Finite Group Databasebeta

← #6537300

Subgroups — #6537300

Order 512.

1875 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34316C211yesmaximal, characteristic
3#29980C211yesmaximal, characteristic, complemented
4#29955C212yesmaximal, complemented
5#29255C211yesmaximal, characteristic, complemented
6#29165C212yesmaximal, complemented
7#29141C211yesmaximal, characteristic, complemented
8#26466C212yesmaximal, complemented
9#26431C214yesmaximal, complemented
10#26024C212yesmaximal, complemented
11#25975C214yesmaximal, complemented
12#24302C212yesmaximal, complemented
13#23774C211yesmaximal, characteristic, complemented
14#23452C212yesmaximal, complemented
15#22713C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22431C212yesmaximal
18C2×D4×Q8—21no—
19C2×#547C2211yescharacteristic
20C22×C4⋊Q8C2212yes—
21C2×#477C2214yes—
22C2×#436C2212yes—
23C2×#417C2214yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2551—21no—
27#2551C2212yes—
28#2546—21no—
29#2440—22no—
30#2440C2214yes—
31#2434—22no—
32#2374—22no—
33#2365C2211yescharacteristic
34#2363—21no—
35#2362C2212yes—
36#2354C2211yescharacteristic
37#2354C2211yescharacteristic
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40#2345—21no—
41#2341C2212yes—
42#2335—22no—
43#2324—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8C2212yes—
47#2297C2212yes—
48#2293C2211yescharacteristic
49#2274—21no—
50C42×D4C2211yescharacteristic
51#2099C2212yes—
52#2099C2211yescharacteristic
53#2096C2212yes—
54#2093C2211yescharacteristic
55#2093C2212yes—
56#2086C2212yes—
57#2068C2214yes—
58#2067C2212yes—
59#2065C2214yes—
60#2057C2212yes—
61#2057C2211yescharacteristic
62#2024C2212yes—
63#2024C2214yes—
64#2024C2212yes—
65#2024C2212yes—
66#2023C2212yes—
67#2023C2214yes—
68#2023C2211yescharacteristic
69#1987C2214yes—
70#1981C2212yes—
71#1956C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1920C2214yes—
76#1894C2214yes—
77#1872C2214yes—
78#1865C2214yes—
79#1863C2214yes—
80#1854C2214yes—
81#1852C2214yes—
82#1851C2214yes—
83#1834C2212yes—
84#1831C2211yescharacteristic
85#1809C2214yes—
86#1803C2214yes—
87#1802C2212yes—
88#1797C2212yes—
89#1788C2214yes—
90C4×#107C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×#41C2312yes—
96C23×Q8—22no—
97C22×M16C2314yes—
98D4×Q8—48no—
99#547—24no—
100#547Q818yescomplemented
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—48no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2314yes—
110C2×C4⋊Q8C2312yes—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8C2312yes—
115C2×#121—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—21no—
125C2×#117—21no—
126C2×#117C2312yes—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4C2314yes—
130C22×C42C2312yes—
131#477—48no—
132#436—28no—
133C2×#107C2312yes—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2312yes—
140#417—48no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2312yes—
150#398—24no—
151#395—24no—
152#395—24no—
153#391—21no—
154#391—22no—
155#391C2312yes—
156#391C2312yes—
157#390—21no—
158#390—22no—
159#390—22no—
160#390C2312yes—
161#390C2311yescharacteristic
162#390C2311yescharacteristic
163#389—21no—
164#389—22no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389C2312yes—
171#389C2311yescharacteristic
172#389C2314yes—
173#389C2311yescharacteristic
174#388—22no—
175#388—22no—
176#388—22no—
177#388—21no—
178#388—21no—
179#388—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#386—22no—
184#386—22no—
185#385—22no—
186#385—24no—
187#385—24no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#380—22no—
195#380—21no—
196#380—22no—
197#380C2312yes—
198#380C2314yes—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2314yes—
235C2×#41—24no—
236C2×#41—44no—
237C2×#41—44no—
238C2×#41C2×Q818yescomplemented
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—24no—
242C22×Q8—48no—
243C22×Q8—48no—
244C22×Q8—84no—
245C22×Q8C2412yes—
246C22×D4—22no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—24no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C2×M16—28no—
256C2×M16—44no—
257C2×M16—44no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—28no—
