Finite Group Databasebeta

← #6660946

Subgroups — #6660946

Order 512.

2090 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2854C211yesmaximal, characteristic
3#32739C214yesmaximal, complemented
4#30365C214yesmaximal, complemented
5#29967C212yesmaximal, complemented
6#29950C212yesmaximal, complemented
7#29906C212yesmaximal, complemented
8#29639C212yesmaximal, complemented
9#29301C212yesmaximal, complemented
10#29109C212yesmaximal, complemented
11#27603C212yesmaximal, complemented
12#24607C214yesmaximal, complemented
13#24005C212yesmaximal
14#20715C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic, complemented
16C22×#123—21no—
17C22×C22⋊Q8—21no—
18C22×#121C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#481C2214yes—
21C2×#432C2212yescomplemented
22#2857—21no—
23#2857—21no—
24#2857—21no—
25#2857—21no—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yescomplemented
29#2857C2212yescomplemented
30#2854—21no—
31#2854—21no—
32#2854—21no—
33#2854—21no—
34#2854C2212yes—
35#2854C2212yes—
36#2854C2212yes—
37#2854C2212yes—
38C2×#391—21no—
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2212yescomplemented
45C2×#388—21no—
46C2×#388—21no—
47C2×#386—21no—
48C2×#384C2211yescharacteristic, complemented
49C2×#380—21no—
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52#2099C2214yescomplemented
53#2093C2214yescomplemented
54#2088C2218yescomplemented
55#2083C2214yescomplemented
56#2076C2218yescomplemented
57#2071C2214yescomplemented
58#2067C2218yescomplemented
59#2058C2214yescomplemented
60#2058C2214yescomplemented
61#2034C2218yes—
62#2023C2214yescomplemented
63#2023C2214yescomplemented
64#1943C2218yescomplemented
65#1936C2218yescomplemented
66#1874C2214yes—
67#1853C2214yes—
68#1852C2214yes—
69#1852C2214yescomplemented
70#1824C2214yes—
71#1824C2214yescomplemented
72#1817C2212yescomplemented
73#1797C2212yes—
74#1797C2212yescomplemented
75#1786C2214yes—
76#1779C2212yes—
77#1749C2212yes—
78#1749C2212yescomplemented
79C23×Q8—21no—
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4D411yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—44no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#123—22no—
91C2×#123—22no—
92C2×#123—22no—
93C2×#123—22no—
94C2×#123—48no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—48no—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121—44no—
103C2×#121C2312yes—
104C2×#121C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—22no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C42—44no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#481—48no—
137#432—44no—
138C2×#107C2318yes—
139C2×#106C2318yes—
140C2×C4⋊C8C2314yes—
141C2×#102C2318yes—
142C2×#102C2318yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#389—21no—
160#389—21no—
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#389C2312yes—
171#389C2314yes—
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
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#380C2314yes—
202#380C2312yes—
203C2×#95—22no—
204C2×#95—21no—
205C2×#95—22no—
206C2×#95—21no—
207C2×#95—21no—
208#340C2318yes—
209#340C2318yes—
210#340C2318yes—
211#340C2318yes—
212#336C2314yes—
213#336C2314yes—
214C25—21no—
215C25C2×D411yescharacteristic
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—44no—
219C22×Q8—44no—
220C22×Q8—44no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—21no—
224C22×D4—24no—
225C22×D4—42no—
226C22×D4—42no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4C2412yes—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—21no—
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
263C22×C8C2418yes—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268#123—48no—
269#123—48no—
270#123—48no—
271#123—48no—
272C22⋊Q8—48no—
273C22⋊Q8—48no—
274C22⋊Q8—48no—
275C22⋊Q8—48no—
276#121—28no—
277#121—44no—
278#121—44no—
279#121—44no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
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—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2412yes—
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—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—48no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42—42no—
