Finite Group Databasebeta

← #6626296

Subgroups — #6626296

Order 512.

2111 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#32959C212yesmaximal
4#32953C211yesmaximal, characteristic
5#29303C212yesmaximal, complemented
6#29241C212yesmaximal, complemented
7#27950C211yesmaximal, characteristic, complemented
8#27680C211yesmaximal, characteristic, complemented
9#27310C214yesmaximal, complemented
10#26398C214yesmaximal, complemented
11#26041C212yesmaximal, complemented
12#24816C212yesmaximal, complemented
13#22968C212yesmaximal, complemented
14#22671C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21453C212yesmaximal
17C4×#450C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4—21no—
21C2×#478C2214yes—
22C2×#474C2212yes—
23C2×#450C2211yescharacteristic
24C2×#440C2214yes—
25C2×C4×SD16C2212yes—
26#2857C2211yescharacteristic
27#2851C2212yes—
28#2590—22no—
29#2568C2211yescharacteristic, complemented
30#2552C2211yescharacteristic, complemented
31#2511—21no—
32#2508C2212yescomplemented
33#2505—22no—
34#2502—21no—
35#2453C2212yescomplemented
36#2365C2211yescharacteristic, complemented
37#2360—21no—
38#2355—21no—
39Q8×C4⋊C4C2211yescharacteristic, complemented
40D4×C4⋊C4—22no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2214yes—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46C4×C22⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C2×#389C2212yes—
49C2×#385—22no—
50C2×#384C2211yescharacteristic
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2099C2212yescomplemented
54#2096C2212yescomplemented
55#2087C2214yescomplemented
56#2085C2214yescomplemented
57#2083C2214yescomplemented
58#2068C2214yescomplemented
59#2059C2214yescomplemented
60#1990C2212yescomplemented
61#1948C2212yescomplemented
62#1944C2214yescomplemented
63#1943C2214yescomplemented
64#1939C2212yescomplemented
65#1921C2214yescomplemented
66#1917C2214yescomplemented
67#1877C2212yes—
68#1877C2214yes—
69#1872C2212yes—
70#1872C2211yescharacteristic
71#1864C2214yes—
72#1864C2212yes—
73#1863C2214yescomplemented
74#1862C2214yes—
75#1862C2214yes—
76#1862C2214yes—
77#1853C2212yes—
78#1853C2211yescharacteristic
79#1852C2212yes—
80#1852C2212yes—
81#1852C2212yes—
82#1845C2212yescomplemented
83#1804C2214yes—
84#1803C2212yescomplemented
85#1802C2212yes—
86#1797C2211yescharacteristic
87#1797C2212yes—
88#1786C2212yes—
89#1786C2211yescharacteristic
90#1770C2212yescomplemented
91#1762C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4—22no—
98C22×SD16C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#125C2312yes—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C2311yescharacteristic
103C2×C4×Q8—21no—
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2312yes—
106C2×C4×D4—28no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—48no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2314yes—
114C2×C4×D4C2314yes—
115C2×C4×D4C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—48no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—22no—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#478—48no—
132#474—28no—
133#450Q818yescomplemented
134#440—48no—
135C4×SD16—28no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2311yescharacteristic
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155#400—22no—
156#400—22no—
157#400—22no—
158#398—22no—
159#398—22no—
160#398—22no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391C2312yes—
166#391C2311yescharacteristic
167#389—22no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2314yes—
179#389C2311yescharacteristic
180#387—22no—
181#387—21no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—21no—
191#385—24no—
192#385—24no—
193#385—24no—
194#385—24no—
195#384—21no—
196#384—21no—
197#384—21no—
198#384—21no—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#382—22no—
203#382C2312yes—
204#380—21no—
205#380—22no—
206#380—22no—
207#380—24no—
208#380C2311yescharacteristic
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C25—21no—
237C22×Q8—21no—
238C22×Q8C2411yescharacteristic
239C22×D4—24no—
240C22×D4—24no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4C2×Q814yes—
244C22×D4C2412yes—
245C23×C4—28no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—44no—
253C23×C4—48no—
254C23×C4—44no—
255C23×C4—48no—
256C23×C4—84no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16C2×Q818yes—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C4⋊Q8—24no—
269#125—44no—
270C22⋊Q8—22no—
271C22⋊Q8—44no—
272C22⋊Q8C2×Q814yes—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—48no—
