Finite Group Databasebeta

← #6635567

Subgroups — #6635567

Order 512.

1750 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37386C211yesmaximal, characteristic, complemented
2#35541C211yesmaximal, characteristic
3#32914C212yesmaximal, complemented
4#31957C214yesmaximal, complemented
5#31937C214yesmaximal, complemented
6#31002C214yesmaximal, complemented
7#30986C214yesmaximal, complemented
8#29764C212yesmaximal, complemented
9#28311C211yesmaximal, characteristic, complemented
10#28302C211yesmaximal, characteristic, complemented
11#27980C211yesmaximal, characteristic, complemented
12#27975C211yesmaximal, characteristic, complemented
13#24014C212yesmaximal, complemented
14#23926C212yesmaximal
15#21905C211yesmaximal, characteristic, complemented
16C2×#536C2212yes—
17C2×#535C2211yescharacteristic
18C2×#518C2211yescharacteristic
19C2×#484C2212yes—
20C2×#483C2212yes—
21C2×#461C2214yes—
22C2×#448C2214yes—
23C2×#450C2212yes—
24C2×#450C2212yes—
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2734—22no—
28#2724—22no—
29#2717—24no—
30#2621C2212yes—
31#2499C2212yescomplemented
32#2499C2212yescomplemented
33#2447—22no—
34#2445—22no—
35#2434C2212yescomplemented
36#2434C2212yescomplemented
37#2422C2214yes—
38#2420C2214yes—
39#2412—24no—
40#2316C2211yescharacteristic
41#2311C2211yescharacteristic
42#2297C2211yescharacteristic
43C2×#390C2211yescharacteristic
44#2285C2211yescharacteristic
45#2094C2212yes—
46#2093C2212yes—
47#2085C2214yescomplemented
48#2074C2214yescomplemented
49#2072C2214yescomplemented
50#2068C2214yescomplemented
51#2066C2214yescomplemented
52#2043C2214yescomplemented
53#2036C2214yescomplemented
54#2035C2214yescomplemented
55#2013C2212yes—
56#2013C2212yes—
57#2000C2214yes—
58#2000C2214yes—
59#1973C2214yes—
60#1968C2214yes—
61#1960C2212yescomplemented
62#1956C2212yescomplemented
63#1924C2214yescomplemented
64#1923C2214yescomplemented
65#1877C2214yes—
66#1877C2214yes—
67#1875C2214yes—
68#1873C2212yes—
69#1873C2212yes—
70#1873C2212yes—
71#1873C2212yes—
72#1854C2214yescomplemented
73#1836C2212yescomplemented
74#1834C2212yescomplemented
75#1804C2214yes—
76#1798C2212yes—
77#1787C2214yescomplemented
78#1780C2212yes—
79#1749C2211yescharacteristic
80C4×C22⋊C8C2211yescharacteristic
81C22×#41C2311yescharacteristic
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86#536—44no—
87#536—44no—
88#535—44no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2312yes—
93C2×#127—21no—
94C2×#124—21no—
95C2×#124—22no—
96C2×#123—22no—
97C2×#123—22no—
98C2×#123—24no—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×#121C2314yes—
104C2×#120—22no—
105#518—44no—
106C2×C4×D4—22no—
107C2×C4×D4C2314yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—22no—
114C22×C22⋊C4—22no—
115C22×C42C2311yescharacteristic
116#484—44no—
117#483—44no—
118#461—48no—
119#450—44no—
120#450—44no—
121#448—48no—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2312yes—
138C2×C4×C8C2312yes—
139#398—24no—
140#397—24no—
141#397—24no—
142#397—24no—
143#397—24no—
144#397—24no—
145#395—24no—
146#395—24no—
147#394—24no—
148#394—24no—
149#391—21no—
150#391D412yes—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#390—21no—
156#390—22no—
157#390D412yes—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—22no—
162#389—22no—
163#389C2314yes—
164#387—24no—
165#387—24no—
166#386—22no—
167#386—24no—
168#386D414yes—
169#385—24no—
170#385—24no—
171#385—24no—
172#384C2312yes—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#382C2312yes—
176#382C2312yes—
177#381—24no—
178#380—22no—
179#380—22no—
180#380—22no—
181#380—22no—
182#380C2314yes—
183C4×C4⋊C4—22no—
184C4×C22⋊C4C2312yes—
185C4×C22⋊C4C2312yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2312yes—
192#336C2312yes—
193C25—21no—
194C2×#41—24no—
195C2×#41—42no—
196C2×#41—42no—
197C22×Q8—22no—
198C22×Q8—42no—
199C22×Q8—42no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8C2×D411yescharacteristic
203C22×Q8C2412yes—
204C22×Q8C2412yes—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—21no—
208C22×D4—24no—
209C22×D4—42no—
210C22×D4C2×D412yes—
211C22×D4C2×D411yescharacteristic
212C22×D4C2412yes—
213C23×C4—21no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4C2×D412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C2×SD16—28no—
220C2×SD16—28no—
221C2×SD16—44no—
222C2×SD16—44no—
