Finite Group Databasebeta

← #6468419

Subgroups — #6468419

Order 512.

1908 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#29288C212yesmaximal, complemented
4#29276C211yesmaximal, characteristic, complemented
5#29272C211yesmaximal, characteristic, complemented
6#29258C211yesmaximal, characteristic, complemented
7#29254C211yesmaximal, characteristic, complemented
8#29236C214yesmaximal, complemented
9#28286C212yesmaximal, complemented
10#27683C212yesmaximal, complemented
11#24304C212yesmaximal, complemented
12#23781C212yesmaximal, complemented
13#23780C212yesmaximal, complemented
14#21617C214yesmaximal, complemented
15#21393C212yesmaximal
16#21364C212yesmaximal, complemented
17#20810C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×C4×#41—21no—
21C2×#478C2214yes—
22C2×#432C2212yes—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×#127C2211yescharacteristic
30C4×#127C2211yescharacteristic
31C4×#125C2212yes—
32C4×#125C2212yes—
33C4×#124—21no—
34C4×#124—21no—
35C4×C22⋊Q8—21no—
36C4×C22⋊Q8—21no—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8—21no—
39C4×#121—21no—
40C4×#121—21no—
41C4×#121C2212yes—
42C4×#121C2212yes—
43#2297—21no—
44#2296C2212yes—
45#2295—21no—
46#2295—21no—
47#2293—21no—
48#2293C2212yes—
49#2285C2211yescharacteristic
50#2285C2211yescharacteristic
51C42×Q8—21no—
52C42×Q8—21no—
53C42×D4—21no—
54C42×D4C2212yes—
55C4×#117C2212yes—
56C2×C43C2211yescharacteristic
57#2100C2212yes—
58#2099C2212yes—
59#2094C2212yes—
60#2093C2212yes—
61#2088C2214yes—
62#2087C2214yes—
63#2067C2214yes—
64#2065C2214yes—
65#2058C2212yes—
66#2057C2212yes—
67#2023C2212yes—
68#2023C2212yes—
69#1989C2214yes—
70#1988C2212yes—
71#1988C2212yes—
72#1982C2212yes—
73#1982C2212yes—
74#1949C2214yes—
75#1948C2212yes—
76#1948C2212yes—
77#1947C2212yes—
78#1947C2212yes—
79#1920C2214yes—
80#1918C2214yes—
81#1875C2214yes—
82#1875C2214yes—
83#1864C2214yes—
84#1862C2214yes—
85#1854C2214yes—
86#1854C2214yes—
87#1842C2212yes—
88#1831C2211yescharacteristic
89#1831C2211yescharacteristic
90#1804C2214yes—
91#1797C2212yes—
92#1797C2212yes—
93#1776C2214yes—
94#1776C2214yes—
95#1767C2212yes—
96#1767C2212yes—
97#1762C2214yes—
98#1762C2214yes—
99C22×#41—21no—
100#533—28no—
101#532—24no—
102#532Q818yescomplemented
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#125C2312yes—
108C2×#124—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×#121—21no—
112C2×#121C2312yes—
113C4×#41—48no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×#117—21no—
133C2×#117—22no—
134C2×#117C2312yes—
135C2×#117C2312yes—
136C22×C42—22no—
137C22×C42D411yescharacteristic
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#478—216no—
145#432—28no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2314yes—
159C2×#97C2314yes—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390C2312yes—
171#390C2312yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#386—22no—
193#386—22no—
194#385—22no—
195#385—22no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#382—21no—
201#382C2312yes—
202#380—21no—
203#380—21no—
204#380—21no—
205#380C2312yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236#376—22no—
237C43—22no—
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#336C2312yes—
255#336C2312yes—
256#336C2312yes—
257#336C2312yes—
258C2×#41—28no—
259C2×#41—44no—
260C2×#41—44no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×D4—22no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4C2412yes—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C22×C8C2414yes—
279C22×C8C2414yes—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—22no—
283C4⋊Q8C2×Q814yes—
284#127—22no—
285#127C2×Q814yes—
286#125—24no—
287#125#4118yescomplemented
288#124—28no—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291#121—28no—
292#121—24no—
293#121C2×Q818yes—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—44no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×D4—28no—
303C4×D4—24no—
304C4×D4—28no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—44no—
309C4×D4—48no—
310C4×D4—48no—
311C4×D4—44no—
312C4×D4C2×Q818yes—
313#117—24no—
314#117—24no—
315#117—48no—
316#117—42no—
