Finite Group Databasebeta

← #6388299

Subgroups — #6388299

Order 512.

1904 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3108C211yesmaximal, characteristic
2#34243C211yesmaximal, characteristic
3C2×#2355C211yesmaximal, characteristic, complemented
4#23107C214yesmaximal, complemented
5#23078C214yesmaximal, complemented
6#22814C212yesmaximal, complemented
7#22805C212yesmaximal, complemented
8#22771C212yesmaximal, complemented
9#22762C212yesmaximal, complemented
10#22050C212yesmaximal, complemented
11#22041C212yesmaximal
12#21981C211yesmaximal, characteristic, complemented
13#21867C211yesmaximal, characteristic
14#21578C212yesmaximal
15C4×#483C212yesmaximal, complemented
16#20740C211yesmaximal, characteristic, complemented
17#20535C211yesmaximal, characteristic, complemented
18C2×#529C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20#3108C4116yescomplemented
21C2×#483C2211yescharacteristic
22C2×#480C2211yescharacteristic
23C2×#479C2212yes—
24C2×#464C2212yes—
25C2×#451C2211yescharacteristic
26C2×#448C2211yescharacteristic
27C2×C4×Q16C2212yes—
28C2×C4×SD16C2212yes—
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2370C2212yes—
32#2367C2212yes—
33#2359—22no—
34#2355—21no—
35#2355—22no—
36#2355—21no—
37#2355C2212yes—
38#2355C2211yescharacteristic
39#2355C2212yes—
40#2355C2214yes—
41#2354C2211yescharacteristic
42#2352—21no—
43Q8×C4⋊C4C2211yescharacteristic
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2211yescharacteristic
46#2334—22no—
47#2314—21no—
48C4×C22⋊Q8C2212yes—
49C4×#121C2212yes—
50C2×#391C2212yes—
51C2×#387—22no—
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#382—21no—
55#2274—21no—
56C2×C4×C4⋊C4—21no—
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C4⋊C4C2211yescharacteristic
59#1992C2212yes—
60#1971C2214yes—
61#1968C2214yes—
62#1959C2212yes—
63#1948C2214yes—
64#1935C2214yes—
65#1933C2214yes—
66#1927C2214yes—
67#1927C2214yes—
68#1927C2214yes—
69#1926C2214yes—
70#1921C2214yes—
71#1920C2214yes—
72#1878C2214yes—
73#1877C2214yes—
74#1874C2212yes—
75#1872C2212yes—
76#1853C2212yes—
77#1853C2212yes—
78#1844C2212yes—
79#1834C2212yes—
80#1823C2212yes—
81#1803C2212yes—
82#1803C2212yes—
83#1803C2212yes—
84#1802C2212yes—
85#1787C2212yes—
86#1776C2214yes—
87C4×#106C2214yes—
88#1771C2214yes—
89#1770C2212yes—
90C8×C4⋊C4C2212yes—
91#1763C2214yes—
92#1762C2214yes—
93C4×#103C2214yes—
94#1749C2212yes—
95C4×C22⋊C8C2212yes—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×SD16C2312yes—
99#529—24no—
100#529C2×C418yes—
101C2×#126—21no—
102C2×#125C2311yescharacteristic
103C2×#124C2311yescharacteristic
104C2×#123—21no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×#121C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—24no—
110C2×C4×Q8C2×C418yescomplemented
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×#117—21no—
120C2×#117—21no—
121C2×#117—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C42—21no—
131C22×C42—22no—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#483C2×C418yescomplemented
135#480C2×C418yes—
136#479C2×C4116yescomplemented
137#464C2×C4116yescomplemented
138#451C2×C418yescomplemented
139#448C2×C418yes—
140C4×Q16C2×C4116yescomplemented
141C4×SD16C2×C4116yescomplemented
142#432—24no—
143#431C2×C418yescomplemented
144C2×#107C2312yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2311yescharacteristic
161C2×#97C2312yes—
162C2×#97C2314yes—
163C2×C4×C8C2314yes—
164C2×C4×C8C2312yes—
165#391—22no—
166#391—22no—
167#391C2314yes—
168#391C2314yes—
169#391C2312yes—
170#390C2312yes—
171#389—22no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#386—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#382—21no—
199#382—22no—
200#382—22no—
201#382—24no—
202#382—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—24no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2314yes—
222C4×C4⋊C4C2314yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C2×#95—22no—
232C2×#95—22no—
233C2×#95—22no—
234C2×#95D412yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8C22×C414yes—
248C22×Q8C2411yescharacteristic
249C22×Q8C2412yes—
250C22×D4—21no—
251C22×D4C2411yescharacteristic
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C2×Q16—42no—
263C2×Q16—42no—
264C2×Q16C22×C418yes—
265C2×SD16—42no—
266C2×SD16—42no—
267C2×SD16C22×C418yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271#126—28no—
272#125—22no—
273#125C22×C414yes—
274#124—22no—
275#124C22×C414yes—
276#123—28no—
277C22⋊Q8—22no—
278C22⋊Q8C22×C414yes—
279#121—22no—
