Finite Group Databasebeta

← #6578965

Subgroups — #6578965

Order 512.

2415 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
1C2×#3110C212yesmaximal
2#36133C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#33009C212yesmaximal
5#32905C211yesmaximal, characteristic
6#27910C211yesmaximal, characteristic, complemented
7#27648C211yesmaximal, characteristic, complemented
8#26384C214yesmaximal, complemented
9#26182C214yesmaximal, complemented
10#26040C212yesmaximal, complemented
11#24730C212yesmaximal, complemented
12#24606C212yesmaximal, complemented
13#24064C212yesmaximal, complemented
14#23002C212yesmaximal, complemented
15#22671C212yesmaximal, complemented
16#22603C211yesmaximal, characteristic, complemented
17C4×#450C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4—21no—
22#3110—216no—
23C2×#484C2212yes—
24C2×#484C2212yes—
25C2×#484C2212yes—
26C2×#474C2214yes—
27C2×#461C2214yes—
28C2×#461C2212yes—
29C2×#450C2212yes—
30C2×#450C2211yescharacteristic
31C2×#440C2214yes—
32C2×#440C2214yes—
33C2×C4×SD16C2212yes—
34C2×#432C2212yes—
35#2857C2211yescharacteristic
36#2855C2212yes—
37#2551C2211yescharacteristic, complemented
38#2546C2211yescharacteristic, complemented
39#2506—21no—
40#2462—22no—
41#2451C2212yescomplemented
42#2446C2212yescomplemented
43#2440—21no—
44#2438—22no—
45#2365—21no—
46#2365C2211yescharacteristic, complemented
47Q8×C4⋊C4C2211yescharacteristic, complemented
48D4×C4⋊C4—21no—
49D4×C4⋊C4—22no—
50D4×C4⋊C4—21no—
51D4×C4⋊C4C2214yes—
52D4×C4⋊C4C2212yes—
53D4×C4⋊C4C2212yes—
54#2344—21no—
55C4×C22⋊Q8C2211yescharacteristic
56C2×#389C2212yes—
57C2×#385—22no—
58C2×#384C2211yescharacteristic
59C2×#380—21no—
60C2×C4×C4⋊C4C2211yescharacteristic
61#2095C2214yescomplemented
62#2086C2214yescomplemented
63#2083C2212yescomplemented
64#2072C2212yescomplemented
65#2066C2214yescomplemented
66#2057C2214yescomplemented
67#2043C2214yescomplemented
68#1956C2212yescomplemented
69#1947C2212yescomplemented
70#1944C2214yescomplemented
71#1939C2212yescomplemented
72#1935C2214yescomplemented
73#1917C2214yescomplemented
74#1877C2214yes—
75#1877C2214yes—
76#1877C2212yes—
77#1877C2214yes—
78#1872C2211yescharacteristic
79#1872C2212yes—
80#1864C2212yes—
81#1863C2214yescomplemented
82#1863C2214yescomplemented
83#1862C2214yes—
84#1862C2214yes—
85#1853C2211yescharacteristic
86#1845C2212yescomplemented
87#1824C2212yes—
88#1817C2212yes—
89#1803C2212yescomplemented
90#1802C2212yes—
91#1797C2211yescharacteristic
92#1786C2211yescharacteristic
93#1770C2212yescomplemented
94C4×#102C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C24×C4—22no—
99C22×SD16C2314yes—
100C22×SD16C2312yes—
101#533—28no—
102#533—44no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#125—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C4×Q8—22no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×D4—28no—
116C2×C4×D4—28no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—44no—
122C2×C4×D4C2312yes—
123C2×C4×D4C2314yes—
124C2×C4×D4C2314yes—
125C2×C4×D4C2312yes—
126C2×#117—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—24no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—48no—
134C22×C4⋊C4D411yescharacteristic
135C22×C4⋊C4D411yescharacteristic
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—21no—
139C22×C22⋊C4—22no—
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#484—28no—
143#484—28no—
144#484—44no—
145#474—216no—
146#461—216no—
147#461—28no—
148#450—28no—
149#450D418yes—
150#440—216no—
151#440—216no—
152C4×SD16—28no—
153#432—28no—
154C2×#107C2314yes—
155C2×#107C2314yes—
156C2×#107C2314yes—
157C2×#106C2312yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2311yescharacteristic
171C2×C22⋊C8C2311yescharacteristic
172C2×#97C2314yes—
173C2×C4×C8C2312yes—
174#400—22no—
175#398—22no—
176#398—22no—
177#398—22no—
178#395—22no—
179#395—22no—
180#391—22no—
181#391C2312yes—
182#391C2311yescharacteristic
183#391C2312yes—
184#391C2312yes—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—22no—
