Finite Group Databasebeta

← #6470960

Subgroups — #6470960

Order 512.

2098 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic, complemented
2C2×#2361C211yesmaximal, characteristic
3#33151C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic, complemented
5#32953C211yesmaximal, characteristic, complemented
6#29968C211yesmaximal, characteristic, complemented
7#29966C211yesmaximal, characteristic, complemented
8#29956C212yesmaximal, complemented
9#26438C214yesmaximal, complemented
10#26436C214yesmaximal, complemented
11#26354C214yesmaximal, complemented
12#26352C214yesmaximal, complemented
13#21934C211yesmaximal, characteristic, complemented
14#21767C212yesmaximal
15#20844C212yesmaximal, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4—21no—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#476C2212yescomplemented
21C2×#474C2212yescomplemented
22C2×#428C2214yescomplemented
23C2×#407C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2851C2212yes—
27#2626—21no—
28#2626—21no—
29#2608—22no—
30#2607—22no—
31#2441C2212yescomplemented
32#2441C2212yescomplemented
33#2439C2212yescomplemented
34#2439C2212yescomplemented
35#2361—22no—
36#2361—22no—
37#2361C2214yes—
38#2361C2214yes—
39#2329—21no—
40#2329—21no—
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#386—21no—
45C2×#385—21no—
46#2280C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2100C2212yescomplemented
50#2099C2212yescomplemented
51#2092C2214yescomplemented
52#2088C2214yescomplemented
53#2086C2214yescomplemented
54#2067C2214yescomplemented
55#2065C2214yescomplemented
56#2058C2212yescomplemented
57#2057C2212yescomplemented
58#2025C2214yescomplemented
59#1989C2214yescomplemented
60#1949C2214yescomplemented
61#1921C2214yescomplemented
62#1919C2214yescomplemented
63#1895C2214yes—
64#1895C2214yes—
65#1875C2214yes—
66#1875C2214yes—
67#1864C2214yescomplemented
68#1862C2214yescomplemented
69#1853C2212yes—
70#1853C2212yes—
71#1853C2214yescomplemented
72#1852C2212yes—
73#1852C2212yes—
74#1852C2214yescomplemented
75#1851C2214yescomplemented
76#1851C2214yescomplemented
77#1823C2212yes—
78#1823C2212yes—
79#1802C2212yescomplemented
80#1802C2212yescomplemented
81#1797C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1786C2212yes—
85#1786C2212yes—
86#1736C2214yescomplemented
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4—21no—
91C24×C4C2311yescharacteristic
92C22×M16C2314yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#125C2312yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×#121—22no—
99C2×#121—24no—
100C2×#121Q818yescomplemented
101C2×#121C2314yes—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4—48no—
107C2×C4×D4—44no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117C2312yes—
111C2×#117C2312yes—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—44no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#476—44no—
130#474—44no—
131#428—48no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143#407—28no—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2314yes—
147#399—22no—
148#399—22no—
149#396—22no—
150#396—22no—
151#396—22no—
152#396—22no—
153#393—22no—
154#393—22no—
155#391—21no—
156#391—21no—
157#391—22no—
158#391—22no—
159#390—22no—
160#390C2314yes—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—24no—
174#388—24no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#386—24no—
182#386—24no—
183#385—22no—
184#385—22no—
185#385—24no—
186#385—24no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382C2312yes—
190#381—22no—
191#380—21no—
192#380—21no—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#379—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4C2314yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4C2312yes—
205C4×C22⋊C4C2312yes—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25—21no—
222C25C2×D411yescharacteristic
223C22×Q8—21no—
224C22×Q8—21no—
225C22×D4—28no—
226C22×D4—24no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—42no—
230C22×D4—42no—
231C22×D4—44no—
232C22×D4—44no—
233C22×D4C2×Q814yes—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—24no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×Q814yes—
256C23×C4#4114yes—
257C23×C4C2411yescharacteristic
258C23×C4C2412yes—
259C2×M16—28no—
260C2×M16—44no—
261C2×M16—44no—
262C22×C8C2414yes—
263C22×C8C2414yes—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266#125—44no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269#121—216no—
270#121—28no—
271#121C2×Q8116yes—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—416no—
277C4×D4—416no—
278C4×D4—44no—
279C4×D4—44no—
280#117—44no—
