Finite Group Databasebeta

← #6661061

Subgroups — #6661061

Order 512.

2104 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic, complemented
3#32739C214yesmaximal, complemented
4#30298C212yesmaximal, complemented
5#30197C212yesmaximal, complemented
6#29968C212yesmaximal, complemented
7#29939C212yesmaximal, complemented
8#29754C212yesmaximal, complemented
9#29690C212yesmaximal, complemented
10#27644C212yesmaximal, complemented
11#24843C214yesmaximal, complemented
12#24558C214yesmaximal, complemented
13#24299C212yesmaximal, complemented
14#20626C211yesmaximal, characteristic, complemented
15C2×#532C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#525—21no—
18C22×C22⋊Q8—21no—
19C23×C42C2211yescharacteristic
20C2×#480C2214yes—
21C2×#476C2214yescomplemented
22C2×#423C2212yes—
23C2×#407C2212yes—
24#2857—21no—
25#2857—21no—
26#2857—21no—
27#2857—21no—
28#2857C2212yes—
29#2857C2212yes—
30#2857C2212yes—
31#2857C2212yes—
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2856C2211yescharacteristic
36#2854—21no—
37#2854C2212yes—
38#2850—21no—
39#2850—21no—
40#2847—21no—
41#2847C2212yes—
42C4×#121—21no—
43C4×#121—21no—
44C4×#121C2212yes—
45C4×#121C2212yes—
46#2297—21no—
47C2×#391—21no—
48C2×#389—21no—
49C2×#389C2212yes—
50#2285C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53C2×#380C2212yes—
54#2100C2214yescomplemented
55#2088C2218yescomplemented
56#2083C2214yescomplemented
57#2083C2214yescomplemented
58#2076C2218yes—
59#2071C2214yescomplemented
60#2067C2218yescomplemented
61#2058C2214yescomplemented
62#2058C2214yescomplemented
63#2041C2214yescomplemented
64#2034C2218yes—
65#2023C2214yescomplemented
66#1987C2214yescomplemented
67#1947C2214yescomplemented
68#1895C2218yes—
69#1874C2214yes—
70#1853C2214yes—
71#1823C2214yes—
72#1809C2218yes—
73#1802C2214yescomplemented
74#1797C2212yes—
75#1797C2212yes—
76#1797C2212yes—
77#1797C2212yes—
78#1728C2212yes—
79C22×#41—21no—
80C23×Q8—21no—
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C22×M16C2314yes—
85#532—44no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—44no—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2312yes—
92C2×#127C2311yescharacteristic
93C2×#124—21no—
94#525—48no—
95C2×#123—22no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—48no—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121—21no—
108C2×#121C2312yes—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#120—22no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×#117—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4—22no—
124C22×C42—44no—
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#480—48no—
138#476—48no—
139C2×#107C2318yes—
140C2×#106C2318yes—
141#423—44no—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×#102C2318yes—
145C2×#102C2318yes—
146#407—44no—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149#391—21no—
150#391—21no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391—22no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390—21no—
159#390C2312yes—
160#390C2312yes—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389C2312yes—
170#389C2314yes—
171#389C2314yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#384—21no—
184#384—21no—
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382—21no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2314yes—
201#380C2312yes—
202#380C2314yes—
203#380C2312yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C2×#95—22no—
209C2×#95—22no—
210#340C2318yes—
211#340C2318yes—
212#340C2318yes—
213#340C2318yes—
214#336C2314yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C2×#41—44no—
218C2×#41—44no—
219C2×#41—44no—
220C22×Q8—21no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2412yes—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×M16—28no—
264C2×M16—44no—
265C2×M16—44no—
266C22×C8C2418yes—
267C4⋊Q8—28no—
268C4⋊Q8—44no—
269C4⋊Q8—42no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272#127—42no—
273#124—44no—
274#123—48no—
275C22⋊Q8—48no—
276C22⋊Q8—48no—
277C22⋊Q8—48no—
278C22⋊Q8—48no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281#121—24no—
282#121—28no—
283#121—24no—
284#121—44no—
285#121—44no—
286#121—44no—
287#121—44no—
288#121—44no—
289#121—44no—
290#120—48no—
291C4×D4—28no—
292C4×D4—44no—
293C4×D4—48no—
294C4×D4—48no—
