Finite Group Databasebeta

← #6537303

Subgroups — #6537303

Order 512.

1931 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34330C211yesmaximal, characteristic
3#29983C211yesmaximal, characteristic, complemented
4#29955C212yesmaximal, complemented
5#29254C211yesmaximal, characteristic, complemented
6#29163C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic, complemented
8#26464C212yesmaximal, complemented
9#26427C214yesmaximal, complemented
10#26026C212yesmaximal, complemented
11#25978C214yesmaximal, complemented
12#24299C212yesmaximal, complemented
13#23776C211yesmaximal, characteristic, complemented
14#23452C212yesmaximal, complemented
15#22715C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22432C212yesmaximal
18C2×#550C2211yescharacteristic
19C2×D4×Q8—21no—
20C22×C4⋊Q8C2212yes—
21C2×#476C2214yes—
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#2366—21no—
33#2366C2211yescharacteristic
34#2364—22no—
35#2363C2212yes—
36#2357—21no—
37#2355C2211yescharacteristic
38#2354C2211yescharacteristic
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2212yes—
41#2339C2212yes—
42#2333—22no—
43#2328—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#121C2212yes—
47#2297C2212yes—
48#2295C2211yescharacteristic
49#2275—21no—
50C42×D4C2211yescharacteristic
51#2099C2212yes—
52#2099C2211yescharacteristic
53#2096C2211yescharacteristic
54#2096C2212yes—
55#2093C2212yes—
56#2088C2212yes—
57#2068C2214yes—
58#2066C2212yes—
59#2065C2214yes—
60#2058C2212yes—
61#2058C2211yescharacteristic
62#2024C2212yes—
63#2024C2214yes—
64#2024C2211yescharacteristic
65#2023C2212yes—
66#2023C2214yes—
67#2023C2212yes—
68#2023C2212yes—
69#1987C2214yes—
70#1982C2212yes—
71#1957C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1921C2214yes—
76#1895C2214yes—
77#1874C2214yes—
78#1864C2214yes—
79#1862C2214yes—
80#1854C2214yes—
81#1853C2214yes—
82#1851C2214yes—
83#1834C2212yes—
84#1831C2211yescharacteristic
85#1809C2214yes—
86#1803C2212yes—
87#1802C2214yes—
88#1797C2212yes—
89#1788C2214yes—
90C4×#106C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93#1736C2214yes—
94#1728C2212yes—
95C22×#41C2312yes—
96C23×Q8—22no—
97C22×M16C2314yes—
98#550—24no—
99#550Q818yescomplemented
100D4×Q8—48no—
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⋊Q8C2314yes—
109C2×C4⋊Q8C2312yes—
110C2×#127—21no—
111C2×#127C2311yescharacteristic
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—24no—
114C2×#121—22no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×#117C2312yes—
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2314yes—
132C22×C42C2312yes—
133#476—48no—
134#436—28no—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142#417—48no—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152#398—24no—
153#395—24no—
154#395—24no—
155#391—21no—
156#391—22no—
157#391C2311yescharacteristic
158#391C2312yes—
159#391C2311yescharacteristic
160#390—21no—
161#390—21no—
162#390—22no—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#390C2312yes—
167#389—22no—
168#389—22no—
169#389—21no—
170#389—22no—
171#389C2312yes—
172#389C2311yescharacteristic
173#389C2314yes—
174#389C2311yescharacteristic
175#388—22no—
176#388—22no—
177#388—22no—
178#387—21no—
179#387—21no—
180#387—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382—21no—
197#381—22no—
198#380—22no—
199#380C2312yes—
200#380C2314yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—21no—
219C2×#95—22no—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
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×Q8C2411yescharacteristic
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C2×M16—28no—
260C2×M16—44no—
261C2×M16—44no—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—28no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—48no—
270C4⋊Q8—48no—
271C4⋊Q8—48no—
272C4⋊Q8—44no—
273C4⋊Q8—82no—
274#127—28no—
275#127—22no—
276#127C2×Q814yescomplemented
277C22⋊Q8—416no—
278C22⋊Q8—48no—
279#121—24no—
