Finite Group Databasebeta

← #6459072

Subgroups — #6459072

Order 512.

2180 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#27973C211yesmaximal, characteristic, complemented
4#27960C211yesmaximal, characteristic, complemented
5#27948C211yesmaximal, characteristic, complemented
6#27909C211yesmaximal, characteristic, complemented
7#27890C212yesmaximal, complemented
8#27878C212yesmaximal, complemented
9#27862C214yesmaximal, complemented
10#24558C214yesmaximal, complemented
11#23761C214yesmaximal, complemented
12#21578C214yesmaximal, complemented
13#21410C212yesmaximal, complemented
14#20824C212yesmaximal, complemented
15C4×#407C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#525—21no—
18C22×#117—21no—
19C23×C42C2211yescharacteristic
20C2×#480C2214yes—
21C2×#407C2212yes—
22C2×C4×M16C2212yes—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854—21no—
28#2854C2212yes—
29#2850—21no—
30#2850—21no—
31#2847—21no—
32#2847C2212yes—
33C4×#121—21no—
34C4×#121—21no—
35C4×#121C2212yes—
36C4×#121C2212yes—
37#2297—21no—
38#2285—21no—
39#2285—21no—
40#2285—21no—
41#2285—21no—
42#2285C2211yescharacteristic
43#2285C2212yes—
44#2285C2212yes—
45#2285C2212yes—
46#2285C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#382—21no—
50C2×#380—21no—
51C2×#380C2212yes—
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2212yes—
54#2099C2214yescomplemented
55#2093C2214yescomplemented
56#2084C2214yescomplemented
57#2076C2218yes—
58#2071C2214yescomplemented
59#2034C2218yes—
60#1990C2212yescomplemented
61#1990C2212yescomplemented
62#1957C2212yescomplemented
63#1957C2212yescomplemented
64#1926C2218yescomplemented
65#1921C2218yescomplemented
66#1895C2218yes—
67#1874C2214yes—
68#1853C2214yes—
69#1845C2212yescomplemented
70#1844C2212yescomplemented
71#1834C2212yescomplemented
72#1834C2212yescomplemented
73#1823C2214yes—
74#1809C2218yes—
75#1776C2218yescomplemented
76#1762C2218yescomplemented
77#1749C2212yes—
78#1749C2212yes—
79C4×C22⋊C8C2211yescharacteristic
80C4×C22⋊C8C2211yescharacteristic
81C4×C22⋊C8C2212yes—
82#1728C2212yes—
83C22×#41—21no—
84C24×C4D411yescharacteristic
85C24×C4D411yescharacteristic
86C24×C4C2311yescharacteristic
87C22×M16C2314yes—
88#532—28no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127C2311yescharacteristic
91C2×#124—21no—
92#525—48no—
93C2×#123—22no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×#121—21no—
97C2×#121—21no—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121—21no—
101C2×#121C2312yes—
102C2×#121C2312yes—
103C2×#121C2312yes—
104C2×#120—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—21no—
108C2×C4×D4C2312yes—
109C2×#117—22no—
110C2×#117—22no—
111C2×#117—22no—
112C2×#117—22no—
113C2×#117—21no—
114C2×#117—48no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4—22no—
122C22×C42—28no—
123C22×C42—22no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#480—216no—
137C2×#107C2318yes—
138C2×#106C2318yes—
139C2×#102C2318yes—
140C2×#102C2318yes—
141#407—28no—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C4×M16—28no—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2312yes—
149#391—21no—
150#391—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390C2312yes—
156#390C2312yes—
157#389—21no—
158#389—21no—
159#389C2312yes—
160#389C2312yes—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#386—22no—
166#386—22no—
167#386—22no—
168#386—22no—
169#386—22no—
170#386—22no—
171#384—21no—
172#384—21no—
173#384—21no—
174#384—21no—
175#384C2312yes—
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2312yes—
179#384C2312yes—
180#384C2312yes—
181#382—21no—
182#382—22no—
183#382—22no—
184#382—22no—
185#382—22no—
186#380—21no—
187#380—21no—
188#380—21no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2314yes—
197#380C2312yes—
198#380C2314yes—
199#380C2312yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C2×#95—22no—
213C2×#95—22no—
214#340C2318yes—
215#340C2318yes—
216#340C2318yes—
217#340C2318yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C2×#41—44no—
223C2×#41—44no—
224C2×#41—44no—
225C22×Q8—21no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2412yes—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
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×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×M16—28no—
263C2×M16—28no—
264C2×M16—44no—
265C22×C8C2414yes—
266C22×C8C2414yes—
267C4⋊Q8—24no—
268#127—24no—
269#124—44no—