287C4×D4#4118yes—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—416no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—48no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—28no—
356#107—44no—
357#107—44no—
358#106—24no—
359#106—44no—
360C4⋊C8—24no—
361C4⋊C8—44no—
362#103—24no—
363#103—44no—
364#102—24no—
365#102—28no—
366#102—28no—
367#102—24no—
368#102C2×Q818yes—
369C22⋊C8—22no—
370C22⋊C8—24no—
371C22⋊C8C2×Q814yes—
372#97—44no—
373C4×C8—24no—
374#95—24no—
375#95—24no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—22no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24C22×D411yescharacteristic
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8C22×Q812yes—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—216no—
414C2×D4—216no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4C22×Q814yes—
420C2×D4C2×#4114yes—
421C2×D4H(5,𝔽2)14yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—28no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—432no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4—832no—
463C22×C4AGL(1,ℤ/8)14yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×Q812yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476SD16—216no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8C22×Q814yes—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—42no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4C22×Q812yes—
522C4⋊C4C2×#4114yes—
523C22⋊C4—28no—
524C22⋊C4—24no—
525C22⋊C4—28no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—24no—
535C42—42no—
536C42—42no—
537C42—44no—
538C42—42no—
539C42—42no—
540C42—82no—
541C42C2×#4114yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—216no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—48no—
554C23—48no—
555C23—42no—
556C23—48no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23D4×Q812yes—
561C23C2×AGL(1,ℤ/8)12yes—
562C23#57812yes—
563C23C23×D411yescharacteristic
564Q8—24no—
565Q8—42no—
566Q8—84no—
567D4—216no—
568D4—24no—
569D4—24no—
570D4—24no—
571D4#55418yes—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—44no—
600C2×C4—416no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—416no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—84no—
626C2×C4—816no—
627C2×C4—816no—
628C2×C4—88no—
629C2×C4#52512yes—
630C2×C4#52512yes—
631C2×C4D4×Q812yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×AGL(1,ℤ/8)12yes—
634C2×C4#57512yes—
635C2×C4C23×Q811yescharacteristic
636C2×C4C22×#4112yes—
637C2×C4C2×H(5,𝔽2)12yes—
638C2×C4C2×2−1+411yescharacteristic
639C8—24no—
640C8—42no—
641C8—44no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—28no—
649C22—432no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22#338614yes—
657C22C2×#52511yescharacteristic
658C22C2×D4×Q811yescharacteristic
659C22#348411yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C2×#57511yescharacteristic
662C22C2×#57811yescharacteristic
663C22#358511yescharacteristic
664C4—22no—
665C4—22no—
666C4—22no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—41no—
673C4—42no—
674C4—42no—
675C4—82no—
676C4—82no—
677C4—84no—
678C4—81no—
679C4—81no—
680C4#311414yes—
681C4C2×#55412yes—
682C4#349212yes—
683C2—24no—
684C2—48no—
685C2#3654611yescharacteristic
686C2C2×#311411yescharacteristic
687C2#6058911yescharacteristic
688C2C2×#338611yescharacteristic
689C2#6136711yescharacteristic
690C2#6158211yescharacteristic
691C2#6159511yescharacteristic
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. #36552C2semidirect
5. #29303C2semidirect
6. #29241C2semidirect
7. #27950C2semidirect
8. #27680C2semidirect
9. #27310C2semidirect
10. #26398C2semidirect
11. #26041C2semidirect
12. #24816C2semidirect
13. #22968C2semidirect
14. #22671C2semidirect
15. #22629C2semidirect
29. #2568C22semidirect
30. #2552C22semidirect
32. #2508C22semidirect
35. #2453C22semidirect
36. #2365C22semidirect
39. Q8×C4⋊C4C22semidirect
53. #2099C22semidirect
54. #2096C22semidirect
55. #2087C22semidirect
56. #2085C22semidirect
57. #2083C22semidirect
58. #2068C22semidirect
59. #2059C22semidirect
60. #1990C22semidirect
61. #1948C22semidirect
62. #1944C22semidirect
63. #1943C22semidirect
64. #1939C22semidirect
65. #1921C22semidirect
66. #1917C22semidirect
73. #1863C22semidirect
82. #1845C22semidirect
84. #1803C22semidirect
90. #1770C22semidirect
133. #450Q8semidirect

Maximal subgroups

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

subgroupindex
#365522
C2×D4×C4⋊C42
#329592
#329532
#293032
#292412
#279502
#276802
#273102
#263982
#260412
#248162
#229682
#226712
#226292
#214532
C4×#4502