Finite Group Databasebeta

← #6606559

Subgroups — #6606559

Order 512.

2235 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3335C212yesmaximal, complemented
2#36179C211yesmaximal, characteristic
3C4×#554C211yesmaximal, characteristic, complemented
4#33196C211yesmaximal, characteristic, complemented
5#33107C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28280C211yesmaximal, characteristic, complemented
8#27647C212yesmaximal, complemented
9#25451C212yesmaximal, complemented
10#25204C214yesmaximal, complemented
11#24753C212yesmaximal, complemented
12#24678C212yesmaximal, complemented
13#24578C214yesmaximal, complemented
14D4×C4⋊C8C211yesmaximal, characteristic, complemented
15#22485C212yesmaximal, complemented
16C4×#460C211yesmaximal, characteristic, complemented
17C2×#1948C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×#547—21no—
20C2×#547—21no—
21C2×#547C2212yes—
22#3335—216no—
23C2×#498C2212yes—
24C2×#481C2214yes—
25C2×#483C2214yes—
26C2×#475C2212yes—
27C2×#476C2212yes—
28C2×#465C2214yes—
29C2×#460C2211yescharacteristic
30C2×C4×SD16C2212yes—
31C2×C8×D4C2212yes—
32C22×#106C2214yes—
33C22×C4⋊C8C2212yes—
34#2868C2211yescharacteristic
35#2856C2212yes—
36#2542—21no—
37#2542—21no—
38#2542C2212yes—
39#2501—22no—
40#2501—22no—
41#2501C2214yes—
42#2370—21no—
43#2369C2211yescharacteristic
44#2368—22no—
45#2360C2212yes—
46#2355C2211yescharacteristic
47#2349—21no—
48D4×C4⋊C4C2211yescharacteristic
49#2345C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×#125—21no—
53C4×C22⋊Q8—22no—
54C4×C22⋊Q8C2212yes—
55C2×#384C2212yes—
56C42×Q8C2211yescharacteristic
57C42×D4—21no—
58C42×D4C2211yescharacteristic
59C2×C4×C4⋊C4C2212yes—
60#2084C2214yes—
61#2084C2214yes—
62#2072C2214yes—
63#2065C2214yes—
64#2065C2214yes—
65#2055C2214yes—
66#1990C2212yes—
67#1988C2212yes—
68#1982C2211yescharacteristic
69#1982C2212yes—
70#1948C2212yes—
71#1948C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1948C2214yes—
76#1948C2212yes—
77#1948C2211yescharacteristic
78#1939C2212yes—
79#1926C2212yes—
80#1926C2214yes—
81#1926C2214yes—
82#1893C2214yes—
83#1880C2212yes—
84#1874C2214yes—
85#1850C2214yes—
86#1842C2211yescharacteristic
87#1823C2214yes—
88#1803C2212yes—
89#1803C2212yes—
90#1803C2211yescharacteristic
91#1802C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C2×#340C2214yes—
96C22×#41—22no—
97C22×SD16C2312yes—
98C23×C8C2314yes—
99#554—24no—
100#554D418yescomplemented
101#547—28no—
102#547—28no—
103#547—48no—
104#547—48no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#125—21no—
111C2×#123—24no—
112C2×#123—24no—
113C2×#123—24no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8C2312yes—
119C2×#121—22no—
120C2×#121—22no—
121C2×#121C2314yes—
122C2×C4×Q8—21no—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4D411yescharacteristic
131C2×C4×D4D411yescharacteristic
132C2×C4×D4C2312yes—
133C2×C4×D4C2311yescharacteristic
134C22×C4⋊C4—22no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C42—22no—
140C22×C42C2312yes—
141#498—28no—
142#483—216no—
143#481—216no—
144#476—28no—
145#475—28no—
146#465—216no—
147#460—24no—
148C4×SD16—28no—
149C8×D4—28no—
150C2×#106—216no—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×#106C2314yes—
155C2×#106C2312yes—
156C2×C4⋊C8—28no—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2311yescharacteristic
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2314yes—
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2311yescharacteristic
163C2×#103C2314yes—