295#117—44no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—48no—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
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—48no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42—42no—
346C2×C42—42no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42—42no—
351C2×C42—42no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—48no—
369#106—48no—
370C4⋊C8—28no—
371C4⋊C8—44no—
372#102—48no—
373#102—48no—
374C22⋊C8—28no—
375C22⋊C8—44no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—41no—
393C24—41no—
394C24C22×D411yescharacteristic
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397#41—816no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—816no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—22no—
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—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
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—21no—
437C22×C4—22no—
438C22×C4—21no—
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—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—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
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—416no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—416no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4AGL(1,ℤ/8)14yes—
497C22×C4AGL(1,ℤ/8)14yes—
498C22×C4#13714yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4H(5,𝔽2)12yes—
511C22×C42−1+414yes—
512C22×C42−1+412yes—
513C22×C4C2511yescharacteristic
514M16—88no—
515C2×C8—28no—
516C2×C8—28no—
517C2×C8—28no—
518C4⋊C4—24no—
519C4⋊C4—28no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—44no—
571C42H(5,𝔽2)14yes—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—28no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23#53612yes—
603C23#53712yes—
604C23C2×AGL(1,ℤ/8)12yes—
605C23C2×AGL(1,ℤ/8)12yes—
606C23C2×#13712yes—
607C23C23×D411yescharacteristic
608Q8—88no—
609Q8—88no—
610Q8—84no—
611D4—44no—
612D4—44no—
613D4—84no—
614D4—84no—
615D4—88no—
616D4—88no—
617D4—88no—
618C2×C4—28no—
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—28no—
636C2×C4—28no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—88no—
677C2×C4—88no—
678C2×C4—88no—
679C2×C4#53412yes—
680C2×C4#53412yes—
681C2×C4#53612yes—
682C2×C4#53712yes—
683C2×C4C2×AGL(1,ℤ/8)12yes—
684C2×C4C2×AGL(1,ℤ/8)12yes—
685C2×C4C2×AGL(1,ℤ/8)12yes—
686C2×C4C2×AGL(1,ℤ/8)12yes—
687C2×C4C2×#13712yes—
688C2×C4C2×#13712yes—
689C2×C4#57512yes—
690C2×C4C2×H(5,𝔽2)11yescharacteristic
691C2×C4C2×H(5,𝔽2)12yes—
692C2×C4C2×2−1+412yes—
693C2×C4C2×2−1+411yescharacteristic
694C8—48no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—416no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22#325314yes—
724C22C2×#53411yescharacteristic
725C22C2×#53611yescharacteristic
726C22C2×#53711yescharacteristic
727C22C22×AGL(1,ℤ/8)11yescharacteristic
728C22C22×AGL(1,ℤ/8)11yescharacteristic
729C22C22×#13711yescharacteristic
730C22C2×#57511yescharacteristic
731C4—22no—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—41no—
740C4—41no—
741C4—82no—
742C4—84no—
743C4—82no—
744C4—84no—
745C4—82no—
746C4#319314yes—
747C4#353612yes—
748C4#353812yes—
749C2—24no—
750C2—42no—
751C2—42no—
752C2—162no—
753C2#3744211yescharacteristic
754C2C2×#319311yescharacteristic
755C2#6075011yescharacteristic
756C2#6086011yescharacteristic
757C2#6086411yescharacteristic
758C2C2×#325311yescharacteristic
759C2#6091811yescharacteristic
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×#2857C2semidirect
3. #32739C2semidirect
4. #30298C2semidirect
5. #30197C2semidirect
6. #29968C2semidirect
7. #29939C2semidirect
8. #29754C2semidirect
9. #29690C2semidirect
10. #27644C2semidirect
11. #24843C2semidirect
12. #24558C2semidirect
13. #24299C2semidirect
14. #20626C2semidirect
21. C2×#476C22semidirect
54. #2100C22semidirect
55. #2088C22semidirect
56. #2083C22semidirect
57. #2083C22semidirect
59. #2071C22semidirect
60. #2067C22semidirect
61. #2058C22semidirect
62. #2058C22semidirect
63. #2041C22semidirect
65. #2023C22semidirect
66. #1987C22semidirect
67. #1947C22semidirect
73. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#28572
#327392
#302982
#301972
#299682
#299392
#297542
#296902
#276442
#248432
#245582
#242992
#206262