Finite Group Databasebeta

← #6450683

Subgroups — #6450683

Order 512.

1796 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2#34121C211yesmaximal, characteristic
3#26104C214yesmaximal, complemented
4#26102C214yesmaximal, complemented
5#26092C212yesmaximal, complemented
6#26090C212yesmaximal, complemented
7#26088C212yesmaximal, complemented
8#26086C212yesmaximal, complemented
9#23734C212yesmaximal
10#23617C212yesmaximal
11#21171C212yesmaximal, complemented
12#21170C211yesmaximal, characteristic, complemented
13#21146C211yesmaximal, characteristic, complemented
14#21034C212yesmaximal, complemented
15#20800C211yesmaximal, characteristic, complemented
16#20648C211yesmaximal, characteristic
17C4×#417C211yesmaximal, characteristic
18C22×#125—21no—
19C2×C4×#41C2211yescharacteristic
20C2×#428C2212yes—
21C2×#428C2212yes—
22C2×#428C2212yes—
23C2×#423C2212yes—
24C2×#423C2211yescharacteristic
25C2×#417C2211yescharacteristic
26C2×#417C2211yescharacteristic
27C2×#417C2212yes—
28C2×C4×M16C2211yescharacteristic
29#2552—21no—
30#2552—21no—
31#2552—22no—
32#2552C2212yes—
33#2552C2212yes—
34#2552C2214yes—
35C2×#398—22no—
36#2357C2212yes—
37#2356C2212yes—
38#2355C2211yescharacteristic
39#2355C2211yescharacteristic
40#2354C2211yescharacteristic
41#2354C2211yescharacteristic
42#2353—22no—
43#2352—21no—
44#2352—21no—
45#2298—22no—
46#2297C2212yes—
47C2×#389—21no—
48C2×#389C2212yes—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#384C2212yes—
52#2283—21no—
53C4×#117—21no—
54C4×#117C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#2096C2214yes—
58#2095C2214yes—
59#2094C2214yes—
60#2093C2214yes—
61#2068C2214yes—
62#2067C2214yes—
63#2066C2214yes—
64#2065C2214yes—
65#1949C2214yes—
66#1948C2212yes—
67#1948C2214yes—
68#1948C2214yes—
69#1947C2212yes—
70#1947C2214yes—
71#1947C2214yes—
72#1940C2214yes—
73#1939C2212yes—
74#1939C2212yes—
75#1921C2214yes—
76#1920C2214yes—
77#1919C2214yes—
78#1918C2214yes—
79#1847C2214yes—
80#1845C2212yes—
81#1844C2212yes—
82#1803C2212yes—
83#1803C2212yes—
84#1802C2212yes—
85#1802C2212yes—
86#1763C2212yes—
87#1762C2212yes—
88C4×#103C2212yes—
89C4×#102C2212yes—
90#1736C2211yescharacteristic
91#1736C2212yes—
92#1736C2212yes—
93#1736C2211yescharacteristic
94#1728C2212yes—
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C22×M16C2311yescharacteristic
98C22×M16C2311yescharacteristic
99C2×#125—22no—
100C2×#125—24no—
101C2×#125—22no—
102C2×#125—48no—
103C4×#41—24no—
104C4×#41Q818yes—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2311yescharacteristic
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4C2311yescharacteristic
110C2×C4×D4C2311yescharacteristic
111C2×#117—21no—
112C2×#117—21no—
113C2×#117—21no—
114C2×#117C2312yes—
115C2×#117C2311yescharacteristic
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C42—21no—
127C22×C42—21no—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#428—28no—
131#428—44no—
132#428—44no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139#423—28no—
140#423—24no—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145#417—24no—
146#417—24no—
147#417—44no—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C4×M16—24no—
155C2×#97C2312yes—
156C2×C4×C8C2312yes—
157#398—24no—
158#398—24no—
159#398—24no—
160#398—24no—
161#391C2312yes—
162#391C2312yes—
163#390C2312yes—
164#390C2312yes—
165#389—22no—
166#389—24no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2314yes—
171#389C2314yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#386—22no—
181#386—22no—
182#384—22no—
183#384—22no—
184#384—21no—
185#384—21no—
186#384C2312yes—
187#384C2312yes—
188#384C2314yes—
189#384C2314yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#383—22no—
193#382—21no—
194#382—21no—
195#382C2312yes—
196#381—22no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224#376—22no—
225#376—22no—
226C2×#95—21no—
227C2×#95—22no—
228C2×#95D412yes—
229C43—22no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#340C2312yes—
237#340C2312yes—
238#336C2314yes—
239#336C2314yes—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41C2×Q814yes—
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16#4114yes—
268C2×M16#4114yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272#125—416no—
