Finite Group Databasebeta

← #6436251

Subgroups — #6436251

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#30166C212yesmaximal, complemented
4#30165C212yesmaximal, complemented
5#30093C212yesmaximal, complemented
6#30077C212yesmaximal, complemented
7#29982C212yesmaximal, complemented
8#29944C212yesmaximal, complemented
9#29870C212yesmaximal, complemented
10#29694C212yesmaximal
11#23839C211yesmaximal, characteristic, complemented
12#23839C211yesmaximal, characteristic, complemented
13#23822C211yesmaximal, characteristic, complemented
14#23822C211yesmaximal, characteristic, complemented
15#23764C214yesmaximal, complemented
16#21617C214yesmaximal, complemented
17#20778C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×C4×#41—21no—
20C2×#478C2214yes—
21#2864C2212yes—
22#2863—21no—
23#2863—21no—
24#2862C2211yescharacteristic
25#2859C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2855—21no—
29#2855—21no—
30#2851C2212yes—
31C4×C4⋊Q8C2211yescharacteristic
32C4×C4⋊Q8C2211yescharacteristic
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×#127C2211yescharacteristic
36C4×#127C2211yescharacteristic
37C4×#125C2212yes—
38C4×#124—21no—
39C4×#124—21no—
40C4×C22⋊Q8—21no—
41C4×C22⋊Q8—21no—
42C4×#121—21no—
43C4×#121—21no—
44C4×#121C2212yes—
45C4×#121C2212yes—
46#2297—21no—
47#2295—21no—
48#2295—21no—
49#2293—21no—
50#2293C2212yes—
51#2285C2211yescharacteristic
52#2283C2212yes—
53C42×D4—21no—
54C42×D4C2212yes—
55C2×C43C2211yescharacteristic
56#2101C2214yes—
57#2100C2212yes—
58#2100C2212yes—
59#2092C2214yes—
60#2085C2214yes—
61#2084C2212yes—
62#2084C2212yes—
63#2083C2212yes—
64#2083C2212yes—
65#2077C2214yes—
66#2075C2214yes—
67#2073C2214yes—
68#2071C2212yes—
69#2071C2212yes—
70#2058C2212yes—
71#2058C2212yes—
72#2057C2212yes—
73#2057C2212yes—
74#2041C2212yes—
75#2041C2212yes—
76#2034C2214yes—
77#2033C2214yes—
78#2023C2212yes—
79#2023C2212yes—
80#1989C2214yes—
81#1949C2214yes—
82#1920C2214yes—
83#1918C2214yes—
84#1864C2214yes—
85#1862C2214yes—
86#1839C2212yes—
87#1831C2212yes—
88#1817C2212yes—
89#1804C2214yes—
90#1797C2212yes—
91#1776C2214yes—
92#1776C2214yes—
93#1767C2212yes—
94#1762C2214yes—
95#1762C2214yes—
96#1749C2212yes—
97#1745C2211yescharacteristic
98#1745C2211yescharacteristic
99C22×#41—21no—
100#532—28no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#125C2312yes—
106C2×#124—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×#121—21no—
110C2×#121C2312yes—
111C4×#41—48no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×#117—21no—
127C2×#117—22no—
128C22×C42—22no—
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#478—216no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2312yes—
148C2×#97C2312yes—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2312yes—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#386—22no—
194#386—22no—
195#385—22no—
196#385—22no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#383—22no—
202#382—21no—
203#382C2312yes—
204#382C2312yes—
205#382C2312yes—
206#381—22no—
207#380—21no—
208#380—21no—
209#380—21no—
210#380C2312yes—
211#380C2312yes—
212#380C2312yes—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232#376—22no—
233C43D411yescharacteristic
234C43D411yescharacteristic
235C43D411yescharacteristic
236C43D411yescharacteristic
237C43C2311yescharacteristic
238C43C2311yescharacteristic
239C43C2311yescharacteristic
240C43C2311yescharacteristic
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2312yes—
250#336C2312yes—
251#336C2312yes—
252#336C2312yes—
253#336C2314yes—
254#336C2314yes—
255C2×#41—28no—
256C2×#41—44no—
257C2×#41—44no—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4C2412yes—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—22no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2411yescharacteristic
275C22×C8C2414yes—
276C22×C8C2414yes—
277C4⋊Q8—24no—
278C4⋊Q8—42no—
279C4⋊Q8—42no—
280#127—24no—
281#125—28no—
282#124—28no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285#121—28no—
286#121—24no—
287#121—24no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×D4—28no—
293C4×D4—24no—
294C4×D4—28no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—48no—
301C4×D4—48no—
302C4×D4#4118yes—
303#117—48no—
304#117—44no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
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—22no—
341C2×C22⋊C4—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—42no—
351C2×C42—42no—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42—48no—
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×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—28no—
384#107—44no—
385#106—28no—
386#106—44no—
387C4⋊C8—28no—
388#102—28no—
389#102—28no—
390#102—44no—
391#102—44no—
392C22⋊C8—44no—
393#97—24no—
394#97—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415C24—22no—
416C24C22×D411yescharacteristic
417#41—432no—
418C2×Q8—24no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×D4—24no—
428C2×D4—28no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C2×#4114yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—28no—
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—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4AGL(1,ℤ/8)14yes—
477C22×C4C22×D411yescharacteristic
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×C4C22×D411yescharacteristic
490C22×C4C2511yescharacteristic
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4C2×#4114yes—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
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—41no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—44no—
564C42—44no—
565C42—44no—
566C42—44no—
567C42—44no—
568C42C2×#4114yes—
569C422−1+414yes—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—24no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23C2×AGL(1,ℤ/8)12yes—
584C23C23×D411yescharacteristic
585Q8—48no—
586Q8—84no—
587Q8—84no—
588D4—28no—
589D4—48no—
590D4—48no—
591D4—416no—
592D4—44no—
593C2×C4—28no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
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—24no—
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—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—416no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—88no—
653C2×C4—88no—
654C2×C4#52512yes—
655C2×C4#52512yes—
656C2×C4#53612yes—
657C2×C4#53612yes—
658C2×C4C2×AGL(1,ℤ/8)12yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4#57512yes—
661C2×C4#57512yes—
662C2×C4#57512yes—
663C2×C4C22×#4112yes—
664C2×C4C2×2−1+412yes—
665C2×C4#58512yes—
666C8—44no—
667C8—44no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22C2×#52511yescharacteristic
683C22C2×#53611yescharacteristic
684C22#344911yescharacteristic
685C22C22×AGL(1,ℤ/8)11yescharacteristic
686C22C2×#57511yescharacteristic
687C22C2×#57511yescharacteristic
688C22C2×#57511yescharacteristic
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—41no—
699C4—41no—
700C4—44no—
701C4—41no—
702C4—41no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4#346114yes—
710C2—42no—
711C2—84no—
712C2#3462711yescharacteristic
713C2#6054611yescharacteristic
714C2#6055811yescharacteristic
715C2#6086211yescharacteristic
716C2#6086411yescharacteristic
717C2#6095811yescharacteristic
718C2#6097011yescharacteristic
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. #30166C2semidirect
4. #30165C2semidirect
5. #30093C2semidirect
6. #30077C2semidirect
7. #29982C2semidirect
8. #29944C2semidirect
9. #29870C2semidirect
11. #23839C2semidirect
12. #23839C2semidirect
13. #23822C2semidirect
14. #23822C2semidirect
15. #23764C2semidirect
16. #21617C2semidirect
17. #20778C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C4×#5322
#301662
#301652
#300932
#300772
#299822
#299442
#298702
#296942
#238392
#238222
#237642
#216172
#207782