Finite Group Databasebeta

← #6489344

Subgroups — #6489344

Order 512.

1792 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#36635C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#32958C212yesmaximal, complemented
4#32955C212yesmaximal, complemented
5#30080C212yesmaximal, complemented
6#30077C212yesmaximal, complemented
7#28721C212yesmaximal, complemented
8#28510C212yesmaximal, complemented
9#28206C211yesmaximal, characteristic, complemented
10#28205C211yesmaximal, characteristic, complemented
11#28202C211yesmaximal, characteristic, complemented
12#28201C211yesmaximal, characteristic, complemented
13#27951C212yesmaximal, complemented
14#27590C214yesmaximal, complemented
15#25858C212yesmaximal
16#22222C214yesmaximal, complemented
17#20777C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×C4×#41C2212yes—
20C2×#488C2214yes—
21C2×#477C2212yes—
22C2×#475C2212yes—
23C2×#476C2212yes—
24C2×#474C2212yes—
25#2864C2212yes—
26#2863—21no—
27#2863—21no—
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2855—21no—
33#2855—21no—
34#2851C2212yes—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×#127C2211yescharacteristic
40C4×#127C2211yescharacteristic
41C4×#125C2212yes—
42C4×#124C2212yes—
43C4×#124C2212yes—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8—21no—
46C4×#121—21no—
47C4×#121—21no—
48C4×#121—21no—
49C4×#121—21no—
50#2297C2212yes—
51#2295—21no—
52#2295—21no—
53#2293—21no—
54#2293—21no—
55#2285C2211yescharacteristic
56#2283C2212yes—
57C42×D4—21no—
58C42×D4—21no—
59C2×C43C2211yescharacteristic
60#2097C2214yes—
61#2092C2214yes—
62#2084C2212yes—
63#2084C2212yes—
64#2083C2212yes—
65#2083C2212yes—
66#2074C2214yes—
67#2073C2214yes—
68#2043C2212yes—
69#2043C2212yes—
70#2041C2212yes—
71#2041C2212yes—
72#1991C2214yes—
73#1958C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1926C2214yes—
77#1925C2214yes—
78#1865C2214yes—
79#1864C2214yes—
80#1863C2214yes—
81#1862C2214yes—
82#1854C2214yes—
83#1854C2214yes—
84#1846C2214yes—
85#1839C2212yes—
86#1831C2211yescharacteristic
87#1831C2211yescharacteristic
88#1817C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1802C2212yes—
92#1802C2212yes—
93#1797C2212yes—
94#1788C2214yes—
95#1788C2214yes—
96C4×C4⋊C8C2212yes—
97C4×C22⋊C8C2212yes—
98#1744C2212yes—
99C22×#41C2312yes—
100#532—28no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#125C2312yes—
106C2×#124C2312yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×#121—21no—
110C2×#121—21no—
111C4×#41—44no—
112C4×#41—44no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×#117—22no—
128C2×#117C2312yes—
129C22×C42—22no—
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#488—216no—
138#477—44no—
139#476—44no—
140#475—44no—
141#474—44no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C4×C8C2314yes—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391C2312yes—
167#391C2312yes—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#386—22no—
203#386—22no—
204#385—22no—
205#385—22no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#383—22no—
211#382C2312yes—
212#382C2312yes—
213#382C2312yes—
214#382C2312yes—
215#381—22no—
216#380—21no—
217#380—21no—
218#380—21no—
219#380—21no—
220#380C2312yes—
221#380C2312yes—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2312yes—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4—22no—
241#376—22no—
242C43D411yescharacteristic
243C43D411yescharacteristic
244C43D411yescharacteristic
245C43D411yescharacteristic
246C43C2311yescharacteristic
247C43C2311yescharacteristic
248C43C2311yescharacteristic
249C43C2311yescharacteristic
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258#336C2312yes—
259#336C2312yes—
260C2×#41—24no—
261C2×#41—24no—
262C2×#41—24no—
263C2×#41—24no—
264C2×#41—42no—
265C2×#41—42no—
266C22×Q8—21no—
267C22×Q8—21no—
268C22×Q8C2412yes—
269C22×D4—22no—
270C22×D4—21no—
271C22×D4—21no—
272C22×D4C2412yes—
273C23×C4—22no—
274C23×C4—22no—
275C23×C4—22no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C22×C8C2414yes—
284C22×C8C2414yes—
285C4⋊Q8—24no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290#127—24no—
291#125—28no—
292#124—24no—
