Finite Group Databasebeta

← #6482019

Subgroups — #6482019

Order 512.

2114 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35165C211yesmaximal, characteristic
2C2×#2352C211yesmaximal, characteristic, complemented
3#32981C212yesmaximal, complemented
4#32979C212yesmaximal, complemented
5#29891C212yesmaximal, complemented
6#29890C212yesmaximal, complemented
7#26312C214yesmaximal, complemented
8#26246C214yesmaximal, complemented
9#25026C214yesmaximal, complemented
10#24981C214yesmaximal, complemented
11#21986C211yesmaximal, characteristic, complemented
12#21865C211yesmaximal, characteristic, complemented
13#21790C211yesmaximal, characteristic, complemented
14#21395C211yesmaximal, characteristic, complemented
15#21365C211yesmaximal, characteristic, complemented
16C2×#535—21no—
17C2×#532C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#481C2214yescomplemented
22C2×#482C2214yescomplemented
23C2×#432C2211yescharacteristic
24C2×#432C2211yescharacteristic
25#2847C2212yes—
26#2847C2212yes—
27#2617—21no—
28#2617—21no—
29#2547C2212yes—
30#2544—22no—
31#2541C2212yes—
32#2537C2212yes—
33#2537C2212yes—
34#2535—22no—
35#2352—22no—
36#2352—22no—
37#2352C2214yes—
38#2352C2214yes—
39#2311—21no—
40C2×#388—21no—
41C2×#387—21no—
42C2×#386—21no—
43#2285C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49#2077C2214yescomplemented
50#2076C2214yescomplemented
51#2075C2214yescomplemented
52#2075C2214yescomplemented
53#2034C2214yescomplemented
54#2033C2214yescomplemented
55#2033C2214yescomplemented
56#2033C2214yescomplemented
57#1993C2212yescomplemented
58#1987C2212yescomplemented
59#1982C2212yescomplemented
60#1961C2212yescomplemented
61#1955C2214yes—
62#1955C2214yes—
63#1949C2214yescomplemented
64#1948C2212yescomplemented
65#1947C2212yescomplemented
66#1875C2214yescomplemented
67#1874C2212yes—
68#1874C2212yes—
69#1874C2212yes—
70#1874C2212yes—
71#1874C2214yescomplemented
72#1854C2214yescomplemented
73#1853C2212yes—
74#1853C2212yes—
75#1853C2214yescomplemented
76#1852C2212yes—
77#1852C2212yes—
78#1847C2212yescomplemented
79#1836C2212yescomplemented
80#1825C2214yescomplemented
81#1824C2214yescomplemented
82#1804C2214yescomplemented
83#1779C2212yes—
84#1779C2212yes—
85#1776C2214yescomplemented
86#1762C2214yescomplemented
87C22×#41—21no—
88C23×D4D411yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92#535—48no—
93#532—24no—
94#532—24no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127—22no—
97C2×#127C2311yescharacteristic
98C2×#124—22no—
99C2×#124—21no—
100C2×#121—22no—
101C2×#121—22no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#120—22no—
109C2×#120—22no—
110C2×C4×D4—21no—
111C2×C4×D4—24no—
112C2×C4×D4—24no—
113C2×C4×D4D412yes—
114C2×C4×D4D412yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C2×#117—22no—
120C2×#117—44no—
121C2×#117C2314yes—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—44no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4D411yescharacteristic
135C22×C22⋊C4D411yescharacteristic
136C22×C42—21no—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#482—48no—
140#481—48no—
141#432—24no—
142#432—24no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×#97C2312yes—
160C2×#97C2312yes—
161#397—22no—
162#397—22no—
163#394—22no—
164#394—22no—
165#394—22no—
166#394—22no—
167#392—22no—
168#392—22no—
169#391—22no—
170#390—22no—
171#388—24no—
172#388—24no—
173#387—24no—
174#387—24no—
175#386—24no—
176#386—24no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#382—21no—
186#382—21no—
187#382—21no—
188#382C2312yes—
189#382C2312yes—
190#380—22no—
191#380—22no—
192#380—21no—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2312yes—
197#380C2314yes—
198#380C2312yes—
199#380C2314yes—
200#380C2312yes—
201#380C2312yes—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—24no—
209C4×C22⋊C4—24no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223C25C2×D411yescharacteristic
224C25C2×D411yescharacteristic
225C2×#41—44no—
226C2×#41—44no—
227C2×#41—44no—
228C22×Q8—21no—
229C22×D4—24no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—24no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—42no—
239C22×D4C2×D412yes—
240C22×D4C2×D412yes—
241C22×D4C2412yes—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—24no—
251C23×C4—22no—
252C23×C4—42no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4—42no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—82no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2412yes—
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272#127—22no—
273#127—22no—
274#127—48no—
275#124—48no—
276#124—44no—
277#121—24no—
278#121—24no—
279#121—48no—
280#121—48no—
281#121—44no—
282#121—44no—
283#121—44no—
284#121—44no—
285#121—44no—
286#120—48no—
287#120—48no—
288C4×D4—28no—
289C4×D4—28no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—44no—
297#117—28no—
298#117—44no—
299#117—48no—
300#117—44no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—48no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—82no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—44no—
