C2semidirect
7. #33096C2semidirect
8. #32979C2semidirect
9. #32978C2semidirect
10. #21580C2semidirect
11. #21578C2semidirect
12. #21395C2semidirect
13. #21365C2semidirect
14. #20715C2semidirect
53. #1987C22semidirect
54. #1982C22semidirect
55. #1948C22semidirect
56. #1947C22semidirect
57. #1926C22semidirect
58. #1925C22semidirect
59. #1921C22semidirect
60. #1920C22semidirect
75. #1776C22semidirect
76. #1762C22semidirect
313. C2×C4⋊C4#41semidirect
314. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
#366242
C22×C4×C4⋊C42
#331492
#331482
#331072
#331062
#330962
#329792
#329782
#215802
#215782
#213952
#213652
#207152