Finite Group Databasebeta

← #6593465

Subgroups — #6593465

Order 512.

1941 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2C4×D4×Q8C211yesmaximal, characteristic, complemented
3#33198C211yesmaximal, characteristic, complemented
4#32903C212yesmaximal, complemented
5#29141C211yesmaximal, characteristic, complemented
6#28252C211yesmaximal, characteristic, complemented
7#27631C212yesmaximal, complemented
8#27239C214yesmaximal, complemented
9#26741C214yesmaximal, complemented
10#24965C212yesmaximal, complemented
11#24661C212yesmaximal, complemented
12#24239C212yesmaximal, complemented
13#23100C212yesmaximal, complemented
14#22799C211yesmaximal, characteristic, complemented
15#22759C212yesmaximal, complemented
16C4×#460C211yesmaximal, characteristic, complemented
17#20740C212yesmaximal, complemented
18C2×#555—21no—
19C2×D4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C2×#477C2212yes—
22C2×#460C2211yescharacteristic
23C2×#460C2212yes—
24C2×#461C2212yes—
25C2×#448C2214yes—
26C2×C4×SD16C2212yes—
27C2×#432C2212yes—
28#2868C2211yescharacteristic
29#2856C2212yes—
30#2563—21no—
31#2537—21no—
32#2537C2212yes—
33#2500—22no—
34#2491—22no—
35#2491C2214yes—
36#2365—21no—
37#2365C2211yescharacteristic
38#2360—22no—
39#2360C2212yes—
40#2354—21no—
41#2354C2212yes—
42Q8×C4⋊C4C2211yescharacteristic
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2211yescharacteristic
45Q8×C22⋊C4C2212yes—
46C4×C4⋊Q8—21no—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C4×C22⋊Q8—22no—
50C4×C22⋊Q8C2212yes—
51#2285C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4—21no—
54C42×D4C2211yescharacteristic
55#2075C2214yes—
56#2055C2214yes—
57#2035C2214yes—
58#2033C2214yes—
59#2024C2214yes—
60#2000C2214yes—
61#1987C2212yes—
62#1982C2211yescharacteristic
63#1973C2214yes—
64#1956C2212yes—
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1947C2212yes—
68#1947C2212yes—
69#1936C2214yes—
70#1928C2214yes—
71#1926C2212yes—
72#1924C2214yes—
73#1919C2214yes—
74#1880C2214yes—
75#1880C2212yes—
76#1880C2214yes—
77#1874C2214yes—
78#1842C2211yescharacteristic
79#1834C2212yes—
80#1823C2214yes—
81#1803C2211yescharacteristic
82#1803C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1797C2212yes—
89#1787C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93#1763C2214yes—
94C4×#102C2212yes—
95C23×Q8C2312yes—
96C22×SD16C2312yes—
97C22×SD16C2314yes—
98#555—48no—
99D4×Q8—24no—
100D4×Q8—24no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#124—24no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121C2314yes—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—28no—
117C2×C4×Q8—44no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×C4×D4C2312yes—
131C2×C4×D4C2311yescharacteristic
132C2×#117—22no—
133C2×#117—22no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4C2312yes—
136C22×C42—22no—
137C22×C42C2312yes—
138#477—44no—
139#461—28no—
140#460—24no—
141#460—44no—
142#448—48no—
143C4×SD16—28no—
144#432—44no—
145C2×#107C2314yes—
146C2×#106C2312yes—
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2314yes—
163C2×#97C2314yes—
164C2×C4×C8C2312yes—
165#394—24no—
166#394—24no—
167#393—24no—
168#391—22no—
169#391—22no—
170#391C2312yes—
171#391C2312yes—
172#390—21no—
173#390—21no—
174#390—22no—
175#390—22no—
176#390C2312yes—
177#390C2314yes—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389C2312yes—
185#388—24no—
186#388—24no—
187#387—22no—
188#387—22no—
189#387—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#382—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2312yes—
206#380C2314yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2314yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C43—21no—
230C43C2311yescharacteristic
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2312yes—
238#336C2314yes—
239#336C2312yes—
240C22×Q8—24no—
241C22×Q8—24no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—42no—
247C22×Q8—42no—
248C22×Q8C2412yes—
249C22×Q8C2411yescharacteristic
250C22×D4—24no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2412yes—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—28no—
265C2×SD16—44no—
266C2×SD16—44no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C4⋊Q8—28no—
271C4⋊Q8—22no—
272C4⋊Q8—42no—
273C4⋊Q8—42no—
274C4⋊Q8—44no—
275C4⋊Q8—82no—
276C4⋊Q8#4114yes—
277#127—22no—
278#127—22no—
279#124—416no—
280C22⋊Q8—216no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283#121—48no—
284#121—48no—
285#121—44no—
286#121—44no—
287#121—84no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—24no—
291C4×Q8—22no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—48no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×Q8—41no—
299C4×Q8—48no—
300C4×Q8#4114yes—
301C4×Q8#4114yes—
