Finite Group Databasebeta

← #6617013

Subgroups — #6617013

Order 512.

1813 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#33100C212yesmaximal, complemented
4#32953C211yesmaximal, characteristic
5#27948C211yesmaximal, characteristic, complemented
6#27681C211yesmaximal, characteristic, complemented
7#27588C212yesmaximal
8#27491C214yesmaximal, complemented
9#27365C214yesmaximal, complemented
10#26091C212yesmaximal, complemented
11#24681C212yesmaximal, complemented
12#24501C212yesmaximal
13#23060C212yesmaximal, complemented
14#22805C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16C4×#450C211yesmaximal, characteristic
17#21361C212yesmaximal, complemented
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#474C2212yes—
21C2×#450C2211yescharacteristic
22C2×C4×SD16C2212yes—
23C2×#431C2212yes—
24#2857C2211yescharacteristic
25#2855C2212yes—
26#2571C2211yescharacteristic, complemented
27#2568C2211yescharacteristic, complemented
28#2533—22no—
29#2514—21no—
30#2514C2212yescomplemented
31#2509C2212yescomplemented
32#2503—21no—
33#2493—22no—
34#2365C2211yescharacteristic, complemented
35#2360—21no—
36#2356—21no—
37#2355C2211yescharacteristic, complemented
38#2354—21no—
39#2354—21no—
40#2354—22no—
41#2354C2212yes—
42#2354C2214yes—
43#2354C2212yes—
44C4×C22⋊Q8C2211yescharacteristic
45C2×#390C2212yes—
46C2×#388—22no—
47C2×#384C2211yescharacteristic
48#2283C2212yes—
49C2×#380—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2100C2212yescomplemented
53#2095C2212yescomplemented
54#2092C2214yes—
55#2084C2214yescomplemented
56#2073C2214yes—
57#2065C2214yes—
58#2055C2214yes—
59#1990C2212yescomplemented
60#1968C2214yes—
61#1949C2214yescomplemented
62#1949C2214yescomplemented
63#1947C2212yescomplemented
64#1939C2212yescomplemented
65#1934C2214yes—
66#1928C2214yescomplemented
67#1927C2214yescomplemented
68#1926C2214yes—
69#1925C2214yes—
70#1919C2214yescomplemented
71#1918C2214yescomplemented
72#1916C2214yescomplemented
73#1877C2212yes—
74#1872C2212yes—
75#1872C2211yescharacteristic
76#1864C2212yes—
77#1862C2214yes—
78#1853C2212yes—
79#1853C2212yes—
80#1853C2211yescharacteristic
81#1852C2212yes—
82#1852C2212yes—
83#1844C2212yescomplemented
84#1817C2212yes—
85#1803C2212yescomplemented
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1786C2212yes—
89#1786C2211yescharacteristic
90C4×#107C2214yescomplemented
91C4×#106C2214yescomplemented
92C8×C4⋊C4C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×SD16C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8C2311yescharacteristic
101C2×C4×Q8—21no—
102C2×C4×Q8C2311yescharacteristic
103C2×C4×D4—28no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—24no—
109C2×#117—22no—
110C2×#117—22no—
111C2×#117—48no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—22no—
120C22×C22⋊C4—22no—
121C22×C42—22no—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#474—28no—
125#450—24no—
126C4×SD16—28no—
127#431—28no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2312yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2311yescharacteristic
144C2×C22⋊C8C2311yescharacteristic
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#401—22no—
148#400—22no—
149#400—22no—
150#400—22no—
151#400—22no—
152#400—22no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391C2312yes—
158#391C2312yes—
159#391C2311yescharacteristic
160#390—22no—
161#390—22no—
162#390C2314yes—
163#390C2314yes—
164#389—21no—
165#389—22no—
166#389—21no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2311yescharacteristic
171#388—24no—
172#388—24no—
173#388—24no—
174#388—24no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—21no—
179#387—22no—
180#387—22no—
181#387—22no—
182#385—22no—
183#385—21no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2312yes—
189#383—22no—
190#383—22no—
191#382—21no—
192#382—21no—
193#382C2312yes—
194#382C2312yes—
195#381—22no—
196#381—22no—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—24no—
201#380C2312yes—
202#380C2311yescharacteristic
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220#376—22no—
221C2×#95—22no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C25—21no—
235C22×Q8—21no—
236C22×Q8C2411yescharacteristic
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—42no—
