Finite Group Databasebeta

← #6493626

Subgroups — #6493626

Order 512.

2124 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#33108C212yesmaximal, complemented
4#33106C212yesmaximal, complemented
5#29923C212yesmaximal, complemented
6#29921C212yesmaximal, complemented
7#27978C212yesmaximal, complemented
8#27914C212yesmaximal, complemented
9#27895C214yesmaximal, complemented
10#27584C214yesmaximal, complemented
11#24004C214yesmaximal, complemented
12#21988C212yesmaximal, complemented
13#21809C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C2×#525C2212yes—
17C22×#117—21no—
18C23×C42C2211yescharacteristic
19C2×#483C2214yes—
20C2×#480C2214yes—
21C2×#432C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2856C2211yescharacteristic
26#2854—21no—
27#2854—21no—
28#2850C2212yes—
29#2850C2212yes—
30#2847—21no—
31#2847—21no—
32C4×#121—21no—
33C4×#121—21no—
34C4×#121—21no—
35C4×#121—21no—
36#2297C2212yes—
37#2285—21no—
38#2285—21no—
39#2285—21no—
40#2285—21no—
41#2285C2212yes—
42#2285C2212yes—
43#2285C2212yes—
44#2285C2212yes—
45#2285C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#382—21no—
49C2×#380—21no—
50C2×#380C2212yes—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2212yes—
53#2095C2214yescomplemented
54#2094C2214yescomplemented
55#2072C2214yescomplemented
56#2071C2214yescomplemented
57#2036C2218yes—
58#2034C2218yes—
59#1982C2214yescomplemented
60#1981C2214yescomplemented
61#1959C2214yescomplemented
62#1959C2214yescomplemented
63#1948C2214yescomplemented
64#1947C2214yescomplemented
65#1927C2218yescomplemented
66#1925C2218yescomplemented
67#1874C2214yes—
68#1872C2214yes—
69#1854C2218yes—
70#1847C2214yescomplemented
71#1836C2214yescomplemented
72#1817C2212yes—
73#1797C2212yes—
74#1797C2212yes—
75#1788C2218yes—
76#1779C2212yes—
77#1749C2212yes—
78#1749C2212yes—
79C22×#41C2312yes—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83#532—28no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#124C2312yes—
87#525—48no—
88C2×#123—22no—
89C2×C22⋊Q8C2312yes—
90C2×C22⋊Q8C2312yes—
91C2×#121—21no—
92C2×#121—21no—
93C2×#121—21no—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121C2312yes—
98C2×#121C2312yes—
99C2×#120—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117—48no—
109C2×#117C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—22no—
117C22×C42—28no—
118C22×C42—22no—
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#483—48no—
132#480—48no—
133#432—28no—
134C2×#107C2318yes—
135C2×#106C2318yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2318yes—
138C2×#103C2318yes—
139C2×#102C2318yes—
140C2×#102C2318yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×#97C2314yes—
145#391C2312yes—
146#391C2312yes—
147#390—21no—
148#390—21no—
149#390—21no—
150#390—21no—
151#390C2312yes—
152#390C2312yes—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#386—22no—
162#386—22no—
163#386—22no—
164#386—22no—
165#386—22no—
166#386—22no—
167#384—21no—
168#384—21no—
169#384—21no—
170#384—21no—
171#384C2312yes—
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#384C2312yes—
175#384C2312yes—
176#384C2312yes—
177#382—22no—
178#382—22no—
179#382—22no—
180#382—22no—
181#382C2312yes—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—21no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380C2314yes—
195#380C2314yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2314yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C2×#95—22no—
209C2×#95—22no—
210#340C2318yes—
211#340C2318yes—
212#336C2314yes—
213#336C2314yes—
214C25C2×D411yescharacteristic
215C2×#41—28no—
216C2×#41—44no—
217C2×#41—44no—
218C22×Q8C2412yes—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C2412yes—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
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×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C22×C8C2418yes—
256C4⋊Q8—24no—
257#127—24no—
258#124—28no—
259#123—48no—
260C22⋊Q8—28no—
261C22⋊Q8—44no—
262#121—28no—