302C4×D4—28no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—44no—
310C4×D4—44no—
311C4×D4—44no—
312#117—48no—
313#117—48no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—48no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—48no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D412yes—
373C2×C42C2×D412yes—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—44no—
382#106—24no—
383C4⋊C8—22no—
384C4⋊C8—24no—
385C4⋊C8—22no—
386C4⋊C8—24no—
387C4⋊C8—42no—
388#103—24no—
389#103—28no—
390#103—28no—
391#103—44no—
392#103—44no—
393#102—24no—
394#102—24no—
395#102—42no—
396#102—44no—
397C22⋊C8—44no—
398C22⋊C8—44no—
399#97—44no—
400C4×C8—24no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—24no—
412#95C2×D412yes—
413#95C2×D412yes—
414#95C2×D412yes—
415C24—22no—
416C24C22×D412yes—
417C2×Q8—28no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—48no—
432C2×Q8—48no—
433C2×Q8—82no—
434C2×Q8—82no—
435C2×Q8C2×#4114yes—
436C2×Q8C2×#4112yes—
437C2×Q8C2×#4112yes—
438C2×Q8H(5,𝔽2)12yes—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—48no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—84no—
455C2×D4—84no—
456C2×D4AGL(1,ℤ/8)14yes—
457C22×C4—21no—
458C22×C4—28no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—82no—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2511yescharacteristic
496SD16—48no—
497SD16—88no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—22no—
507C4⋊C4—28no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—24no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—81no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4C2×#4112yes—
541C4⋊C4C2×#4112yes—
542C22⋊C4—28no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—48no—
547C22⋊C4—44no—
548C22⋊C4—48no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C42—24no—
555C42—24no—
556C42—24no—
557C42—24no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—21no—
563C42—21no—
564C42—41no—
565C42—44no—
566C42—44no—
567C42—42no—
568C42—48no—
569C42—44no—
570C42—42no—
571C42—42no—
572C42—81no—
573C42#13714yes—
574C42C2×#4112yes—
575C42C2×#4112yes—
576C42C2×#4112yes—
577C42H(5,𝔽2)12yes—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23C2×AGL(1,ℤ/8)14yes—
592C23C23×D411yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—22no—
596Q8—24no—
597Q8—24no—
598Q8—416no—
599Q8—42no—
600Q8—42no—
601Q8—164no—
602Q8#55514yescomplemented
603D4—28no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—84no—
609D4—84no—
610D4—168no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—48no—
636C2×C4—416no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—48no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—44no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4#52512yes—
663C2×C4#52512yes—
664C2×C4#54612yes—
665C2×C4#54612yes—
666C2×C4C2×AGL(1,ℤ/8)12yes—
667C2×C4C2×#13712yes—
668C2×C4C2×#13712yes—
669C2×C4C2×#13712yes—
670C2×C4#57712yes—
671C2×C4C22×#4111yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C2×H(5,𝔽2)11yescharacteristic
675C2×C4C2×H(5,𝔽2)11yescharacteristic
676C2×C4#58511yescharacteristic
677C8—42no—
678C8—42no—
679C8—44no—
680C22—24no—
681C22—24no—
682C22—24no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22C2×#52511yescharacteristic
695C22C2×#54611yescharacteristic
696C22#348811yescharacteristic
697C22C22×AGL(1,ℤ/8)11yescharacteristic
698C22C22×#13711yescharacteristic
699C22C2×#57711yescharacteristic
700C22#358611yescharacteristic
701C4—22no—
702C4—24no—
703C4—22no—
704C4—22no—
705C4—41no—
706C4—42no—
707C4—44no—
708C4—42no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—81no—
713C4—84no—
714C4—82no—
715C4—82no—
716C4#311214yes—
717C4#332914yes—
718C4C2×#55511yescharacteristic
719C4#347511yescharacteristic
720C4#350611yescharacteristic
721C4#351911yescharacteristic
722C2—44no—
723C2—162no—
724C2#3616011yescharacteristic
725C2C2×#311211yescharacteristic
726C2#6059911yescharacteristic
727C2C2×#332911yescharacteristic
728C2#6116111yescharacteristic
729C2#6196611yescharacteristic
730C2#6198311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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
2. C4×D4×Q8C2semidirect
3. #33198C2semidirect
4. #32903C2semidirect
5. #29141C2semidirect
6. #28252C2semidirect
7. #27631C2semidirect
8. #27239C2semidirect
9. #26741C2semidirect
10. #24965C2semidirect
11. #24661C2semidirect
12. #24239C2semidirect
13. #23100C2semidirect
14. #22799C2semidirect
15. #22759C2semidirect
16. C4×#460C2semidirect
17. #20740C2semidirect
602. Q8#555semidirect

Maximal subgroups

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

subgroupindex
#361872
C4×D4×Q82
#331982
#329032
#291412
#282522
#276312
#272392
#267412
#249652
#246612
#242392
#231002
#227992
#227592
C4×#4602
#207402