Finite Group Databasebeta

← #6468985

Subgroups — #6468985

Order 512.

2008 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#29370C212yesmaximal, complemented
4#29366C212yesmaximal, complemented
5#29262C212yesmaximal, complemented
6#28164C212yesmaximal, complemented
7#27685C212yesmaximal, complemented
8#27587C212yesmaximal, complemented
9#24216C212yesmaximal
10#23935C212yesmaximal, complemented
11#23823C211yesmaximal, characteristic, complemented
12#23822C211yesmaximal, characteristic, complemented
13#23805C211yesmaximal, characteristic, complemented
14#23803C211yesmaximal, characteristic, complemented
15#21508C214yesmaximal, complemented
16#21457C214yesmaximal, complemented
17#20778C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×C4×#41C2212yes—
21C2×#464C2214yes—
22C2×#452C2214yes—
23C2×#441C2214yes—
24C2×#440C2214yes—
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×C4⋊Q8C2211yescharacteristic
30C4×C4⋊Q8C2211yescharacteristic
31C4×#127C2211yescharacteristic
32C4×#127C2211yescharacteristic
33C4×#125—21no—
34C4×#125—21no—
35C4×#124C2212yes—
36C4×#124C2212yes—
37C4×C22⋊Q8—21no—
38C4×C22⋊Q8—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C4×#121—21no—
42C4×#121—21no—
43C4×#121—21no—
44C4×#121—21no—
45#2297C2212yes—
46#2296—21no—
47#2295—21no—
48#2295C2212yes—
49#2293—21no—
50#2293—21no—
51#2285C2211yescharacteristic
52#2285C2211yescharacteristic
53C42×Q8—21no—
54C42×Q8C2212yes—
55C42×D4—21no—
56C42×D4—21no—
57C4×#117—21no—
58C2×C43C2211yescharacteristic
59#2100C2212yes—
60#2099C2212yes—
61#2096C2212yes—
62#2096C2212yes—
63#2095C2212yes—
64#2093C2212yes—
65#2072C2212yes—
66#2071C2212yes—
67#2043C2212yes—
68#2041C2212yes—
69#2024C2212yes—
70#2023C2212yes—
71#2016C2212yes—
72#2016C2212yes—
73#2014C2212yes—
74#2013C2212yes—
75#1983C2214yes—
76#1949C2214yes—
77#1928C2214yes—
78#1926C2214yes—
79#1880C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1877C2214yes—
83#1873C2214yes—
84#1864C2214yes—
85#1863C2214yes—
86#1854C2214yes—
87#1839C2212yes—
88#1831C2212yes—
89#1817C2212yes—
90#1805C2214yes—
91#1797C2212yes—
92#1788C2214yes—
93#1767C2212yes—
94#1763C2214yes—
95#1762C2214yes—
96#1749C2212yes—
97#1745C2211yescharacteristic
98#1745C2211yescharacteristic
99C22×#41C2312yes—
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102#533—28no—
103#532—28no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#125—21no—
109C2×#124C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2312yes—
112C2×#121—21no—
113C2×#121—21no—
114C4×#41—48no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117—21no—
135C2×#117—22no—
136C2×#117C2312yes—
137C22×C42—22no—
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42D411yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#464—216no—
146#452—216no—
147#441—48no—
148#440—48no—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2314yes—
161C2×#97C2312yes—
162C2×#97C2312yes—
163#391—21no—
164#391—21no—
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390C2312yes—
174#390C2312yes—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389C2312yes—
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#386—22no—
196#386—22no—
197#385—22no—
198#385—22no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382—21no—
204#382C2312yes—
205#380—21no—
206#380—21no—
207#380—21no—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239#376—22no—
240C43—22no—
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43D411yescharacteristic
244C43D411yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247C43C2311yescharacteristic
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#336C2312yes—
254#336C2312yes—
255#336C2312yes—
256#336C2312yes—
257#336C2314yes—
258#336C2314yes—
259C2×#41—24no—
260C2×#41—24no—
261C2×#41—28no—
262C2×#41—44no—
263C22×Q8—21no—
264C22×Q8C2412yes—
265C22×Q8C2412yes—
266C22×D4—22no—
267C22×D4—21no—
268C22×D4—21no—
269C22×D4C2412yes—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2411yescharacteristic
280C2×Q16—28no—
281C2×Q16—28no—
282C2×Q16—44no—
283C2×SD16—28no—
284C2×SD16—28no—
285C2×SD16—44no—
286C22×C8C2414yes—
287C22×C8C2414yes—
288C4⋊Q8—24no—
289C4⋊Q8—24no—
290C4⋊Q8—24no—
291#127—24no—
292#125—28no—
293#124—24no—
294#124#4118yes—
295C22⋊Q8—28no—
296C22⋊Q8—24no—