192#389—22no—
193#389—22no—
194#389C2314yes—
195#389C2311yescharacteristic
196#389C2312yes—
197#389C2312yes—
198#389C2314yes—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—21no—
204#387—22no—
205#387—22no—
206#385—22no—
207#385—21no—
208#385—24no—
209#385—24no—
210#385—24no—
211#385—24no—
212#384—21no—
213#384—21no—
214#384—21no—
215#384C2311yescharacteristic
216#384C2312yes—
217#384C2312yes—
218#381—22no—
219#380—21no—
220#380—21no—
221#380—22no—
222#380—24no—
223#380—22no—
224#380C2311yescharacteristic
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4D411yescharacteristic
229C4×C4⋊C4D412yes—
230C4×C4⋊C4D411yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2311yescharacteristic
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2311yescharacteristic
235C4×C22⋊C4—21no—
236C2×#95—22no—
237C2×#95D412yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2312yes—
243#336C2312yes—
244#336C2314yes—
245#336C2314yes—
246#336C2312yes—
247C25—21no—
248C22×Q8—22no—
249C22×Q8C2412yes—
250C22×Q8C2411yescharacteristic
251C22×D4—24no—
252C22×D4—24no—
253C22×D4—21no—
254C22×D4—22no—
255C22×D4C2×D414yes—
256C22×D4C2412yes—
257C23×C4—28no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—24no—
263C23×C4—24no—
264C23×C4—44no—
265C23×C4—48no—
266C23×C4—48no—
267C23×C4—44no—
268C23×C4—84no—
269C23×C4C2×D412yes—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D412yes—
272C23×C4C2×D411yescharacteristic
273C23×C4C2411yescharacteristic
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—28no—
277C2×SD16—28no—
278C2×SD16—28no—
279C2×SD16C2×D418yes—
280C22×C8C2414yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—42no—
287C4⋊Q8—42no—
288C4⋊Q8—44no—
289#125—48no—
290C22⋊Q8—22no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—48no—
294C22⋊Q8C2×D414yes—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—42no—
299C4×Q8—42no—
300C4×Q8—48no—
301C4×D4—28no—
302C4×D4—28no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—28no—
306C4×D4—28no—
307C4×D4—24no—
308C4×D4—44no—
309C4×D4—44no—
310C4×D4—48no—
311C4×D4—48no—
312C4×D4#4118yes—
313#117—48no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—416no—
343C2×C4⋊C4—48no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4—48no—
346C2×C4⋊C4—48no—
347C2×C4⋊C4—82no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2414yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—28no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—48no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—24no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—28no—
385#107—28no—
386#107—44no—
387#106—24no—
388C4⋊C8—28no—
389C4⋊C8—24no—
390#103—28no—
391#103—28no—
392#103—24no—
393#103—28no—
394#103—44no—
395#102—28no—
396#102—24no—
397#102—24no—
398#102C2×D418yes—
399C22⋊C8—24no—
400C22⋊C8—22no—
401C22⋊C8C2×D414yes—
402#97—28no—
403C4×C8—24no—
404#95—24no—
405#95—24no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—22no—
410#95—22no—
411#95—21no—
412#95—21no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95C2×D412yes—
420#95C2×D412yes—
421#95C2×D412yes—
422#95C2×D412yes—
423C24—24no—
424C24—24no—
425C24—24no—
426C24—22no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24—42no—
431C24C22×D411yescharacteristic
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—82no—
442C2×Q8C22×D412yes—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—216no—
446C2×D4—216no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4C22×D414yes—
452C2×D4C2×#4114yes—
453C2×D42−1+414yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—28no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—28no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—41no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—432no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—84no—
493C22×C4—84no—
494C22×C4—832no—
495C22×C4#13714yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4H(5,𝔽2)12yes—
