Finite Group Databasebeta

← #6433943

Subgroups — #6433943

Order 512.

1861 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic, complemented
3#30363C214yesmaximal, complemented
4#30362C214yesmaximal, complemented
5#30361C214yesmaximal, complemented
6#30360C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23733C211yesmaximal, characteristic, complemented
9#23642C212yesmaximal, complemented
10#23553C211yesmaximal, characteristic, complemented
11#23482C212yesmaximal
12#23453C211yesmaximal, characteristic, complemented
13#21159C212yesmaximal, complemented
14#21093C212yesmaximal, complemented
15C4×#407C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#417C2212yes—
22C2×#417C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25#2855C2212yes—
26#2855C2212yes—
27#2854C2212yes—
28#2854C2212yes—
29#2853—22no—
30#2851—22no—
31#2851—22no—
32#2851C2214yes—
33#2851C2214yes—
34#2849—22no—
35C4×#123—22no—
36C4×#123—22no—
37#2297C2211yescharacteristic
38#2297C2211yescharacteristic
39#2296C2211yescharacteristic
40#2296C2211yescharacteristic
41C2×#389C2212yes—
42C2×#387—22no—
43C2×#385—22no—
44C2×#382C2211yescharacteristic
45#2280C2211yescharacteristic
46C2×#380C2212yes—
47#2101C2214yescomplemented
48#2101C2214yescomplemented
49#2097C2214yescomplemented
50#2092C2214yescomplemented
51#2077C2214yes—
52#2077C2214yes—
53#2076C2214yes—
54#2075C2214yes—
55#2059C2214yescomplemented
56#2059C2214yescomplemented
57#2036C2214yes—
58#2035C2214yes—
59#2034C2214yes—
60#2033C2214yes—
61#2026C2214yescomplemented
62#2025C2214yescomplemented
63#1944C2214yescomplemented
64#1943C2214yescomplemented
65#1942C2214yescomplemented
66#1941C2214yescomplemented
67#1936C2214yescomplemented
68#1935C2214yescomplemented
69#1934C2214yescomplemented
70#1933C2214yescomplemented
71#1749C2214yes—
72C4×C22⋊C8C2214yes—
73#1736C2212yes—
74#1736C2212yes—
75#1736C2212yes—
76#1736C2212yes—
77#1728C2211yescharacteristic
78#1728C2211yescharacteristic
79C22×#41C2311yescharacteristic
80C24×C4D412yes—
81C24×C4C2311yescharacteristic
82C22×M16C2312yes—
83C22×M16C2312yes—
84C2×#125—21no—
85C2×#125—21no—
86C2×#125—44no—
87C2×#125C2314yes—
88#525—44no—
89C2×#123—22no—
90C2×#123—22no—
91C2×#123—21no—
92C2×C22⋊Q8C2312yes—
93C2×#121C2312yes—
94C2×#120—21no—
95C2×C4×D4—22no—
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C22×C4⋊C4—22no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—22no—
104C22×C42—44no—
105C22×C42D412yes—
106C22×C42D412yes—
107C22×C42D412yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112#428—28no—
113#428—44no—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118#417—28no—
119#417—44no—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124#407—28no—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127C4×M16—44no—
128C2×#97C2314yes—
129C2×C4×C8C2314yes—
130#391C2312yes—
131#391C2311yescharacteristic
132#391C2311yescharacteristic
133#390C2312yes—
134#390C2311yescharacteristic
135#390C2311yescharacteristic
136#389—22no—
137#389—22no—
138#389C2312yes—
139#389C2312yes—
140#389C2314yes—
141#389C2314yes—
142#389C2312yes—
143#389C2312yes—
144#388—22no—
145#388—22no—
146#388—22no—
147#388—22no—
148#387—24no—
149#387—24no—
150#387—24no—
151#387—24no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#386—22no—
157#386—22no—
158#386—22no—
159#386—22no—
160#385—22no—
161#385—22no—
162#385—22no—
163#385—22no—
164#385—24no—
165#385—24no—
166#385—24no—
167#385—24no—
168#382—21no—
169#382—21no—
170#382C2314yes—
171#381—22no—
172#381—22no—
173#381—21no—
174#380—22no—
175#380—22no—
176#380C2312yes—
177#380C2314yes—
178#380C2312yes—
179#380C2314yes—
180#379—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—22no—
188C2×#95—22no—
189C2×#95—22no—
190C2×#95—22no—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25C2×D411yescharacteristic
202C2×#41—24no—
203C2×#41—24no—
204C2×#41—22no—
205C2×#41—22no—
206C22×Q8C2411yescharacteristic
207C22×D4—22no—
208C22×D4—21no—
209C22×D4C2411yescharacteristic