297C22⋊Q8#4118yes—
298#121—28no—
299#121—28no—
300C4×Q8—28no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—28no—
304C4×Q8—28no—
305C4×Q8—28no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×D4—28no—
310C4×D4—28no—
311C4×D4—28no—
312C4×D4—44no—
313C4×D4—44no—
314C4×D4—44no—
315C4×D4—44no—
316C4×D4—48no—
317C4×D4—48no—
318C4×D4—44no—
319#117—28no—
320#117—48no—
321#117—44no—
322#117—44no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—42no—
374C2×C42—42no—
375C2×C42—42no—
376C2×C42—48no—
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×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405#107—44no—
406#106—44no—
407C4⋊C8—28no—
408#103—28no—
409#103—28no—
410#103—44no—
411#103—44no—
412#103—44no—
413#102—28no—
414#102—28no—
415#102—44no—
416C22⋊C8—28no—
417#97—24no—
418#97—24no—
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—
432#95—22no—
433#95—22no—
434#95—22no—
435C24—22no—
436C24C22×D411yescharacteristic
437#41—416no—
438#41—416no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8C2×#4114yes—
449C2×Q8C2×#4114yes—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4—28no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4C2×#4114yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—28no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
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×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C2×#4114yes—
513C22×C4C2511yescharacteristic
514Q16—416no—
515SD16—416no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—28no—
527C4⋊C4—24no—
528C4⋊C4—28no—
529C4⋊C4—22no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4C2×#4114yes—
549C22⋊C4—28no—
550C22⋊C4—28no—
551C22⋊C4—28no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C42—28no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42—44no—
587C42—44no—
588C42—44no—
589C42—44no—
590C42—44no—
591C42—44no—
592C42—44no—
593C42—44no—
594C42#13714yes—
595C42C2×#4114yes—
596C42H(5,𝔽2)14yes—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—24no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23#52512yes—
611C23C23×D411yescharacteristic
612Q8—28no—
613Q8—28no—
614Q8—48no—
615Q8—44no—
616Q8—44no—
617D4—28no—
618D4—48no—
619D4—48no—
620D4—416no—
621D4—44no—
622C2×C4—28no—
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—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—28no—
646C2×C4—21no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—416no—
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—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—88no—
680C2×C4—88no—
681C2×C4#52512yes—
682C2×C4#52512yes—
683C2×C4#52512yes—
684C2×C4#53512yes—
685C2×C4#53512yes—
686C2×C4C2×#13712yes—
687C2×C4C2×#13712yes—
688C2×C4C2×#13712yes—
689C2×C4#57512yes—
690C2×C4#57512yes—
691C2×C4#57512yes—
692C2×C4C22×#4112yes—
693C2×C4C22×#4112yes—
694C2×C4C2×H(5,𝔽2)12yes—
695C8—44no—
696C8—44no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22C2×#52511yescharacteristic
712C22C2×#52511yescharacteristic
713C22C2×#53511yescharacteristic
714C22C22×#13711yescharacteristic
715C22C2×#57511yescharacteristic
716C22C2×#57511yescharacteristic
717C22C2×#57511yescharacteristic
718C4—22no—
719C4—22no—
720C4—22no—
721C4—22no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—22no—
726C4—22no—
727C4—22no—
728C4—22no—
729C4—41no—
730C4—41no—
731C4—44no—
732C4—44no—
733C4—44no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4#348114yes—
739C2—42no—
740C2—84no—
741C2#3496311yescharacteristic
742C2#6054611yescharacteristic
743C2#6054611yescharacteristic
744C2#6056011yescharacteristic
745C2#6056211yescharacteristic
746C2#6080111yescharacteristic
747C2#6080411yescharacteristic
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. #29370C2semidirect
4. #29366C2semidirect
5. #29262C2semidirect
6. #28164C2semidirect
7. #27685C2semidirect
8. #27587C2semidirect
10. #23935C2semidirect
11. #23823C2semidirect
12. #23822C2semidirect
13. #23805C2semidirect
14. #23803C2semidirect
15. #21508C2semidirect
16. #21457C2semidirect
17. #20778C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
C4×#5322
#293702
#293662
#292622
#281642
#276852
#275872
#242162
#239352
#238232
#238222
#238052
#238032
#215082
#214572
#207782