357C2×C42—82no—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42#4114yes—
361C2×C42#4114yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—44no—
365#107—44no—
366#106—28no—
367#106—44no—
368C4⋊C8—24no—
369C4⋊C8—24no—
370#102—28no—
371#102—44no—
372#102—44no—
373#102—44no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376C22⋊C8—44no—
377C22⋊C8—44no—
378C22⋊C8—44no—
379C22⋊C8—44no—
380#97—24no—
381#97—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—41no—
403C24—41no—
404C24—41no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24AGL(1,ℤ/8)14yes—
411C24C22×D412yes—
412C24C22×D412yes—
413C24C22×D411yescharacteristic
414#41—816no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—416no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—81no—
441C2×D4—81no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4—82no—
445C2×D4—82no—
446C2×D4—82no—
447C2×D4—82no—
448C2×D4—82no—
449C2×D4—82no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—42no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—48no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—44no—
504C22×C4—84no—
505C22×C4—816no—
506C22×C4#12314yes—
507C22×C4#12314yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C2×#4112yes—
516C22×C4C2×#4112yes—
517C22×C4C2511yescharacteristic
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—48no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—81no—
544C4⋊C4—81no—
545C4⋊C4H(5,𝔽2)14yes—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—48no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—84no—
564C42—48no—
565C42—44no—
566C42—82no—
567C42—82no—
568C42#12314yes—
569C42#12314yes—
570C42C2×#4112yes—
571C42C2×#4112yes—
572C42C2×#4112yes—
573C42C2×#4112yes—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—22no—
579C23—22no—
580C23—28no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—82no—
609C23—82no—
610C23—82no—
611C23—82no—
612C23C2×#12312yes—
613C23C2×AGL(1,ℤ/8)14yes—
614C23C2×AGL(1,ℤ/8)12yes—
615C23#57512yes—
616C23C23×D411yescharacteristic
617Q8—84no—
618D4—48no—
619D4—48no—
620D4—44no—
621D4—44no—
622D4—84no—
623D4—84no—
624D4—88no—
625D4—164no—
626D4—164no—
627D4—164no—
628D4—164no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—28no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—44no—
650C2×C4—48no—
651C2×C4—48no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—48no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
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—42no—
679C2×C4—88no—
680C2×C4—816no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—84no—
684C2×C4C2×#12312yes—
685C2×C4C2×#12312yes—
686C2×C4C2×#12312yes—
687C2×C4C2×#12312yes—
688C2×C4C2×#12312yes—
689C2×C4C22×#4111yescharacteristic
690C2×C4C22×#4111yescharacteristic
691C2×C4C2×H(5,𝔽2)12yes—
692C2×C4#58512yes—
693C8—44no—
694C8—44no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—24no—
702C22—24no—
703C22—24no—
704C22—416no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—88no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22#309414yes—
725C22C22×#12311yescharacteristic
726C22#348411yescharacteristic
727C22#348511yescharacteristic
728C22C22×AGL(1,ℤ/8)11yescharacteristic
729C22C2×#57511yescharacteristic
730C22#358511yescharacteristic
731C22#358611yescharacteristic
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—42no—
738C4—42no—
739C4—44no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4#346012yes—
747C4#350612yes—
748C2—24no—
749C2—44no—
750C2—82no—
751C2—162no—
752C2#3515211yescharacteristic
753C2C2×#309411yescharacteristic
754C2#6051711yescharacteristic
755C2#6157211yescharacteristic
756C2#6157611yescharacteristic
757C2#6170711yescharacteristic
758C2#6171111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35165C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2352C2semidirect
3. #32981C2semidirect
4. #32979C2semidirect
5. #29891C2semidirect
6. #29890C2semidirect
7. #26312C2semidirect
8. #26246C2semidirect
9. #25026C2semidirect
10. #24981C2semidirect
11. #21986C2semidirect
12. #21865C2semidirect
13. #21790C2semidirect
14. #21395C2semidirect
15. #21365C2semidirect
21. C2×#481C22semidirect
22. C2×#482C22semidirect
49. #2077C22semidirect
50. #2076C22semidirect
51. #2075C22semidirect
52. #2075C22semidirect
53. #2034C22semidirect
54. #2033C22semidirect
55. #2033C22semidirect
56. #2033C22semidirect
57. #1993C22semidirect
58. #1987C22semidirect
59. #1982C22semidirect
60. #1961C22semidirect
63. #1949C22semidirect
64. #1948C22semidirect
65. #1947C22semidirect
66. #1875C22semidirect
71. #1874C22semidirect
72. #1854C22semidirect
75. #1853C22semidirect
78. #1847C22semidirect
79. #1836C22semidirect
80. #1825C22semidirect
81. #1824C22semidirect
82. #1804C22semidirect
85. #1776C22semidirect
86. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#351652
C2×#23522
#329812
#329792
#298912
#298902
#263122
#262462
#250262
#249812
#219862
#218652
#217902
#213952
#213652