Finite Group Databasebeta

← #6800074

Subgroups — #6800074

Order 512.

1993 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic, complemented
3#32953C211yesmaximal, characteristic, complemented
4#32103C214yesmaximal, complemented
5#30899C214yesmaximal, complemented
6#29953C212yesmaximal, complemented
7#28414C212yesmaximal, complemented
8#27776C212yesmaximal, complemented
9#25864C212yesmaximal, complemented
10#25856C211yesmaximal, characteristic, complemented
11#25275C212yesmaximal, complemented
12#24727C212yesmaximal, complemented
13#24215C212yesmaximal
14#24209C211yesmaximal, characteristic, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22047C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C2×#554—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2212yes—
21C22×C4×D4C2211yescharacteristic
22C2×#489C2214yes—
23C2×#487C2212yescomplemented
24C2×#474C2212yes—
25C2×#463C2214yes—
26C2×#461C2212yescomplemented
27C2×#448C2212yes—
28C2×#447C2211yescharacteristic
29C2×C4×D8C2212yes—
30#2857C2211yescharacteristic
31#2857C2212yes—
32#2564—22no—
33#2556—22no—
34#2551C2211yescharacteristic, complemented
35#2547—21no—
36#2547—21no—
37#2547—22no—
38#2547C2214yes—
39#2547C2212yes—
40#2547C2212yes—
41#2542C2211yescharacteristic, complemented
42#2506C2212yescomplemented
43#2492—21no—
44#2452C2212yescomplemented
45#2417—21no—
46C2×#397—22no—
47#2365C2211yescharacteristic, complemented
48#2360—21no—
49D4×C4⋊C4C2211yescharacteristic, complemented
50C4×#121C2212yes—
51C4×#121C2211yescharacteristic
52C2×#391C2212yes—
53C2×#380—21no—
54C2×#380—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56#2095C2214yescomplemented
57#2068C2214yescomplemented
58#2068C2214yescomplemented
59#2043C2214yescomplemented
60#2043C2212yescomplemented
61#2042C2214yes—
62#2042C2214yes—
63#2042C2214yes—
64#2024C2214yescomplemented
65#2013C2212yescomplemented
66#2012C2214yes—
67#1983C2214yes—
68#1981C2212yescomplemented
69#1973C2214yescomplemented
70#1959C2214yes—
71#1956C2212yescomplemented
72#1939C2212yescomplemented
73#1935C2214yescomplemented
74#1928C2214yescomplemented
75#1917C2214yescomplemented
76#1879C2212yes—
77#1877C2214yescomplemented
78#1874C2211yescharacteristic
79#1862C2212yes—
80#1862C2214yes—
81#1852C2211yescharacteristic
82#1852C2212yes—
83#1802C2212yes—
84#1797C2212yes—
85#1797C2211yescharacteristic
86#1787C2212yes—
87#1787C2212yes—
88#1787C2212yes—
89#1786C2211yescharacteristic
90#1786C2212yes—
91#1770C2212yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2314yes—
99C22×D8C2312yes—
100#554—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#125—22no—
110C2×#124—22no—
111C2×#124—22no—
112C2×#124—48no—
113C2×#124C2314yes—
114C2×#124C2314yes—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—21no—
120C2×#121C2311yescharacteristic
121C2×C4×Q8—21no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—44no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4D411yescharacteristic
135C22×C4⋊C4D411yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—21no—
138C22×C22⋊C4—22no—
139C22×C22⋊C4—22no—
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#489—48no—
143#487—44no—
144#474—44no—
145#463—48no—
146#461—44no—
147#448—28no—
148#447—24no—
149C4×D8—44no—
150C2×#107C2312yes—
151C2×#107C2314yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×C22⋊C8C2311yescharacteristic
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2311yescharacteristic
170C2×#97C2314yes—
171C2×C4×C8C2312yes—
172#400—22no—
173#399—22no—
174#398—22no—
175#397—24no—
176#397—24no—
177#397—24no—