270#123—48no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273#121—28no—
274#121—28no—
275#121—44no—
276#121—44no—
277#121—44no—
278#121—44no—
279#121—44no—
280#121—44no—
281#120—48no—
282C4×D4—28no—
283C4×D4—48no—
284C4×D4—48no—
285C4×D4—44no—
286#117—48no—
287#117—48no—
288#117—48no—
289#117—48no—
290#117—44no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
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—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
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×C22⋊C4—22no—
332C2×C22⋊C4—48no—
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—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—48no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42#4118yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365#107—216no—
366#106—48no—
367#102—216no—
368#102—48no—
369C22⋊C8—24no—
370C22⋊C8—28no—
371C22⋊C8#4118yescomplemented
372#97—28no—
373C4×C8—24no—
374C4×C8—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—41no—
392C24—41no—
393C24C22×D411yescharacteristic
394C24C22×D411yescharacteristic
395C24C22×D411yescharacteristic
396#41—816no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C22×C4—232no—
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—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—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4#12314yes—
483C22×C4#12314yes—
484C22×C4AGL(1,ℤ/8)14yes—
485C22×C4AGL(1,ℤ/8)14yes—
486C22×C4#13714yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C2×#4114yes—
499C22×C4C2×#4114yes—
500C22×C4C2511yescharacteristic
501M16—416no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8C2×#4118yes—
508C4⋊C4—28no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—28no—
512C4⋊C4—24no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—48no—
556C42—48no—
557C42—44no—
558C42#12314yes—
559C42#12314yes—
560C42C2×#4114yes—
561C42C2×#4114yes—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—28no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23C2×#12312yes—
593C23#53212yes—
594C23C2×AGL(1,ℤ/8)12yes—
595C23C2×AGL(1,ℤ/8)12yes—
596C23C2×#13712yes—
597C23C23×D411yescharacteristic
598Q8—84no—
599D4—48no—
600D4—88no—
601D4—88no—
602D4—88no—
603D4—84no—
604D4—84no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—88no—
659C2×C4—88no—
660C2×C4—88no—
661C2×C4C2×#12312yes—
662C2×C4C2×#12312yes—
663C2×C4C2×#12312yes—
664C2×C4C2×#12312yes—
665C2×C4C2×#12312yes—
666C2×C4#53212yes—
667C2×C4C2×AGL(1,ℤ/8)12yes—
668C2×C4C2×AGL(1,ℤ/8)12yes—
669C2×C4C2×AGL(1,ℤ/8)12yes—
670C2×C4C2×AGL(1,ℤ/8)12yes—
671C2×C4C2×#13712yes—
672C2×C4C2×#13712yes—
673C2×C4#57512yes—
674C2×C4C22×#4112yes—
675C2×C4C22×#4112yes—
676C2×C4#58512yes—
677C8—28no—
678C8—44no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—416no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22#318414yes—
708C22C22×#12311yescharacteristic
709C22C2×#53211yescharacteristic
710C22#344911yescharacteristic
711C22C22×AGL(1,ℤ/8)11yescharacteristic
712C22C22×AGL(1,ℤ/8)11yescharacteristic
713C22C22×#13711yescharacteristic
714C22C2×#57511yescharacteristic
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—41no—
723C4—41no—
724C4—84no—
725C4—82no—
726C4—84no—
727C4—82no—
728C4—82no—
729C4#309214yes—
730C4#309414yes—
731C4#347514yes—
732C2—24no—
733C2—42no—
734C2—42no—
735C2—162no—
736C2#3481011yescharacteristic
737C2C2×#309211yescharacteristic
738C2C2×#309411yescharacteristic
739C2C2×#318411yescharacteristic
740C2#6070111yescharacteristic
741C2#6096511yescharacteristic
742C2#6096911yescharacteristic
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. C2×#2285C2semidirect
3. #27973C2semidirect
4. #27960C2semidirect
5. #27948C2semidirect
6. #27909C2semidirect
7. #27890C2semidirect
8. #27878C2semidirect
9. #27862C2semidirect
10. #24558C2semidirect
11. #23761C2semidirect
12. #21578C2semidirect
13. #21410C2semidirect
14. #20824C2semidirect
15. C4×#407C2semidirect
54. #2099C22semidirect
55. #2093C22semidirect
56. #2084C22semidirect
58. #2071C22semidirect
60. #1990C22semidirect
61. #1990C22semidirect
62. #1957C22semidirect
63. #1957C22semidirect
64. #1926C22semidirect
65. #1921C22semidirect
69. #1845C22semidirect
70. #1844C22semidirect
71. #1834C22semidirect
72. #1834C22semidirect
75. #1776C22semidirect
76. #1762C22semidirect
371. C22⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#22852
#279732
#279602
#279482
#279092
#278902
#278782
#278622
#245582
#237612
#215782
#214102
#208242
C4×#4072