263C4⋊Q8—22no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—48no—
267C4⋊Q8—48no—
268C4⋊Q8—48no—
269C4⋊Q8—44no—
270C4⋊Q8—82no—
271C4⋊Q8C2×Q814yescomplemented
272#123—216no—
273C22⋊Q8—24no—
274C22⋊Q8—416no—
275C22⋊Q8—48no—
276C22⋊Q8C2×Q818yescomplemented
277#121—28no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—48no—
285C4×D4C2×Q814yes—
286C4×D4C2×Q814yescomplemented
287C4×D4#4114yes—
288#117—44no—
289#117—44no—
290#117—44no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—24no—
358#107—42no—
359#107—44no—
360#106—44no—
361C4⋊C8—24no—
362C4⋊C8—42no—
363C4⋊C8—44no—
364#103—28no—
365#103—44no—
366#103—44no—
367#103—44no—
368#102—24no—
369#102—42no—
370#102—44no—
371C22⋊C8—28no—
372C4×C8—24no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—24no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C24C22×D412yes—
391#41—416no—
392#41—88no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—48no—
398C2×Q8—42no—
399C2×Q8—48no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—84no—
405C2×Q8—832no—
406C2×Q8C22×Q814yes—
407C2×D4—28no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4C22⋊Q814yes—
416C2×D4C22×Q812yes—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—48no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—48no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—84no—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×Q814yes—
452C22×C4C2×#4114yes—
453C22×C4C2511yescharacteristic
454M16—88no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4C22⋊Q814yes—
496C4⋊C4#13714yes—
497C4⋊C4C22×Q812yes—
498C4⋊C4H(5,𝔽2)14yes—
499C22⋊C4—28no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—48no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4C22⋊Q818yes—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—42no—
519C42—42no—
520C42—41no—
521C42—42no—
522C42—44no—
523C42—44no—
524C42—82no—
525C42—82no—
526C42—82no—
527C42AGL(1,ℤ/8)14yes—
528C42C22×Q812yes—
529C42C2×#4112yes—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23C2×C22⋊Q814yes—
544C23C23×D411yescharacteristic
545Q8—28no—
546Q8—48no—
547Q8—44no—
548Q8—88no—
549Q8—816no—
550D4—28no—
551D4—24no—
552D4—42no—
553D4—88no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—48no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—88no—
602C2×C4—816no—
603C2×C4C2×C22⋊Q812yes—
604C2×C4C2×C22⋊Q814yes—
605C2×C4C2×C22⋊Q812yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4C2×#13714yes—
610C2×C4C2×#13712yes—
611C2×C4#57812yes—
612C2×C4C23×Q811yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×H(5,𝔽2)12yes—
615C2×C4#58512yes—
616C8—42no—
617C8—44no—
618C8—42no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C22×C22⋊Q811yescharacteristic
634C22#348411yescharacteristic
635C22#350011yescharacteristic
636C22C22×AGL(1,ℤ/8)11yescharacteristic
637C22C22×#13711yescharacteristic
638C22C2×#57811yescharacteristic
639C22#358511yescharacteristic
640C4—22no—
641C4—22no—
642C4—24no—
643C4—22no—
644C4—22no—
645C4—24no—
646C4—42no—
647C4—44no—
648C4—41no—
649C4—42no—
650C4—44no—
651C4—81no—
652C4—82no—
653C4—82no—
654C4—81no—
655C4—84no—
656C4#307914yes—
657C4#350112yes—
658C4#351412yes—
659C2—44no—
660C2—82no—
661C2#3570211yescharacteristic
662C2C2×#307911yescharacteristic
663C2#6049811yescharacteristic
664C2#6158211yescharacteristic
665C2#6160211yescharacteristic
666C2#6253811yescharacteristic
667C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. D4×C4⋊Q8C2semidirect
3. #29980C2semidirect
4. #29955C2semidirect
5. #29255C2semidirect
6. #29165C2semidirect
7. #29141C2semidirect
8. #26466C2semidirect
9. #26431C2semidirect
10. #26024C2semidirect
11. #25975C2semidirect
12. #24302C2semidirect
13. #23774C2semidirect
14. #23452C2semidirect
15. #22713C2semidirect
16. #22700C2semidirect
100. #547Q8semidirect
238. C2×#41C2×Q8semidirect
271. C4⋊Q8C2×Q8semidirect
276. C22⋊Q8C2×Q8semidirect
286. C4×D4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#343162
#299802
#299552
#292552
#291652
#291412
#264662
#264312
#260242
#259752
#243022
#237742
#234522
#227132
#227002
#224312