178#397—24no—
179#396—22no—
180#395—22no—
181#393—22no—
182#391—21no—
183#391—21no—
184#391—22no—
185#391—22no—
186#391—22no—
187#391C2314yes—
188#391C2314yes—
189#390C2311yescharacteristic
190#390C2312yes—
191#389—21no—
192#389—22no—
193#389—21no—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2311yescharacteristic
198#389C2312yes—
199#388—22no—
200#388—21no—
201#387—22no—
202#387—22no—
203#386—21no—
204#386—22no—
205#385—22no—
206#385—22no—
207#384C2311yescharacteristic
208#384C2312yes—
209#380—21no—
210#380—22no—
211#380—21no—
212#380—22no—
213#380—24no—
214#380—22no—
215#380—22no—
216#380—22no—
217#380—24no—
218#380C2312yes—
219#380C2311yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D412yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—21no—
229C4×C22⋊C4—22no—
230C2×#95—22no—
231C2×#95—22no—
232C2×#95D412yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238#336C2312yes—
239#336C2314yes—
240C25—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8C2414yes—
249C22×D4—21no—
250C22×D4—24no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—42no—
255C22×D4C2411yescharacteristic
256C22×D4C2412yes—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—44no—
264C23×C4—44no—
265C23×C4—44no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×SD16—28no—
272C2×SD16—44no—
273C2×SD16—44no—
274C2×D8—24no—
275C2×D8—24no—
276C2×D8—42no—
277C22×C8C2412yes—
278C22×C8C2414yes—
279C22×C8C2412yes—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286C4⋊Q8—44no—
287C4⋊Q8—44no—
288#125—48no—
289#124—28no—
290#124—28no—
291#124—44no—
292#124—44no—
293#124—48no—
294#124—48no—
295C22⋊Q8—44no—
296C22⋊Q8—44no—
297C22⋊Q8—48no—
298C22⋊Q8—48no—
299C22⋊Q8—44no—
300#121—22no—
301#121#4114yes—
302C4×Q8—44no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—42no—
307C4×D4—42no—
308C4×D4—44no—
309C4×D4—44no—
310C4×D4—44no—
311C4×D4—48no—
312C4×D4—44no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—44no—
363C2×C22⋊C4—48no—
364C2×C22⋊C4—48no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2412yes—
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377#107—42no—
378#107—44no—
379C4⋊C8—24no—
380C4⋊C8—44no—
381#103—28no—
382#103—44no—
383#103—44no—
384#103—44no—
385#103—44no—
386#103—44no—
387#103—44no—
388#102—28no—
389#102—24no—
390#102—24no—
391#102—42no—
392#102—42no—
393#102—44no—
394C22⋊C8—22no—
395C22⋊C8—24no—
396C22⋊C8—22no—
397#97—44no—
398C4×C8—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—21no—
405#95—21no—
406#95—22no—
407#95—21no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95C2×D412yes—
413#95C2×D412yes—
414#95C2×D412yes—
415#95C2×D412yes—
416C24—24no—
417C24—21no—
418C24—22no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—42no—
425C24C22×D411yescharacteristic
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—42no—
436C2×Q8—44no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—82no—
440C2×Q8—82no—
441C2×Q8—88no—
442C2×Q8—88no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—48no—
454C2×D4—48no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4C2×#4112yes—
460C2×D4H(5,𝔽2)14yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—41no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—82no—
498C22×C4—82no—
499C22×C4—816no—
500C22×C4AGL(1,ℤ/8)14yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C2×#4112yes—
511C22×C42−1+412yes—
512C22×C4C2511yescharacteristic
513SD16—88no—
514D8—48no—
515C2×C8—22no—
516C2×C8—24no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—22no—
523C2×C8—22no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4—81no—