507C22×C4C2511yescharacteristic
508SD16—216no—
509SD16—416no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—24no—
513C2×C8—22no—
514C2×C8—24no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—24no—
518C2×C8C22×D414yes—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—48no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—48no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—81no—
551C4⋊C4—81no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C4⋊C4AGL(1,ℤ/8)14yes—
556C4⋊C4C22×D412yes—
557C4⋊C4C2×#4114yes—
558C22⋊C4—28no—
559C22⋊C4—24no—
560C22⋊C4—28no—
561C22⋊C4—48no—
562C22⋊C4—48no—
563C22⋊C4—44no—
564C22⋊C4—42no—
565C42—22no—
566C42—22no—
567C42—24no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—44no—
575C42—44no—
576C42—42no—
577C42—42no—
578C42C2×#4114yes—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—216no—
584C23—24no—
585C23—24no—
586C23—24no—
587C23—48no—
588C23—48no—
589C23—48no—
590C23—48no—
591C23—48no—
592C23—42no—
593C23—48no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23D4212yes—
598C23C2×#13712yes—
599C23#57812yes—
600C23C23×D411yescharacteristic
601Q8—24no—
602Q8—42no—
603Q8—44no—
604Q8—44no—
605Q8—88no—
606D4—216no—
607D4—24no—
608D4—24no—
609D4—24no—
610D4#54718yes—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—416no—
640C2×C4—48no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—48no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—416no—
649C2×C4—41no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—41no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—816no—
665C2×C4—816no—
666C2×C4—88no—
667C2×C4#52512yes—
668C2×C4#52512yes—
669C2×C4D4212yes—
670C2×C4C2×AGL(1,ℤ/8)14yes—
671C2×C4C2×AGL(1,ℤ/8)12yes—
672C2×C4C2×#13712yes—
673C2×C4C2×#13712yes—
674C2×C4C23×D411yescharacteristic
675C2×C4C22×#4112yes—
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C2×C4C2×2−1+412yes—
678C8—24no—
679C8—44no—
680C8—42no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—28no—
688C22—432no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22#327614yes—
696C22C2×#52511yescharacteristic
697C22C2×D4211yescharacteristic
698C22#348611yescharacteristic
699C22C22×AGL(1,ℤ/8)11yescharacteristic
700C22C22×#13711yescharacteristic
701C22C2×#57811yescharacteristic
702C22#358511yescharacteristic
703C4—22no—
704C4—22no—
705C4—24no—
706C4—22no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—44no—
711C4—42no—
712C4—42no—
713C4—41no—
714C4—82no—
715C4—84no—
716C4—82no—
717C4—81no—
718C4—81no—
719C4#311014yescomplemented
720C4C2×#54712yes—
721C4#349212yes—
722C2—24no—
723C2—48no—
724C2#3610311yescharacteristic
725C2C2×#311011yescharacteristic
726C2#6058911yescharacteristic
727C2C2×#327611yescharacteristic
728C2D4×AGL(1,ℤ/8)11yescharacteristic
729C2#6182611yescharacteristic
730C2#6183711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×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
2. #36133C2semidirect
6. #27910C2semidirect
7. #27648C2semidirect
8. #26384C2semidirect
9. #26182C2semidirect
10. #26040C2semidirect
11. #24730C2semidirect
12. #24606C2semidirect
13. #24064C2semidirect
14. #23002C2semidirect
15. #22671C2semidirect
16. #22603C2semidirect
37. #2551C22semidirect
38. #2546C22semidirect
41. #2451C22semidirect
42. #2446C22semidirect
46. #2365C22semidirect
47. Q8×C4⋊C4C22semidirect
61. #2095C22semidirect
62. #2086C22semidirect
63. #2083C22semidirect
64. #2072C22semidirect
65. #2066C22semidirect
66. #2057C22semidirect
67. #2043C22semidirect
68. #1956C22semidirect
69. #1947C22semidirect
70. #1944C22semidirect
71. #1939C22semidirect
72. #1935C22semidirect
73. #1917C22semidirect
81. #1863C22semidirect
82. #1863C22semidirect
86. #1845C22semidirect
89. #1803C22semidirect
93. #1770C22semidirect
719. C4#3110semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
#361332
C2×D4×C4⋊C42
#330092
#329052
#279102
#276482
#263842
#261822
#260402
#247302
#246062
#240642
#230022
#226712
#226032
C4×#4502