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
362#107—48no—
363#106—48no—
364C4⋊C8—44no—
365#102—48no—
366#102—48no—
367C22⋊C8—28no—
368C22⋊C8—28no—
369C22⋊C8—44no—
370#97—44no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—41no—
387C24—41no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24#13714yes—
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398C24C22×D411yescharacteristic
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—816no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—88no—
419C2×D4—816no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
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—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—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—416no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—44no—
492C22×C4—42no—
493C22×C4AGL(1,ℤ/8)14yes—
494C22×C4AGL(1,ℤ/8)14yes—
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×C4H(5,𝔽2)14yes—
507C22×C42−1+414yes—
508C22×C4C2511yescharacteristic
509C2×C8—28no—
510C2×C8—28no—
511C2×C8—28no—
512C4⋊C4—28no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—28no—
516C4⋊C4—24no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—48no—
539C22⋊C4—44no—
540C22⋊C4—48no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—28no—
575C23—21no—
576C23—28no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23—84no—
597C23—84no—
598C23—84no—
599C23#53412yes—
600C23#53612yes—
601C23C2×AGL(1,ℤ/8)12yes—
602C23C2×AGL(1,ℤ/8)12yes—
603C23C2×#13712yes—
604C23C2×#13712yes—
605C23C2×#13712yes—
606C23C23×D411yescharacteristic
607Q8—88no—
608Q8—88no—
609D4—48no—
610D4—84no—
611D4—88no—
612D4—88no—
613C2×C4—28no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—28no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—28no—
630C2×C4—28no—
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—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
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—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—88no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4—88no—
677C2×C4#53412yes—
678C2×C4#53612yes—
679C2×C4C2×AGL(1,ℤ/8)12yes—
680C2×C4C2×AGL(1,ℤ/8)12yes—
681C2×C4C2×AGL(1,ℤ/8)12yes—
682C2×C4C2×AGL(1,ℤ/8)12yes—
683C2×C4#57512yes—
684C2×C4C2×H(5,𝔽2)12yes—
685C2×C4C2×2−1+412yes—
686C2×C4#58512yes—
687C8—48no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—416no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—168no—
716C22#319314yes—
717C22#323914yes—
718C22C2×#53411yescharacteristic
719C22C2×#53611yescharacteristic
720C22#344911yescharacteristic
721C22C22×AGL(1,ℤ/8)11yescharacteristic
722C22C22×AGL(1,ℤ/8)11yescharacteristic
723C22C22×#13711yescharacteristic
724C22C2×#57511yescharacteristic
725C4—22no—
726C4—22no—
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—82no—
737C4—84no—
738C4—82no—
739C4—84no—
740C4—82no—
741C4#353514yes—
742C2—24no—
743C2—24no—
744C2—42no—
745C2—162no—
746C2#3744111yescharacteristic
747C2C2×#319311yescharacteristic
748C2#6075011yescharacteristic
749C2C2×#323911yescharacteristic
750C2#6086411yescharacteristic
751C2#6096011yescharacteristic
752C2#6096811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
1. C2×#2857C2semidirect
3. #32739C2semidirect
4. #30365C2semidirect
5. #29967C2semidirect
6. #29950C2semidirect
7. #29906C2semidirect
8. #29639C2semidirect
9. #29301C2semidirect
10. #29109C2semidirect
11. #27603C2semidirect
12. #24607C2semidirect
14. #20715C2semidirect
15. C22×C4⋊Q8C22semidirect
21. C2×#432C22semidirect
28. #2857C22semidirect
29. #2857C22semidirect
44. C2×#389C22semidirect
48. C2×#384C22semidirect
52. #2099C22semidirect
53. #2093C22semidirect
54. #2088C22semidirect
55. #2083C22semidirect
56. #2076C22semidirect
57. #2071C22semidirect
58. #2067C22semidirect
59. #2058C22semidirect
60. #2058C22semidirect
62. #2023C22semidirect
63. #2023C22semidirect
64. #1943C22semidirect
65. #1936C22semidirect
69. #1852C22semidirect
71. #1824C22semidirect
72. #1817C22semidirect
74. #1797C22semidirect
78. #1749C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#28542
#327392
#303652
#299672
#299502
#299062
#296392
#293012
#291092
#276032
#246072
#240052
#207152