280#121—216no—
281#121C2×Q818yescomplemented
282C4×Q8—22no—
283C4×Q8—44no—
284C4×Q8C2×Q814yescomplemented
285C4×D4—28no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—48no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—82no—
294C4×D4—84no—
295C4×D4C2×Q814yes—
296C4×D4#4114yes—
297#117—44no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—21no—
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—44no—
358#106—24no—
359#106—42no—
360#106—44no—
361C4⋊C8—24no—
362C4⋊C8—42no—
363C4⋊C8—44no—
364#103—24no—
365#103—42no—
366#103—44no—
367#102—28no—
368#102—44no—
369#102—44no—
370#102—44no—
371C22⋊C8—28no—
372C4×C8—24no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—24no—
387#95—24no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—22no—
392C24C22×D412yes—
393#41—416no—
394#41—88no—
395C2×Q8—24no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—48no—
400C2×Q8—42no—
401C2×Q8—48no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—84no—
407C2×Q8—832no—
408C2×Q8C22×Q812yes—
409C2×D4—28no—
410C2×D4—28no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4C22⋊Q814yes—
427C2×D4C22×Q814yes—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—84no—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×Q814yes—
466C22×C4C2×#4114yes—
467C22×C4C2511yescharacteristic
468M16—88no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4C22⋊Q814yes—
506C4⋊C4#13714yes—
507C4⋊C4C22×Q812yes—
508C4⋊C4H(5,𝔽2)14yes—
509C22⋊C4—28no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4C22⋊Q818yes—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—42no—
529C42—42no—
530C42—41no—
531C42—44no—
532C42—42no—
533C42—82no—
534C42—82no—
535C42AGL(1,ℤ/8)14yes—
536C42C22×Q812yes—
537C42C2×#4112yes—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23C2×C22⋊Q814yes—
552C23C23×D411yescharacteristic
553Q8—24no—
554Q8—48no—
555Q8—42no—
556Q8—88no—
557Q8—816no—
558D4—28no—
559D4—28no—
560D4—48no—
561D4—44no—
562D4—84no—
563D4—88no—
564D4—88no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—88no—
612C2×C4—816no—
613C2×C4—164no—
614C2×C4—164no—
615C2×C4C2×C22⋊Q812yes—
616C2×C4C2×C22⋊Q814yes—
617C2×C4C2×C22⋊Q812yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4C2×AGL(1,ℤ/8)12yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4C2×#13714yes—
622C2×C4C2×#13712yes—
623C2×C4#57712yes—
624C2×C4C23×Q811yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4#58512yes—
628C8—42no—
629C8—44no—
630C8—42no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22C22×C22⋊Q811yescharacteristic
646C22#348411yescharacteristic
647C22#350011yescharacteristic
648C22C22×AGL(1,ℤ/8)11yescharacteristic
649C22C22×#13711yescharacteristic
650C22C2×#57711yescharacteristic
651C22#358511yescharacteristic
652C4—22no—
653C4—22no—
654C4—24no—
655C4—22no—
656C4—22no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—41no—
661C4—42no—
662C4—44no—
663C4—82no—
664C4—82no—
665C4—81no—
666C4—84no—
667C4—81no—
668C4#307914yes—
669C4#350112yes—
670C4#351412yes—
671C2—44no—
672C2—84no—
673C2#3570211yescharacteristic
674C2C2×#307911yescharacteristic
675C2#6049511yescharacteristic
676C2#6158211yescharacteristic
677C2#6159311yescharacteristic
678C2#6254111yescharacteristic
679C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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. #29983C2semidirect
4. #29955C2semidirect
5. #29254C2semidirect
6. #29163C2semidirect
7. #29139C2semidirect
8. #26464C2semidirect
9. #26427C2semidirect
10. #26026C2semidirect
11. #25978C2semidirect
12. #24299C2semidirect
13. #23776C2semidirect
14. #23452C2semidirect
15. #22715C2semidirect
16. #22700C2semidirect
99. #550Q8semidirect
238. C2×#41C2×Q8semidirect
276. #127C2×Q8semidirect
281. #121C2×Q8semidirect
284. C4×Q8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#343302
#299832
#299552
#292542
#291632
#291392
#264642
#264272
#260262
#259782
#242992
#237762
#234522
#227152
#227002
#224322