273#125—48no—
274#125—48no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8C2×Q814yes—
278C4×Q8C2×Q814yes—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4C2×Q814yes—
284C4×D4C2×Q814yes—
285#117—22no—
286#117—28no—
287#117—24no—
288#117—24no—
289#117—44no—
290#117—44no—
291#117C2×Q814yes—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—48no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4—84no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—28no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—44no—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×Q814yes—
364C2×C42#4114yes—
365C2×C42C2412yes—
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#107—28no—
370#107—44no—
371#107—44no—
372#106—28no—
373#106—44no—
374#106—44no—
375C4⋊C8—28no—
376C4⋊C8—28no—
377C4⋊C8—24no—
378C4⋊C8—24no—
379#103—24no—
380#103—24no—
381#103—44no—
382#102—24no—
383#102—24no—
384#102—44no—
385#97—24no—
386C4×C8—24no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405C24—21no—
406C24C22×D411yescharacteristic
407#41—28no—
408#41—28no—
409#41—44no—
410#41—44no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8C22×Q812yes—
417C2×D4—24no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4C22×Q812yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4C22⋊Q814yes—
463C22×C4C22⋊Q814yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×Q812yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+412yes—
479C22×C4C2511yescharacteristic
480M16—28no—
481M16—28no—
482M16—48no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8C2×#4114yes—
491C2×C8C2×#4114yes—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—81no—
522C4⋊C4—81no—
523C4⋊C4—84no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C4⋊C4AGL(1,ℤ/8)14yes—
528C4⋊C4#13714yes—
529C4⋊C4C22×Q812yes—
530C4⋊C4C22×Q812yes—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—24no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—44no—
549C42—44no—
550C42—44no—
551C42—44no—
552C42—44no—
553C42—82no—
554C42—82no—
555C42C22⋊Q814yes—
556C42C22⋊Q814yes—
557C42C22×Q812yes—
558C42C22×Q812yes—
559C42C2×#4112yes—
560C42C2×#4112yes—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23C2×C22⋊Q812yes—
573C23#54712yes—
574C23#55012yes—
575C23C23×D411yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580D4—22no—
581D4—22no—
582D4—48no—
583D4—41no—
584D4—41no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4C2×C22⋊Q812yes—
636C2×C4C2×C22⋊Q812yes—
637C2×C4C2×C22⋊Q812yes—
638C2×C4C2×C22⋊Q812yes—
639C2×C4C2×C22⋊Q812yes—
640C2×C4#53314yes—
641C2×C4#54414yes—
642C2×C4#54712yes—
643C2×C4#55012yes—
644C2×C4#55214yes—
645C2×C4#55314yes—
646C2×C4C2×AGL(1,ℤ/8)14yes—
647C2×C4C2×AGL(1,ℤ/8)12yes—
648C2×C4C2×#13714yes—
649C2×C4C2×#13712yes—
650C2×C4C23×Q811yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C8—24no—
657C8—24no—
658C8—44no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22#244114yes—
673C22C22×C22⋊Q811yescharacteristic
674C22C2×#54711yescharacteristic
675C22C2×#55011yescharacteristic
676C22C22×AGL(1,ℤ/8)11yescharacteristic
677C22C22×#13711yescharacteristic
678C22#358511yescharacteristic
679C22#358611yescharacteristic
680C4—22no—
681C4—24no—
682C4—24no—
683C4—22no—
684C4—22no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—84no—
694C4—82no—
695C4#244114yes—
696C4#307914yes—
697C4C2×#53311yescharacteristic
698C4C2×#54411yescharacteristic
699C4C2×#55211yescharacteristic
700C4C2×#55311yescharacteristic
701C2—24no—
702C2—82no—
703C2C2×#244111yescharacteristic
704C2C2×#307911yescharacteristic
705C2#6049311yescharacteristic
706C2#6125211yescharacteristic
707C2#6125411yescharacteristic
708C2#6140211yescharacteristic
709C2#6140411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. C2×#2552C2semidirect
3. #26104C2semidirect
4. #26102C2semidirect
5. #26092C2semidirect
6. #26090C2semidirect
7. #26088C2semidirect
8. #26086C2semidirect
11. #21171C2semidirect
12. #21170C2semidirect
13. #21146C2semidirect
14. #21034C2semidirect
15. #20800C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
#341212
#261042
#261022
#260922
#260902
#260882
#260862
#237342
#236172
#211712
#211702
#211462
#210342
#208002
#206482
C4×#4172