223C2×SD16—44no—
224C2×SD16—44no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—42no—
228C4⋊Q8—42no—
229C4⋊Q8—44no—
230C4⋊Q8—48no—
231#127—44no—
232#124—44no—
233#124—48no—
234#123—48no—
235#123—48no—
236#123—88no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—42no—
242C22⋊Q8—42no—
243C22⋊Q8—42no—
244C22⋊Q8—42no—
245#121—28no—
246#121—44no—
247#120—48no—
248C4×D4—28no—
249C4×D4—44no—
250C4×D4—48no—
251#117—42no—
252#117—42no—
253#117—42no—
254#117—42no—
255#117—42no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—84no—
290C2×C22⋊C4C2×D414yes—
291C2×C22⋊C4C2×D412yes—
292C2×C22⋊C4C2×D414yes—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—22no—
296C2×C42—42no—
297C2×C42C2×D412yes—
298C2×C42C2412yes—
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301#107—44no—
302#106—44no—
303C4⋊C8—44no—
304#103—44no—
305#103—44no—
306#103—44no—
307#103—44no—
308#103—44no—
309#103—44no—
310#102—44no—
311#102—44no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C22⋊C8—44no—
315C22⋊C8—44no—
316#97—42no—
317C4×C8—42no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95C2×D414yes—
328#95C2×D414yes—
329C24—24no—
330C24—22no—
331C24—42no—
332C24—42no—
333C24—82no—
334C24—82no—
335C24C22×D412yes—
336C24C22×D411yescharacteristic
337#41—88no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—44no—
350C2×Q8—82no—
351C2×Q8—82no—
352C2×Q8—88no—
353C2×Q8—88no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—24no—
360C2×D4—416no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—48no—
367C2×D4—42no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4H(5,𝔽2)14yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4C22×D412yes—
396C22×C4C22×D414yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4H(5,𝔽2)14yes—
401C22×C4C2511yescharacteristic
402SD16—88no—
403SD16—88no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—82no—
427C4⋊C4—84no—
428C4⋊C4—82no—
429C4⋊C4—84no—
430C4⋊C4—82no—
431C22⋊C4—28no—
432C22⋊C4—42no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—48no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—44no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—48no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—88no—
469C23—162no—
470C23—162no—
471C23—164no—
472C23#53512yes—
473C23C23×D411yescharacteristic
474Q8—44no—
475Q8—44no—
476Q8—44no—
477Q8—44no—
478Q8—82no—
479Q8—88no—
480D4—28no—
481D4—48no—
482D4—44no—
483D4—82no—
484D4—84no—
485D4—168no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—28no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—84no—
521C2×C4—88no—
522C2×C4—168no—
523C2×C4—162no—
524C2×C4#53512yes—
525C2×C4#57714yes—
526C2×C4C2×H(5,𝔽2)12yes—
527C2×C4C2×H(5,𝔽2)12yes—
528C2×C4#58512yes—
529C8—44no—
530C8—44no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—44no—
537C22—42no—
538C22—44no—
539C22—42no—
540C22—42no—
541C22—48no—
542C22—44no—
543C22—84no—
544C22—82no—
545C22—82no—
546C22—168no—
547C22C2×#53511yescharacteristic
548C22#348411yescharacteristic
549C22#348511yescharacteristic
550C22C2×#57712yes—
551C22#358612yes—
552C4—24no—
553C4—22no—
554C4—42no—
555C4—42no—
556C4—44no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4#353414yes—
565C2—42no—
566C2—82no—
567C2—84no—
568C2#3672111yescharacteristic
569C2#6082011yescharacteristic
570C2#6082811yescharacteristic
571C2#6168012yes—
572C2#6175312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37386C2semidirect
3. #32914C2semidirect
4. #31957C2semidirect
5. #31937C2semidirect
6. #31002C2semidirect
7. #30986C2semidirect
8. #29764C2semidirect
9. #28311C2semidirect
10. #28302C2semidirect
11. #27980C2semidirect
12. #27975C2semidirect
13. #24014C2semidirect
15. #21905C2semidirect
31. #2499C22semidirect
32. #2499C22semidirect
35. #2434C22semidirect
36. #2434C22semidirect
47. #2085C22semidirect
48. #2074C22semidirect
49. #2072C22semidirect
50. #2068C22semidirect
51. #2066C22semidirect
52. #2043C22semidirect
53. #2036C22semidirect
54. #2035C22semidirect
61. #1960C22semidirect
62. #1956C22semidirect
63. #1924C22semidirect
64. #1923C22semidirect
72. #1854C22semidirect
73. #1836C22semidirect
74. #1834C22semidirect
77. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#373862
#355412
#329142
#319572
#319372
#310022
#309862
#297642
#283112
#283022
#279802
#279752
#240142
#239262
#219052