164C2×#103C2312yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#102C2314yes—
170C2×C22⋊C8C2314yes—
171C2×C4×C8C2312yes—
172#399—24no—
173#399—24no—
174#399—24no—
175#391—21no—
176#391—22no—
177#391C2312yes—
178#390—21no—
179#390—21no—
180#390—22no—
181#390—22no—
182#390C2312yes—
183#390C2314yes—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—22no—
188#389C2312yes—
189#389C2312yes—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#385—22no—
196#385—22no—
197#385—22no—
198#384—21no—
199#384—21no—
200#384—22no—
201#384—22no—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2312yes—
206#384C2314yes—
207#382—21no—
208#380—22no—
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4D412yes—
233C4×C22⋊C4C2312yes—
234C2×#95—22no—
235C43—21no—
236C43C2311yescharacteristic
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#336C2314yes—
245#336C2312yes—
246C2×#41—48no—
247C2×#41—48no—
248C2×#41—48no—
249C22×Q8—21no—
250C22×Q8—22no—
251C22×Q8C2411yescharacteristic
252C22×D4—24no—
253C22×D4—21no—
254C22×D4—21no—
255C22×D4C2×D411yescharacteristic
256C22×D4C2412yes—
257C23×C4—22no—
258C23×C4—24no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2412yes—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—42no—
267C22×C8—28no—
268C22×C8—28no—
269C22×C8—44no—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—22no—
275C4⋊Q8—24no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8C2×D414yes—
279#127—22no—
280#127—22no—
281#125—28no—
282#123—416no—
283#123—416no—
284#123—416no—
285C22⋊Q8—24no—
286C22⋊Q8—216no—
287C22⋊Q8—48no—
288C22⋊Q8—48no—
289C22⋊Q8—48no—
290C22⋊Q8C2×D418yes—
291#121—28no—
292#121—28no—
293#121—48no—
294#121—48no—
295C4×Q8—22no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8—41no—
300C4×Q8—82no—
301C4×Q8C2×D414yes—
302C4×D4—28no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—24no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—44no—
309C4×D4SD1618yescomplemented
310C4×D4C2×D414yes—
311C4×D4#4114yes—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—28no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—48no—
339C2×C4⋊C4—48no—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D418yes—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C42—21no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—21no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—84no—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#106—24no—
385#106—28no—
386#106—28no—
387#106—28no—
388#106—28no—
389C4⋊C8—28no—
390C4⋊C8—24no—
391C4⋊C8—24no—
392C4⋊C8—22no—
393C4⋊C8—42no—
394C4⋊C8#4114yescomplemented
395#103—24no—
396#103—28no—
397#102—24no—
398#102—24no—
399#102—28no—
400#102—24no—
401#102—44no—
402C22⋊C8—28no—
403C4×C8—24no—
404#95—21no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—21no—
412#95—21no—
413#95—22no—
414#95—22no—
415#95C2×D412yes—
416#95C2×D412yes—
417#95C2×D412yes—
418C24—22no—
419C24C22×D412yes—
420#41—832no—
421C2×Q8—24no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—44no—
430C2×Q8—81no—
431C2×Q8—81no—
432C2×Q8C22×D412yes—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—48no—
442C2×D4—48no—
443C2×D4—48no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4#12114yes—
449C2×D4C2×SD1614yes—
450C2×D4AGL(1,ℤ/8)14yes—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—28no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4C2×SD1618yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4114yes—
492C22×C4C2511yescharacteristic
493SD16—48no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—416no—