317#117—42no—
318#117—44no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4C2412yes—
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—22no—
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—22no—
367C2×C42—42no—
368C2×C42—42no—
369C2×C42—42no—
370C2×C42—42no—
371C2×C42—48no—
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×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×Q814yes—
395C2×C42#4114yes—
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402#107—28no—
403#107—44no—
404#106—28no—
405#106—44no—
406C4⋊C8—24no—
407C4⋊C8—24no—
408C4⋊C8—28no—
409C4⋊C8—44no—
410#102—28no—
411#102—28no—
412#102—44no—
413#102—44no—
414C22⋊C8—28no—
415#97—28no—
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—
432C24—22no—
433C24C22×D411yescharacteristic
434#41—432no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×D4—28no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4C22×Q814yes—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—28no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4C22⋊Q814yes—
492C22×C4C22⋊Q814yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×Q812yes—
507C22×C4C2×#4112yes—
508C22×C4C2511yescharacteristic
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4C22×Q814yes—
544C4⋊C4C2×#4114yes—
545C4⋊C4C2×#4114yes—
546C22⋊C4—28no—
547C22⋊C4—28no—
548C22⋊C4—28no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C42—28no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—41no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C42—41no—
580C42—44no—
581C42—44no—
582C42—44no—
583C42—44no—
584C42—44no—
585C42—44no—
586C42—44no—
587C42—44no—
588C42C22⋊Q814yes—
589C42C22⋊Q814yes—
590C42AGL(1,ℤ/8)14yes—
591C42C22×Q812yes—
592C42C22×Q812yes—
593C42C2×#4112yes—
594C42C2×#4112yes—
595C42C2×#4114yes—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—24no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23C2×C22⋊Q812yes—
610C23C23×D411yescharacteristic
611Q8—48no—
612Q8—48no—
613Q8—48no—
614D4—28no—
615D4—48no—
616D4—416no—
617D4—48no—
618D4—44no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
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—28no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—416no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
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—88no—
676C2×C4—88no—
677C2×C4C2×C22⋊Q812yes—
678C2×C4C2×C22⋊Q812yes—
679C2×C4C2×C22⋊Q812yes—
680C2×C4C2×C22⋊Q812yes—
681C2×C4C2×C22⋊Q812yes—
682C2×C4#52512yes—
683C2×C4#52512yes—
684C2×C4C2×AGL(1,ℤ/8)12yes—
685C2×C4C2×AGL(1,ℤ/8)12yes—
686C2×C4C2×AGL(1,ℤ/8)12yes—
687C2×C4#57512yes—
688C2×C4#57512yes—
689C2×C4#57512yes—
690C2×C4C23×Q811yescharacteristic
691C2×C4C22×#4112yes—
692C2×C4C22×#4111yescharacteristic
693C2×C4#58512yes—
694C8—44no—
695C8—44no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22C22×C22⋊Q811yescharacteristic
711C22C2×#52511yescharacteristic
712C22#344911yescharacteristic
713C22C22×AGL(1,ℤ/8)11yescharacteristic
714C22C2×#57511yescharacteristic
715C22C2×#57511yescharacteristic
716C22C2×#57511yescharacteristic
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—41no—
728C4—41no—
729C4—44no—
730C4—44no—
731C4—44no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4#307714yes—
737C4Q8×#4112yes—
738C4#348012yes—
739C2—42no—
740C2—84no—
741C2#3495211yescharacteristic
742C2C2×#307711yescharacteristic
743C2#6048411yescharacteristic
744C2#6054611yescharacteristic
745C2#6056111yescharacteristic
746C2#6096111yescharacteristic
747C2#6096511yescharacteristic
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. C4×#533C2semidirect
3. #29288C2semidirect
4. #29276C2semidirect
5. #29272C2semidirect
6. #29258C2semidirect
7. #29254C2semidirect
8. #29236C2semidirect
9. #28286C2semidirect
10. #27683C2semidirect
11. #24304C2semidirect
12. #23781C2semidirect
13. #23780C2semidirect
14. #21617C2semidirect
16. #21364C2semidirect
17. #20810C2semidirect
102. #532Q8semidirect
287. #125#41semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
C4×#5322
#292882
#292762
#292722
#292582
#292542
#292362
#282862
#276832
#243042
#237812
#237802
#216172
#213932
#213642
#208102