263#121—44no—
264#121—44no—
265#121—44no—
266#121—44no—
267#121—44no—
268#121—44no—
269#121—44no—
270#120—48no—
271C4×D4—48no—
272C4×D4—48no—
273C4×D4—44no—
274C4×D4—44no—
275#117—44no—
276#117—48no—
277#117—48no—
278#117—48no—
279#117—48no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
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—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—48no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—48no—
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×C42#4118yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—48no—
356#106—48no—
357C4⋊C8—28no—
358#103—48no—
359#103—48no—
360#102—48no—
361#102—48no—
362C22⋊C8—28no—
363C22⋊C8—28no—
364C22⋊C8—44no—
365#97—28no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24C22×D411yescharacteristic
385C24C22×D411yescharacteristic
386C24C22×D411yescharacteristic
387#41—816no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C22×C4—232no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
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—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—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4#12314yes—
475C22×C4#12314yes—
476C22×C4AGL(1,ℤ/8)14yes—
477C22×C4#13714yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4114yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4C2511yescharacteristic
492C2×C8—28no—
493C2×C8—28no—
494C2×C8—28no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—44no—
545C42—48no—
546C42—48no—
547C42#12314yes—
548C42#12314yes—
549C42C2×#4114yes—
550C42C2×#4114yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—28no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23C2×#12312yes—
582C23#53512yes—
583C23C2×AGL(1,ℤ/8)12yes—
584C23C2×#13712yes—
585C23#57512yes—
586C23C23×D411yescharacteristic
587Q8—48no—
588D4—48no—
589D4—84no—
590D4—88no—
591D4—88no—
592D4—88no—
593D4—84no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—88no—
650C2×C4—88no—
651C2×C4—88no—
652C2×C4C2×#12312yes—
653C2×C4C2×#12312yes—
654C2×C4C2×#12312yes—
655C2×C4C2×#12312yes—
656C2×C4C2×#12312yes—
657C2×C4#53512yes—
658C2×C4C2×AGL(1,ℤ/8)12yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C2×#13712yes—
661C2×C4C2×#13712yes—
662C2×C4#57512yes—
663C2×C4C22×#4112yes—
664C2×C4C2×H(5,𝔽2)12yes—
665C2×C4#58512yes—
666C8—48no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—416no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22#321414yes—
696C22C22×#12311yescharacteristic
697C22C2×#53511yescharacteristic
698C22#344911yescharacteristic
699C22C22×AGL(1,ℤ/8)11yescharacteristic
700C22C22×#13711yescharacteristic
701C22C2×#57511yescharacteristic
702C22C2×#57511yescharacteristic
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—22no—
711C4—41no—
712C4—41no—
713C4—84no—
714C4—84no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4#309214yes—
719C4#349414yes—
720C2—24no—
721C2—42no—
722C2—42no—
723C2—162no—
724C2#3526211yescharacteristic
725C2C2×#309211yescharacteristic
726C2#6051111yescharacteristic
727C2C2×#321411yescharacteristic
728C2#6080411yescharacteristic
729C2#6095111yescharacteristic
730C2#6096311yescharacteristic
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. #33108C2semidirect
4. #33106C2semidirect
5. #29923C2semidirect
6. #29921C2semidirect
7. #27978C2semidirect
8. #27914C2semidirect
9. #27895C2semidirect
10. #27584C2semidirect
11. #24004C2semidirect
12. #21988C2semidirect
13. #21809C2semidirect
14. #20715C2semidirect
53. #2095C22semidirect
54. #2094C22semidirect
55. #2072C22semidirect
56. #2071C22semidirect
59. #1982C22semidirect
60. #1981C22semidirect
61. #1959C22semidirect
62. #1959C22semidirect
63. #1948C22semidirect
64. #1947C22semidirect
65. #1927C22semidirect
66. #1925C22semidirect
70. #1847C22semidirect
71. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#22852
#331082
#331062
#299232
#299212
#279782
#279142
#278952
#275842
#240042
#219882
#218092
#207152