552C4⋊C4—82no—
553C4⋊C4C2×#4112yes—
554C22⋊C4—24no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—42no—
564C22⋊C4—44no—
565C22⋊C4—42no—
566C22⋊C4—42no—
567C22⋊C4—81no—
568C22⋊C4—82no—
569C42—22no—
570C42—22no—
571C42—24no—
572C42—24no—
573C42—22no—
574C42—42no—
575C42—42no—
576C42—44no—
577C42—42no—
578C42H(5,𝔽2)14yes—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—24no—
585C23—48no—
586C23—48no—
587C23—48no—
588C23—41no—
589C23—42no—
590C23—42no—
591C23—41no—
592C23—42no—
593C23—41no—
594C23—41no—
595C23—42no—
596C23—41no—
597C23—42no—
598C23—41no—
599C23—42no—
600C23—84no—
601C23—84no—
602C23—84no—
603C23—84no—
604C23#54812yes—
605C23C2×AGL(1,ℤ/8)12yes—
606C23#57812yes—
607C23C23×D411yescharacteristic
608Q8—44no—
609Q8—44no—
610Q8—44no—
611Q8—44no—
612Q8—88no—
613Q8—84no—
614Q8—164no—
615D4—24no—
616D4—24no—
617D4—24no—
618D4—48no—
619D4—48no—
620D4—42no—
621D4—42no—
622D4—42no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—21no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—88no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4—88no—
674C2×C4—84no—
675C2×C4—88no—
676C2×C4—164no—
677C2×C4#53512yes—
678C2×C4#53512yes—
679C2×C4#54812yes—
680C2×C4C2×AGL(1,ℤ/8)12yes—
681C2×C4C2×AGL(1,ℤ/8)12yes—
682C2×C4#57512yes—
683C2×C4C22×#4111yescharacteristic
684C2×C4C2×H(5,𝔽2)12yes—
685C2×C4C2×H(5,𝔽2)12yes—
686C2×C4C2×2−1+411yescharacteristic
687C8—42no—
688C8—44no—
689C8—42no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—42no—
697C22—42no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—42no—
702C22—816no—
703C22—82no—
704C22—81no—
705C22—82no—
706C22—82no—
707C22—81no—
708C22—82no—
709C22—81no—
710C22—81no—
711C22#336714yes—
712C22C2×#53511yescharacteristic
713C22C2×#54811yescharacteristic
714C22#348411yescharacteristic
715C22C22×AGL(1,ℤ/8)11yescharacteristic
716C22C2×#57511yescharacteristic
717C22C2×#57811yescharacteristic
718C22#358611yescharacteristic
719C4—22no—
720C4—22no—
721C4—22no—
722C4—42no—
723C4—41no—
724C4—42no—
725C4—42no—
726C4—42no—
727C4—42no—
728C4—81no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—81no—
733C4—84no—
734C4#321314yes—
735C4#351912yes—
736C4#353612yes—
737C2—24no—
738C2—82no—
739C2—84no—
740C2#3912011yescharacteristic
741C2C2×#321311yescharacteristic
742C2#6081411yescharacteristic
743C2C2×#336711yescharacteristic
744C2#6132311yescharacteristic
745C2#6158511yescharacteristic
746C2#6159511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36561C2semidirect
2. C2×#2547C2semidirect
3. #32953C2semidirect
4. #32103C2semidirect
5. #30899C2semidirect
6. #29953C2semidirect
7. #28414C2semidirect
8. #27776C2semidirect
9. #25864C2semidirect
10. #25856C2semidirect
11. #25275C2semidirect
12. #24727C2semidirect
14. #24209C2semidirect
15. #22629C2semidirect
16. #22047C2semidirect
23. C2×#487C22semidirect
26. C2×#461C22semidirect
34. #2551C22semidirect
41. #2542C22semidirect
42. #2506C22semidirect
44. #2452C22semidirect
47. #2365C22semidirect
49. D4×C4⋊C4C22semidirect
56. #2095C22semidirect
57. #2068C22semidirect
58. #2068C22semidirect
59. #2043C22semidirect
60. #2043C22semidirect
64. #2024C22semidirect
65. #2013C22semidirect
68. #1981C22semidirect
69. #1973C22semidirect
71. #1956C22semidirect
72. #1939C22semidirect
73. #1935C22semidirect
74. #1928C22semidirect
75. #1917C22semidirect
77. #1877C22semidirect
91. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#25472
#329532
#321032
#308992
#299532
#284142
#277762
#258642
#258562
#252752
#247272
#242152
#242092
#226292
#220472
C4×#4472