Finite Group Databasebeta

← #6434455

Subgroups — #6434455

Order 512.

2536 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic, complemented
3#30126C214yesmaximal, complemented
4#30076C212yesmaximal, complemented
5#30038C212yesmaximal, complemented
6#29952C212yesmaximal, complemented
7#29907C212yesmaximal, complemented
8#29712C212yesmaximal, complemented
9#29622C212yesmaximal, complemented
10#23822C212yesmaximal, complemented
11#23802C212yesmaximal, complemented
12#23682C214yesmaximal, complemented
13#21452C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C22×#127C2211yescharacteristic
16C2×#529—21no—
17C22×C4×D4—21no—
18C22×C4×D4—21no—
19C22×C4×D4C2212yes—
20C2×#447C2214yes—
21C2×#442C2218yescomplemented
22#2861—21no—
23#2860—21no—
24#2860C2212yescomplemented
25#2859C2211yescharacteristic, complemented
26#2858C2211yescharacteristic, complemented
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2854—21no—
30#2854C2212yes—
31#2850—21no—
32C4×C4⋊Q8C2211yescharacteristic, complemented
33C4×#127—21no—
34C4×#127—21no—
35C4×#127—21no—
36C4×#127—21no—
37C4×#127C2212yes—
38C4×#127C2212yes—
39C4×#127C2212yes—
40C4×#127C2212yes—
41C4×#127C2211yescharacteristic, complemented
42C4×#124—21no—
43C4×#121—21no—
44C4×#121C2212yes—
45#2293—21no—
46#2293C2212yescomplemented
47C2×#390—21no—
48C2×#390—21no—
49C2×#390C2212yes—
50C2×#384C2211yescharacteristic
51C2×C43C2211yescharacteristic
52#2100C2214yescomplemented
53#2094C2214yescomplemented
54#2094C2214yescomplemented
55#2084C2214yescomplemented
56#2071C2214yescomplemented
57#2071C2214yescomplemented
58#2065C2218yescomplemented
59#2054C2218yescomplemented
60#2041C2214yes—
61#2041C2214yes—
62#2023C2214yescomplemented
63#2011C2214yes—
64#2011C2214yes—
65#2011C2214yescomplemented
66#1879C2218yescomplemented
67#1874C2214yes—
68#1864C2218yescomplemented
69#1853C2214yes—
70#1839C2212yescomplemented
71#1831C2212yescomplemented
72#1817C2212yes—
73#1797C2212yes—
74#1786C2214yes—
75#1767C2212yescomplemented
76#1762C2218yescomplemented
77#1749C2212yes—
78#1745C2212yescomplemented
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2312yes—
82C24×C4—22no—
83C22×D8C2318yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127—28no—
86C2×#127—21no—
87C2×#127—21no—
88C2×#127C2312yes—
89C2×#127C2312yes—
90C2×#127C2311yescharacteristic
91#529—48no—
92C2×#126—22no—
93C2×#125—21no—
94C2×#124—21no—
95C2×#124—21no—
96C2×#123—22no—
97C2×C22⋊Q8—21no—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121C2312yes—
101C2×C4×Q8—21no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—48no—
116C2×C4×D4—48no—
117C2×C4×D4—48no—
118C2×C4×D4C2314yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2314yes—
121C2×#117—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4—22no—
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#447—216no—
136#442—416no—
137C2×#106C2318yes—
138C2×C4⋊C8C2314yes—
139C2×#102C2318yes—
140C2×#102C2318yes—
141C2×#102C2318yes—
142C2×#102C2318yes—
143C2×C22⋊C8C2314yes—
144C2×#97C2314yes—
145#391—21no—
146#391—21no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390C2314yes—
164#390C2314yes—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#386—22no—
186#386—22no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#382—21no—
194#380—21no—
195#380C2312yes—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208#340C2318yes—
209#340C2318yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214C25—22no—
215C22×Q8—21no—
216C22×D4—28no—
217C22×D4—28no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—24no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4#4118yes—
234C22×D4C2412yes—
235C22×D4C2414yes—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—48no—
241C23×C4—48no—
242C23×C4—48no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×D8—216no—
251C2×D8—48no—
252C2×D8—48no—
253C22×C8C2418yes—
254C4⋊Q8—42no—
255#127—28no—
256#127—28no—
257#127—28no—
258#127—28no—
259#127—42no—
260#126—48no—
261#125—44no—
262#124—44no—
263#124—44no—
264#123—48no—
265C22⋊Q8—44no—
266#121—28no—
267#121—44no—
268#121—44no—
269C4×Q8—44no—
270C4×D4—216no—
271C4×D4—28no—
272C4×D4—48no—
273C4×D4—44no—
274C4×D4—48no—
275C4×D4—48no—
276C4×D4—48no—
277C4×D4—48no—
278C4×D4—48no—
279C4×D4—48no—
280C4×D4—48no—
281C4×D4—48no—
282C4×D4—48no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—48no—
