Finite Group Databasebeta

← #6627949

Subgroups — #6627949

Order 512.

2135 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#32905C211yesmaximal, characteristic
4#32905C212yesmaximal
5#29335C212yesmaximal, complemented
6#29244C212yesmaximal, complemented
7#27972C211yesmaximal, characteristic, complemented
8#27681C211yesmaximal, characteristic, complemented
9#27573C214yesmaximal, complemented
10#26182C214yesmaximal, complemented
11#26106C212yesmaximal, complemented
12#24730C212yesmaximal, complemented
13#23002C212yesmaximal, complemented
14#22762C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21453C212yesmaximal
17C4×#450C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4—21no—
21C2×#461C2212yes—
22C2×#461C2214yes—
23C2×#450C2211yescharacteristic
24C2×#450C2212yes—
25C2×#440C2214yes—
26C2×C4×SD16C2212yes—
27#2857C2211yescharacteristic
28#2857C2212yes—
29#2585—22no—
30#2569—22no—
31#2552C2211yescharacteristic, complemented
32#2546C2211yescharacteristic, complemented
33#2511C2212yescomplemented
34#2509—21no—
35#2446C2212yescomplemented
36#2442—21no—
37#2370—21no—
38#2365C2211yescharacteristic, complemented
39#2355C2211yescharacteristic, complemented
40D4×C4⋊C4—22no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2214yes—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46#2345—21no—
47C4×C22⋊Q8C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49C2×#389C2212yes—
50C2×#385—22no—
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2100C2212yescomplemented
55#2099C2214yescomplemented
56#2095C2212yescomplemented
57#2086C2214yescomplemented
58#2072C2214yescomplemented
59#2066C2214yescomplemented
60#2043C2214yescomplemented
61#1956C2212yescomplemented
62#1947C2212yescomplemented
63#1939C2212yescomplemented
64#1935C2214yescomplemented
65#1935C2214yescomplemented
66#1920C2214yescomplemented
67#1917C2214yescomplemented
68#1877C2212yes—
69#1877C2214yes—
70#1877C2214yes—
71#1877C2214yes—
72#1877C2214yes—
73#1872C2212yes—
74#1872C2212yes—
75#1872C2212yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1864C2212yes—
79#1863C2214yescomplemented
80#1862C2214yes—
81#1853C2211yescharacteristic
82#1852C2212yes—
83#1834C2212yescomplemented
84#1803C2212yescomplemented
85#1802C2214yes—
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1786C2211yescharacteristic
90#1770C2212yescomplemented
91#1762C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4—22no—
98C22×SD16C2312yes—
99C22×SD16C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125—21no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2311yescharacteristic
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2312yes—
107C2×C4×D4—28no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—48no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2314yes—
115C2×C4×D4C2314yes—
116C2×C4×D4C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—48no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#461—28no—
133#461—48no—
134#450—24no—
135#450—28no—
136#440—48no—
137C4×SD16—28no—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2311yescharacteristic
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#400—22no—
158#400—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#391C2311yescharacteristic
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2312yes—
179#389C2314yes—
180#389C2311yescharacteristic
181#387—22no—
182#387—22no—
183#387—21no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#385—21no—
190#385—22no—
191#385—24no—
192#385—24no—
193#385—24no—
194#385—24no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#384C2312yes—
201#382—21no—
202#380—21no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380—22no—
207#380—24no—
208#380—22no—
209#380C2311yescharacteristic
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224C2×#95—22no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C22×Q8C2411yescharacteristic
239C22×Q8C2412yes—
240C22×D4—24no—
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4#4114yes—
245C22×D4C2412yes—
246C23×C4—28no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—44no—
254C23×C4—48no—
255C23×C4—44no—
256C23×C4—48no—
257C23×C4—84no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—28no—
266C2×SD16—28no—
267C2×SD16—28no—
268C2×SD16—24no—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C4⋊Q8—24no—
273C4⋊Q8—44no—
274#125—44no—
275C22⋊Q8—24no—
276C22⋊Q8—22no—
277C22⋊Q8—24no—
278C22⋊Q8—22no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—48no—