281#117—44no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2×Q814yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—42no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42—44no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349#107—28no—
350#107—44no—
351#106—28no—
352#106—44no—
353C4⋊C8—44no—
354C4⋊C8—42no—
355C4⋊C8—42no—
356#102—44no—
357#102—44no—
358#102—44no—
359#102—44no—
360C22⋊C8—24no—
361C22⋊C8—24no—
362C22⋊C8—28no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—22no—
381C24—22no—
382C24—24no—
383C24—22no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24—41no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C22⋊Q814yes—
395C24C22×D412yes—
396C24C22×D411yescharacteristic
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×D4—216no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—48no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—432no—
416C2×D4—48no—
417C2×D4C22×Q814yes—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
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—48no—
435C22×C4—48no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—48no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4—816no—
465C22×C4—816no—
466C22×C4C22⋊Q814yes—
467C22×C4C22⋊Q818yes—
468C22×C4AGL(1,ℤ/8)14yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×Q812yes—
478C22×C4C22×Q814yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4114yes—
481C22×C4C2511yescharacteristic
482M16—416no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—48no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4C22×Q814yes—
515C4⋊C4H(5,𝔽2)14yes—
516C22⋊C4—216no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—84no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—42no—
535C42—41no—
536C42—41no—
537C42—48no—
538C42—82no—
539C42—82no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—28no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23C2×C22⋊Q814yes—
572C23C2×C22⋊Q812yes—
573C23C2×AGL(1,ℤ/8)12yes—
574C23#57512yes—
575C23C23×D411yescharacteristic
576Q8—162no—
577Q8—162no—
578D4—24no—
579D4—24no—
580D4—48no—
581D4—432no—
582D4—42no—
583D4—42no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—42no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—88no—
636C2×C4—88no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—816no—
640C2×C4—84no—
641C2×C4C2×C22⋊Q814yes—
642C2×C4C2×C22⋊Q812yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C23×Q811yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)12yes—
648C2×C4#58512yes—
649C8—44no—
650C8—44no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—416no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—816no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22#307714yes—
674C22C22×C22⋊Q811yescharacteristic
675C22#348411yescharacteristic
676C22#348511yescharacteristic
677C22C22×AGL(1,ℤ/8)11yescharacteristic
678C22C2×#57511yescharacteristic
679C22#358511yescharacteristic
680C22#358611yescharacteristic
681C4—22no—
682C4—22no—
683C4—24no—
684C4—22no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—44no—
689C4—41no—
690C4—41no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#345912yes—
697C4#350112yes—
698C2—24no—
699C2—44no—
700C2—84no—
701C2#3498511yescharacteristic
702C2C2×#307711yescharacteristic
703C2#6049211yescharacteristic
704C2#6157111yescharacteristic
705C2#6158311yescharacteristic
706C2#6171311yescharacteristic
707C2#6171711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35033C2semidirect
3. #33151C2semidirect
4. #32956C2semidirect
5. #32953C2semidirect
6. #29968C2semidirect
7. #29966C2semidirect
8. #29956C2semidirect
9. #26438C2semidirect
10. #26436C2semidirect
11. #26354C2semidirect
12. #26352C2semidirect
13. #21934C2semidirect
15. #20844C2semidirect
20. C2×#476C22semidirect
21. C2×#474C22semidirect
22. C2×#428C22semidirect
31. #2441C22semidirect
32. #2441C22semidirect
33. #2439C22semidirect
34. #2439C22semidirect
49. #2100C22semidirect
50. #2099C22semidirect
51. #2092C22semidirect
52. #2088C22semidirect
53. #2086C22semidirect
54. #2067C22semidirect
55. #2065C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
58. #2025C22semidirect
59. #1989C22semidirect
60. #1949C22semidirect
61. #1921C22semidirect
62. #1919C22semidirect
67. #1864C22semidirect
68. #1862C22semidirect
71. #1853C22semidirect
74. #1852C22semidirect
75. #1851C22semidirect
76. #1851C22semidirect
79. #1802C22semidirect
80. #1802C22semidirect
86. #1736C22semidirect
100. C2×#121Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#23612
#331512
#329562
#329532
#299682
#299662
#299562
#264382
#264362
#263542
#263522
#219342
#217672
#208442