503C2×C8C2×#4114yes—
504C4⋊C4—28no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—28no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—82no—
535C4⋊C4—81no—
536C4⋊C4—82no—
537C4⋊C4#12114yes—
538C4⋊C4C2×SD1614yes—
539C4⋊C4C22×D412yes—
540C4⋊C4C22×D414yes—
541C4⋊C42−1+412yes—
542C22⋊C4—28no—
543C22⋊C4—24no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—48no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4#12118yes—
552C22⋊C4C2×SD1618yes—
553C42—24no—
554C42—24no—
555C42—24no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—21no—
561C42—21no—
562C42—41no—
563C42—42no—
564C42—42no—
565C42—42no—
566C42—42no—
567C42—44no—
568C42—81no—
569C42—82no—
570C42—82no—
571C42C2×SD1614yes—
572C42C22×D412yes—
573C42C2×#4112yes—
574C42C2×#4112yes—
575C422−1+412yes—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23C2×#12114yes—
588C23C22×SD1614yes—
589C23C2×AGL(1,ℤ/8)14yes—
590C23C23×D411yescharacteristic
591Q8—22no—
592Q8—22no—
593Q8—41no—
594Q8—41no—
595Q8—84no—
596Q8—88no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—84no—
602D4—816no—
603D4—84no—
604D4#460116yescomplemented
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—216no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—48no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—88no—
652C2×C4—88no—
653C2×C4—816no—
654C2×C4—88no—
655C2×C4C2×#12112yes—
656C2×C4C2×#12114yes—
657C2×C4C2×#12112yes—
658C2×C4#54712yes—
659C2×C4#54712yes—
660C2×C4C22×SD1612yes—
661C2×C4C22×SD1614yes—
662C2×C4C22×SD1612yes—
663C2×C4C22×SD1612yes—
664C2×C4C2×AGL(1,ℤ/8)12yes—
665C2×C4#57712yes—
666C2×C4C23×D411yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C2×C4C2×2−1+411yescharacteristic
671C2×C4#58511yescharacteristic
672C8—24no—
673C8—44no—
674C8—42no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22C2×#46018yes—
689C22C22×#12111yescharacteristic
690C22C2×#54711yescharacteristic
691C22#348911yescharacteristic
692C22C23×SD1611yescharacteristic
693C22C22×AGL(1,ℤ/8)11yescharacteristic
694C22C2×#57711yescharacteristic
695C22#358611yescharacteristic
696C4—22no—
697C4—24no—
698C4—22no—
699C4—22no—
700C4—24no—
701C4—42no—
702C4—44no—
703C4—42no—
704C4—42no—
705C4—41no—
706C4—42no—
707C4—81no—
708C4—82no—
709C4—82no—
710C4—84no—
711C4C2×#46014yes—
712C4#333514yescomplemented
713C4D4×#4111yescharacteristic
714C4#348211yescharacteristic
715C4#349911yescharacteristic
716C4#351511yescharacteristic
717C2—28no—
718C2—162no—
719C2#3625911yescharacteristic
720C2C22×#46011yescharacteristic
721C2#6042911yescharacteristic
722C2C2×#333511yescharacteristic
723C2#6125211yescharacteristic
724C2#6209911yescharacteristic
725C2#6212811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
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. C2×#3335C2semidirect
3. C4×#554C2semidirect
4. #33196C2semidirect
5. #33107C2semidirect
6. #28687C2semidirect
7. #28280C2semidirect
8. #27647C2semidirect
9. #25451C2semidirect
10. #25204C2semidirect
11. #24753C2semidirect
12. #24678C2semidirect
13. #24578C2semidirect
14. D4×C4⋊C8C2semidirect
15. #22485C2semidirect
16. C4×#460C2semidirect
17. C2×#1948C2semidirect
100. #554D4semidirect
309. C4×D4SD16semidirect
394. C4⋊C8#41semidirect
604. D4#460semidirect
712. C4#3335semidirect

Maximal subgroups

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

subgroupindex
C2×#33352
#361792
C4×#5542
#331962
#331072
#286872
#282802
#276472
#254512
#252042
#247532
#246782
#245782
D4×C4⋊C82
#224852
C4×#4602
C2×#19482