242C22×D4C2412yes—
243C23×C4—28no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—42no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—44no—
265C4×Q8—24no—
266C4×Q8—44no—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4#4118yes—
273#117—48no—
274#117—416no—
275#117—48no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4—84no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—48no—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340#107—28no—
341#107—44no—
342#106—24no—
343#106—44no—
344#106—44no—
345C4⋊C8—28no—
346C4⋊C8—24no—
347#103—24no—
348#103—44no—
349#102—24no—
350#102—28no—
351#102—24no—
352#102—44no—
353#102—42no—
354C22⋊C8—24no—
355C22⋊C8—22no—
356C22⋊C8#4114yes—
357C4×C8—28no—
358C4×C8—24no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—24no—
385C24—22no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24C22×D411yescharacteristic
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C2×#4114yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—44no—
436C22×C4—432no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4#13514yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4112yes—
456C22×C42−1+412yes—
457C22×C4C2511yescharacteristic
458SD16—48no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8C2×#4114yes—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4AGL(1,ℤ/8)14yes—
498C4⋊C4C2×#4114yes—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—48no—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—48no—
518C42—42no—
519C42—48no—
520C42—42no—
521C42—44no—
522C42—82no—
523C42C2×#4114yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—24no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23#54712yes—
543C23C2×#13512yes—
544C23#57712yes—
545C23C23×D411yescharacteristic
546Q8—42no—
547Q8—42no—
548Q8—84no—
549D4—24no—
550D4—24no—
551D4—48no—
552D4—42no—
553D4—42no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—416no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—416no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—88no—
608C2×C4#52512yes—
609C2×C4#52512yes—
610C2×C4#54712yes—
611C2×C4C2×#13512yes—
612C2×C4C2×#13512yes—
613C2×C4C2×AGL(1,ℤ/8)14yes—
614C2×C4C2×AGL(1,ℤ/8)12yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4112yes—
617C2×C4C2×2−1+411yescharacteristic
618C2×C4#58512yes—
619C8—24no—
620C8—44no—
621C8—42no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—816no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22#334014yes—
637C22C2×#52511yescharacteristic
638C22C2×#54711yescharacteristic
639C22#348711yescharacteristic
640C22C22×#13511yescharacteristic
641C22C22×AGL(1,ℤ/8)11yescharacteristic
642C22C2×#57711yescharacteristic
643C22#358611yescharacteristic
644C4—24no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—41no—
653C4—81no—
654C4—81no—
655C4—82no—
656C4—81no—
657C4—82no—
658C4—81no—
659C4—84no—
660C4#310514yes—
661C4#347512yes—
662C4#349512yes—
663C2—24no—
664C2—84no—
665C2#3642711yescharacteristic
666C2C2×#310511yescharacteristic
667C2#6058511yescharacteristic
668C2C2×#334011yescharacteristic
669C2#6125211yescharacteristic
670C2#6188211yescharacteristic
671C2#6190111yescharacteristic
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. #36177C2semidirect
3. #33100C2semidirect
5. #27948C2semidirect
6. #27681C2semidirect
8. #27491C2semidirect
9. #27365C2semidirect
10. #26091C2semidirect
11. #24681C2semidirect
13. #23060C2semidirect
14. #22805C2semidirect
15. #22599C2semidirect
17. #21361C2semidirect
26. #2571C22semidirect
27. #2568C22semidirect
30. #2514C22semidirect
31. #2509C22semidirect
34. #2365C22semidirect
37. #2355C22semidirect
52. #2100C22semidirect
53. #2095C22semidirect
55. #2084C22semidirect
59. #1990C22semidirect
61. #1949C22semidirect
62. #1949C22semidirect
63. #1947C22semidirect
64. #1939C22semidirect
66. #1928C22semidirect
67. #1927C22semidirect
70. #1919C22semidirect
71. #1918C22semidirect
72. #1916C22semidirect
83. #1844C22semidirect
85. #1803C22semidirect
90. C4×#107C22semidirect
91. C4×#106C22semidirect
92. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#361772
C2×#23542
#331002
#329532
#279482
#276812
#275882
#274912
#273652
#260912
#246812
#245012
#230602
#228052
#225992
C4×#4502
#213612