286#117—48no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
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—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4—48no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
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×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#106—48no—
365C4⋊C8—44no—
366#102—216no—
367#102—48no—
368#102—48no—
369#102—48no—
370C22⋊C8—28no—
371#97—28no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378C24—28no—
379C24—24no—
380C24—22no—
381C24—44no—
382C24—44no—
383C24—44no—
384C24—44no—
385C24—44no—
386C24—44no—
387C24C22×D411yescharacteristic
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—416no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—432no—
408C2×D4—432no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—416no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4C2×#4118yes—
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—24no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—832no—
461C22×C4AGL(1,ℤ/8)14yes—
462C22×C4AGL(1,ℤ/8)14yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4114yes—
477C22×C4C2511yescharacteristic
478D8—432no—
479C2×C8—28no—
480C2×C8—28no—
481C2×C8—28no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—28no—
485C4⋊C4—24no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—44no—
544C42AGL(1,ℤ/8)14yes—
545C42H(5,𝔽2)14yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—28no—
551C23—416no—
552C23—416no—
553C23—416no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—88no—
568C23—88no—
569C23—88no—
570C23—88no—
571C23#52512yes—
572C23C2×AGL(1,ℤ/8)12yes—
573C23C2×AGL(1,ℤ/8)12yes—
574C23C23×D411yescharacteristic
575Q8—84no—
576D4—216no—
577D4—416no—
578D4—416no—
579D4—416no—
580D4—416no—
581D4—416no—
582D4—48no—
583D4—48no—
584D4—84no—
585D4—84no—
586D4—88no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
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—28no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—88no—
635C2×C4—816no—
636C2×C4—816no—
637C2×C4—88no—
638C2×C4#52512yes—
639C2×C4#53512yes—
640C2×C4#53512yes—
641C2×C4C2×AGL(1,ℤ/8)12yes—
642C2×C4C2×AGL(1,ℤ/8)12yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4C2×AGL(1,ℤ/8)12yes—
647C2×C4C2×AGL(1,ℤ/8)12yes—
648C2×C4#57512yes—
649C2×C4C22×#4112yes—
650C2×C4C2×H(5,𝔽2)12yes—
651C2×C4#58512yes—
652C8—48no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—48no—
660C22—48no—
661C22—48no—
662C22—832no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22—84no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22#310914yes—
675C22C2×#52511yescharacteristic
676C22C2×#53511yescharacteristic
677C22#344911yescharacteristic
678C22C22×AGL(1,ℤ/8)11yescharacteristic
679C22C22×AGL(1,ℤ/8)11yescharacteristic
680C22C22×AGL(1,ℤ/8)11yescharacteristic
681C22C2×#57511yescharacteristic
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—41no—
694C4—41no—
695C4—82no—
696C4—82no—
697C4—84no—
698C4—82no—
699C4#345914yes—
700C2—24no—
701C2—88no—
702C2—162no—
703C2#3459311yescharacteristic
704C2C2×#310911yescharacteristic
705C2#6055811yescharacteristic
706C2#6079811yescharacteristic
707C2#6080311yescharacteristic
708C2#6095811yescharacteristic
709C2#6096811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #36631C2semidirect
2. C2×C4×#127C2semidirect
3. #30126C2semidirect
4. #30076C2semidirect
5. #30038C2semidirect
6. #29952C2semidirect
7. #29907C2semidirect
8. #29712C2semidirect
9. #29622C2semidirect
10. #23822C2semidirect
11. #23802C2semidirect
12. #23682C2semidirect
13. #21452C2semidirect
14. #20778C2semidirect
21. C2×#442C22semidirect
24. #2860C22semidirect
25. #2859C22semidirect
26. #2858C22semidirect
32. C4×C4⋊Q8C22semidirect
41. C4×#127C22semidirect
46. #2293C22semidirect
52. #2100C22semidirect
53. #2094C22semidirect
54. #2094C22semidirect
55. #2084C22semidirect
56. #2071C22semidirect
57. #2071C22semidirect
58. #2065C22semidirect
59. #2054C22semidirect
62. #2023C22semidirect
65. #2011C22semidirect
66. #1879C22semidirect
68. #1864C22semidirect
70. #1839C22semidirect
71. #1831C22semidirect
75. #1767C22semidirect
76. #1762C22semidirect
78. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×C4×#1272
#301262
#300762
#300382
#299522
#299072
#297122
#296222
#238222
#238022
#236822
#214522
#207782