291C4×D4—48no—
292C4×D4#4118yes—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—416no—
320C2×C4⋊C4—48no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—48no—
346C2×C42—24no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—44no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—44no—
361#106—24no—
362C4⋊C8—24no—
363C4⋊C8—44no—
364#103—24no—
365#103—28no—
366#103—44no—
367#103—44no—
368#103—44no—
369#102—24no—
370#102—28no—
371#102—24no—
372#102—24no—
373C22⋊C8—24no—
374C22⋊C8—22no—
375C22⋊C8—22no—
376#97—44no—
377C4×C8—24no—
378#95—24no—
379#95—24no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—24no—
400C24—24no—
401C24—24no—
402C24—22no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24C22×D411yescharacteristic
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—216no—
419C2×D4—216no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4C2×#4114yes—
425C2×D4C2×#4114yes—
426C2×D4H(5,𝔽2)14yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—42no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—432no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—832no—
468C22×C4#13714yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C42−1+412yes—
480C22×C4C2511yescharacteristic
481SD16—48no—
482SD16—416no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4AGL(1,ℤ/8)14yes—
529C4⋊C4C2×#4114yes—
530C22⋊C4—28no—
531C22⋊C4—28no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—24no—
542C42—42no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—82no—
547C42—82no—
548C42C2×#4114yes—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—216no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—48no—
561C23—48no—
562C23—42no—
563C23—48no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23#54812yes—
568C23C2×#13712yes—
569C23#57712yes—
570C23C23×D411yescharacteristic
571Q8—42no—
572Q8—44no—
573Q8—42no—
574Q8—44no—
575D4—216no—
576D4—24no—
577D4—24no—
578D4—24no—
579D4#55318yes—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—416no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—416no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—816no—
633C2×C4—84no—
634C2×C4—816no—
635C2×C4—88no—
636C2×C4#52512yes—
637C2×C4#52512yes—
638C2×C4#54812yes—
639C2×C4C2×AGL(1,ℤ/8)14yes—
640C2×C4C2×AGL(1,ℤ/8)12yes—
641C2×C4C2×#13712yes—
642C2×C4C2×#13712yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4112yes—
645C2×C4C2×H(5,𝔽2)12yes—
646C2×C4C2×2−1+411yescharacteristic
647C8—42no—
648C8—44no—
649C8—42no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—28no—
657C22—432no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22#337214yes—
665C22C2×#52511yescharacteristic
666C22C2×#54811yescharacteristic
667C22#348411yescharacteristic
668C22C22×AGL(1,ℤ/8)11yescharacteristic
669C22C22×#13711yescharacteristic
670C22C2×#57711yescharacteristic
671C22#358611yescharacteristic
672C4—22no—
673C4—24no—
674C4—22no—
675C4—22no—
676C4—44no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—41no—
681C4—82no—
682C4—81no—
683C4—81no—
684C4—84no—
685C4—81no—
686C4—81no—
687C4—82no—
688C4#311014yes—
689C4C2×#55312yes—
690C4#349212yes—
691C2—24no—
692C2—48no—
693C2#3658111yescharacteristic
694C2C2×#311011yescharacteristic
695C2#6058511yescharacteristic
696C2C2×#337211yescharacteristic
697C2#6131711yescharacteristic
698C2#6158611yescharacteristic
699C2#6160011yescharacteristic
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
1. #36599C2semidirect
5. #29335C2semidirect
6. #29244C2semidirect
7. #27972C2semidirect
8. #27681C2semidirect
9. #27573C2semidirect
10. #26182C2semidirect
11. #26106C2semidirect
12. #24730C2semidirect
13. #23002C2semidirect
14. #22762C2semidirect
15. #22629C2semidirect
31. #2552C22semidirect
32. #2546C22semidirect
33. #2511C22semidirect
35. #2446C22semidirect
38. #2365C22semidirect
39. #2355C22semidirect
54. #2100C22semidirect
55. #2099C22semidirect
56. #2095C22semidirect
57. #2086C22semidirect
58. #2072C22semidirect
59. #2066C22semidirect
60. #2043C22semidirect
61. #1956C22semidirect
62. #1947C22semidirect
63. #1939C22semidirect
64. #1935C22semidirect
65. #1935C22semidirect
66. #1920C22semidirect
67. #1917C22semidirect
79. #1863C22semidirect
83. #1834C22semidirect
84. #1803C22semidirect
90. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#365992
C2×D4×C4⋊C42
#329052
#293352
#292442
#279722
#276812
#275732
#261822
#261062
#247302
#230022
#227622
#226292
#214532
C4×#4502