280#121C22×C414yes—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—22no—
286C4×Q8—44no—
287C4×Q8C22×C418yes—
288C4×Q8C22×C418yes—
289C4×Q8C22×C414yes—
290C4×D4—28no—
291C4×D4—22no—
292C4×D4—44no—
293C4×D4—84no—
294C4×D4C22×C414yes—
295#117—28no—
296#117—44no—
297#117—82no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
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—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4C22×C414yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C42—28no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—44no—
358C2×C42—84no—
359C2×C42C22×C414yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42#4114yes—
365C2×C42C2412yes—
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#107C22×C418yes—
370#106—44no—
371#106—44no—
372#106C22×C418yes—
373C4⋊C8—24no—
374C4⋊C8C22×C418yes—
375#103—44no—
376#103—44no—
377#103—44no—
378#103C22×C418yes—
379#103C22×C418yes—
380#103C22×C418yes—
381#102—44no—
382#102C22×C418yes—
383C22⋊C8—24no—
384C22⋊C8C22×C414yes—
385C22⋊C8C22×C414yes—
386#97—24no—
387#97—44no—
388C4×C8—44no—
389C4×C8C22×C418yes—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—21no—
409C24C22×D411yescharacteristic
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—48no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—48no—
423C2×Q8C23×C414yes—
424C2×Q8C23×C412yes—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4C23×C412yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4C4×D414yes—
468C22×C4C4×D414yes—
469C22×C4C23×C412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4112yes—
480C22×C4C2511yescharacteristic
481Q16—48no—
482SD16—48no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8C23×C414yes—
491C2×C8C23×C414yes—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4#13714yes—
524C4⋊C4C23×C414yes—
525C4⋊C4C23×C412yes—
526C4⋊C4C23×C412yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C42—28no—
537C42—28no—
538C42—24no—
539C42—24no—
540C42—24no—
541C42—42no—
542C42—42no—
543C42—44no—
544C42—84no—
545C42—84no—
546C42—82no—
547C42—82no—
548C42—82no—
549C42C4×D414yes—
550C42C4×D414yes—
551C42C23×C412yes—
552C42C23×C412yes—
553C42C2×#4112yes—
554C42C2×#4112yes—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23C2×C4×D412yes—
567C23#57712yes—
568C23#57712yes—
569C23C23×D411yescharacteristic
570Q8—24no—
571Q8—24no—
572Q8—24no—
573Q8—48no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577D4—24no—
578D4—48no—
579D4—42no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
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—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—48no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—84no—
632C2×C4—88no—
633C2×C4—164no—
634C2×C4C2×C4×D412yes—
635C2×C4C2×C4×D412yes—
636C2×C4C2×C4×D412yes—
637C2×C4C2×C4×D412yes—
638C2×C4C2×C4×D412yes—
639C2×C4C2×#13714yes—
640C2×C4C2×#13712yes—
641C2×C4#57512yes—
642C2×C4C24×C411yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4#58512yes—
645C2×C4#58512yes—
646C8—24no—
647C8—24no—
648C8—44no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22#295214yes—
663C22C22×C4×D411yescharacteristic
664C22#348511yescharacteristic
665C22#348711yescharacteristic
666C22C22×#13711yescharacteristic
667C22C2×#57511yescharacteristic
668C22C2×#57711yescharacteristic
669C22C2×#57711yescharacteristic
670C4—22no—
671C4—22no—
672C4—24no—
673C4—22no—
674C4—22no—
675C4—44no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—82no—
684C4—84no—
685C4—82no—
686C4C4×#13714yes—
687C4#342812yes—
688C4#349312yes—
689C2—24no—
690C2—82no—
691C2#3420111yescharacteristic
692C2C2×C4×#13711yescharacteristic
693C2C2×#295211yescharacteristic
694C2#6173511yescharacteristic
695C2#6173911yescharacteristic
696C2#6190411yescharacteristic
697C2#6190611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C2×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
3. C2×#2355C2semidirect
4. #23107C2semidirect
5. #23078C2semidirect
6. #22814C2semidirect
7. #22805C2semidirect
8. #22771C2semidirect
9. #22762C2semidirect
10. #22050C2semidirect
12. #21981C2semidirect
15. C4×#483C2semidirect
16. #20740C2semidirect
17. #20535C2semidirect
20. #3108C4semidirect
110. C2×C4×Q8C2×C4semidirect
134. #483C2×C4semidirect
136. #479C2×C4semidirect
137. #464C2×C4semidirect
138. #451C2×C4semidirect
140. C4×Q16C2×C4semidirect
141. C4×SD16C2×C4semidirect
143. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31082
#342432
C2×#23552
#231072
#230782
#228142
#228052
#227712
#227622
#220502
#220412
#219812
#218672
#215782
C4×#4832
#207402
#205352