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2×D412yes—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2×D412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C2×M16—24no—
231C2×M16—24no—
232C2×M16—24no—
233C2×M16—24no—
234C2×M16—24no—
235C2×M16#4118yes—
236C22×C8C2414yes—
237C22×C8C2414yes—
238#125—44no—
239#125—44no—
240#125—44no—
241#125—44no—
242#123—48no—
243#123—48no—
244#123—44no—
245C22⋊Q8—42no—
246C22⋊Q8—42no—
247#121—42no—
248#121—42no—
249#120—44no—
250C4×D4—48no—
251#117—24no—
252#117—24no—
253#117—22no—
254#117—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2414yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2411yescharacteristic
276C2×C4⋊C4C2411yescharacteristic
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—48no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2412yes—
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314#107—28no—
315#107—44no—
316#106—28no—
317#106—44no—
318#103—28no—
319#103—44no—
320#102—28no—
321#102—44no—
322C22⋊C8—28no—
323C22⋊C8—28no—
324#97—44no—
325C4×C8—44no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—21no—
347C24—42no—
348C24C22×D412yes—
349C24C22×D411yescharacteristic
350#41—48no—
351#41—48no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—44no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—432no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4114yes—
422C22×C4H(5,𝔽2)14yes—
423C22×C42−1+412yes—
424C22×C42−1+412yes—
425C22×C4C2511yescharacteristic
426M16—216no—
427M16—48no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8C2×#4118yes—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—42no—
484C23—22no—
485C23—22no—
486C23—28no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—48no—
494C23—48no—
495C23—44no—
496C23—44no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23#53212yes—
503C23#53412yes—
504C23#53612yes—
505C23#57514yes—
506C23C23×D411yescharacteristic
507Q8—42no—
508Q8—42no—
509D4—42no—
510D4—42no—
511D4—88no—
512D4—84no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—48no—
528C2×C4—48no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—88no—
559C2×C4—88no—
560C2×C4—84no—
561C2×C4#53212yes—
562C2×C4#53412yes—
563C2×C4#53612yes—
564C2×C4#53918yes—
565C2×C4#54218yes—
566C2×C4#57514yes—
567C2×C4C22×#4112yes—
568C2×C4C2×H(5,𝔽2)12yes—
569C2×C4C2×2−1+411yescharacteristic
570C2×C4C2×2−1+411yescharacteristic
571C8—28no—
572C8—44no—
573C22—24no—
574C22—24no—
575C22—22no—
576C22—22no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—416no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—82no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22#266314yes—
591C22C2×#53211yescharacteristic
592C22C2×#53411yescharacteristic
593C22C2×#53611yescharacteristic
594C22C2×#57512yes—
595C22C2×#57512yes—
596C4—24no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—81no—
602C4—82no—
603C4—84no—
604C4—84no—
605C4—82no—
606C4—81no—
607C4—81no—
608C4#266314yes—
609C4C2×#53912yes—
610C4C2×#54212yes—
611C2—24no—
612C2—44no—
613C2—161no—
614C2C2×#266311yescharacteristic
615C2#6070212yes—
616C2#6075212yes—
617C2#6086111yescharacteristic
618C2#6086211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2851C2semidirect
3. #30363C2semidirect
4. #30362C2semidirect
5. #30361C2semidirect
6. #30360C2semidirect
7. #23746C2semidirect
8. #23733C2semidirect
9. #23642C2semidirect
10. #23553C2semidirect
12. #23453C2semidirect
13. #21159C2semidirect
14. #21093C2semidirect
15. C4×#407C2semidirect
47. #2101C22semidirect
48. #2101C22semidirect
49. #2097C22semidirect
50. #2092C22semidirect
55. #2059C22semidirect
56. #2059C22semidirect
61. #2026C22semidirect
62. #2025C22semidirect
63. #1944C22semidirect
64. #1943C22semidirect
65. #1942C22semidirect
66. #1941C22semidirect
67. #1936C22semidirect
68. #1935C22semidirect
69. #1934C22semidirect
70. #1933C22semidirect

Maximal subgroups

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

subgroupindex
#366262
C2×#28512
#303632
#303622
#303612
#303602
#237462
#237332
#236422
#235532
#234822
#234532
#211592
#210932
C4×#4072