293#124#4118yes—
294C22⋊Q8—44no—
295C22⋊Q8—44no—
296#121—28no—
297#121—28no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—42no—
301C4×Q8—42no—
302C4×Q8—44no—
303C4×Q8—44no—
304C4×D4—28no—
305C4×D4—28no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—42no—
309C4×D4—42no—
310C4×D4—44no—
311C4×D4—44no—
312C4×D4—44no—
313C4×D4—48no—
314C4×D4—48no—
315C4×D4—44no—
316#117—48no—
317#117—42no—
318#117—42no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—42no—
367C2×C42—42no—
368C2×C42—42no—
369C2×C42—48no—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398#107—44no—
399#107—44no—
400#106—44no—
401#106—44no—
402C4⋊C8—24no—
403C4⋊C8—24no—
404#103—28no—
405#103—28no—
406#103—44no—
407#103—44no—
408#102—28no—
409#102—28no—
410#102—44no—
411#102—44no—
412C22⋊C8—28no—
413C4×C8—28no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434C24—22no—
435C24C22×D411yescharacteristic
436#41—48no—
437#41—48no—
438#41—48no—
439#41—48no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8C2×#4114yes—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—28no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—44no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4C2×#4114yes—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—28no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
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×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C42−1+414yes—
518C22×C4C2511yescharacteristic
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C22⋊C4—28no—
556C22⋊C4—28no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—21no—
583C42—21no—
584C42—22no—
585C42—22no—
586C42—41no—
587C42—41no—
588C42—41no—
589C42—41no—
590C42—41no—
591C42—41no—
592C42—44no—
593C42—44no—
594C42—44no—
595C42—44no—
596C42—44no—
597C42#13514yes—
598C42C2×#4114yes—
599C42H(5,𝔽2)12yes—
600C42H(5,𝔽2)12yes—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—24no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23#53612yes—
615C23C23×D411yescharacteristic
616Q8—24no—
617Q8—24no—
618Q8—42no—
619Q8—42no—
620Q8—84no—
621Q8—84no—
622D4—24no—
623D4—24no—
624D4—48no—
625D4—48no—
626D4—416no—
627D4—42no—
628D4—42no—
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—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—416no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
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—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—88no—
688C2×C4—88no—
689C2×C4#52512yes—
690C2×C4#52512yes—
691C2×C4#53512yes—
692C2×C4#53512yes—
693C2×C4#53612yes—
694C2×C4C2×#13512yes—
695C2×C4C2×#13512yes—
696C2×C4C2×#13512yes—
697C2×C4#57512yes—
698C2×C4#57512yes—
699C2×C4#57512yes—
700C2×C4C22×#4112yes—
701C2×C4C2×H(5,𝔽2)11yescharacteristic
702C2×C4C2×H(5,𝔽2)11yescharacteristic
703C2×C4C2×2−1+412yes—
704C8—44no—
705C8—44no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22C2×#52511yescharacteristic
721C22C2×#53511yescharacteristic
722C22C2×#53611yescharacteristic
723C22C22×#13511yescharacteristic
724C22C2×#57511yescharacteristic
725C22C2×#57511yescharacteristic
726C22C2×#57511yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—41no—
736C4—44no—
737C4—41no—
738C4—41no—
739C4—41no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4#321214yes—
747C4#352012yes—
748C4#352012yes—
749C2—42no—
750C2—84no—
751C2#3524011yescharacteristic
752C2#6053611yescharacteristic
753C2#6054611yescharacteristic
754C2C2×#321211yescharacteristic
755C2#6080511yescharacteristic
756C2#6084611yescharacteristic
757C2#6086311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36635C2semidirect
3. #32958C2semidirect
4. #32955C2semidirect
5. #30080C2semidirect
6. #30077C2semidirect
7. #28721C2semidirect
8. #28510C2semidirect
9. #28206C2semidirect
10. #28205C2semidirect
11. #28202C2semidirect
12. #28201C2semidirect
13. #27951C2semidirect
14. #27590C2semidirect
16. #22222C2semidirect
17. #20777C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C4×#5322
#329582
#329552
#300802
#300772
#287212
#285102
#282062
#282052
#282022
#282012
#279512
#275902
#258582
#222222
#207772