Finite Group Databasebeta

← #6506463

Subgroups — #6506463

Order 512.

2500 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic, complemented
2C2×#2432C211yesmaximal, characteristic
3#29111C212yesmaximal, complemented
4#29103C212yesmaximal, complemented
5#27278C218yesmaximal, complemented
6#27274C218yesmaximal, complemented
7#25864C212yesmaximal, complemented
8#25856C212yesmaximal, complemented
9#21765C212yesmaximal, complemented
10#20601C212yesmaximal
11#20524C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C2×#543—21no—
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C2×#489C2214yescomplemented
18C2×#487C2214yescomplemented
19C2×#417C2214yes—
20C2×#417C2214yes—
21C2×#407C2212yes—
22#2857C2212yes—
23#2857C2212yes—
24#2659—21no—
25#2659—21no—
26#2549—24no—
27#2440C2214yescomplemented
28#2440C2214yescomplemented
29#2432—24no—
30#2432C2218yes—
31C2×#394—21no—
32C2×#394—21no—
33#2315—21no—
34C2×#391C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#385—21no—
37C2×#385—21no—
38C2×#380C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40#2072C2214yescomplemented
41#2071C2214yescomplemented
42#2043C2218yescomplemented
43#2042C2218yescomplemented
44#2024C2214yescomplemented
45#2023C2214yescomplemented
46#1959C2214yescomplemented
47#1956C2214yescomplemented
48#1935C2218yescomplemented
49#1934C2218yescomplemented
50#1894C2218yescomplemented
51#1872C2214yescomplemented
52#1872C2214yescomplemented
53#1823C2214yes—
54#1788C2218yes—
55#1787C2214yes—
56#1787C2214yes—
57#1786C2214yes—
58#1753C2218yescomplemented
59#1749C2212yes—
60C4×C22⋊C8C2212yes—
61C22×#41—22no—
62C22×#41—28no—
63C22×#41—44no—
64C22×#41—44no—
65C22×#41C2314yes—
66C23×Q8C2311yescharacteristic
67C23×Q8C2311yescharacteristic
68C23×D4—21no—
69C23×D4C2311yescharacteristic
70C24×C4—21no—
71C24×C4C2311yescharacteristic
72C22×M16C2314yes—
73#543—48no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—44no—
76C2×C4⋊Q8C2314yes—
77C2×C4⋊Q8C2312yes—
78C2×C4⋊Q8C2312yes—
79C2×#124—22no—
80C2×#124—44no—
81C2×#124C2314yes—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8C2314yes—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4—22no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4D411yescharacteristic
102C22×C22⋊C4D411yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105#489—48no—
106#487—48no—
107#417—48no—
108#417—48no—
109C2×#103C2318yes—
110C2×#103C2318yes—
111C2×#103C2318yes—
112C2×#103C2318yes—
113C2×#102C2318yes—
114C2×#102C2318yes—
115#407—28no—
116C2×C22⋊C8C2314yes—
117C2×C22⋊C8C2314yes—
118C2×#97C2314yes—
119C2×C4×C8C2314yes—
120#398—24no—
121#398—24no—
122#395—24no—
123#395—24no—
124#394—28no—
125#394—28no—
126#391—22no—
127#391—22no—
128#391—22no—
129#391C2314yes—
130#390—22no—
131#390C2314yes—
132#389—22no—
133#389—22no—
134#389C2314yes—
135#389C2314yes—
136#385—24no—
137#385—24no—
138#385—28no—
139#385—28no—
140#384C2312yes—
141#384C2312yes—
142#380—22no—
143#380—22no—
144#380—22no—
145#380—22no—
146#380C2314yes—
147#380C2314yes—
148C4×C22⋊C4D414yes—
149C4×C22⋊C4C2314yes—
150C2×#95—21no—
151C2×#95—21no—
152C2×#95—22no—
153C2×#95D411yescharacteristic
154C2×#95D411yescharacteristic
155#340C2318yes—
156#340C2318yes—
157#336C2314yes—
158#336C2314yes—
159C25—21no—
160C25C2×D411yescharacteristic
161C2×#41—28no—
162C2×#41—28no—
163C2×#41—28no—
164C2×#41—28no—
165C2×#41—44no—
166C2×#41—44no—
167C2×#41—48no—
168C2×#41—48no—
169C2×#41—464no—
170C2×#41—48no—
171C22×Q8—24no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8C2412yes—
182C22×Q8C2412yes—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—21no—
186C22×D4—24no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4C2412yes—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D412yes—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4#4118yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C2×M16—28no—
213C2×M16—44no—
214C2×M16—44no—
215C22×C8C2418yes—
216C4⋊Q8—28no—
217C4⋊Q8—44no—
218C4⋊Q8—44no—
219C4⋊Q8—44no—
220C4⋊Q8—48no—
221#124—28no—
222#124—44no—
223#124—48no—
224C22⋊Q8—28no—
225C22⋊Q8—28no—
226C22⋊Q8—416no—
227C22⋊Q8—416no—
228C22⋊Q8—48no—
229C22⋊Q8—48no—
230C22⋊Q8—48no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2414yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4C2×D414yes—
270C2×C22⋊C4C2×D414yes—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2412yes—
276C2×C42C2412yes—
277#103—216no—
278#103—48no—
279#103—48no—
280#103—48no—
281#102—216no—
282#102—48no—
283C22⋊C8—28no—
284C22⋊C8—28no—
285#97—44no—
286C4×C8—44no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95C2×D414yes—
294#95C2×D414yes—
295C24—22no—
296C24—22no—
297C24—24no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24#12414yes—
310C24C22×D412yes—
311C24C22×D411yescharacteristic
312#41—232no—
313#41—416no—
314#41—416no—
315#41—416no—
316#41—464no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—416no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—44no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—44no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×Q8—82no—
335C2×Q8—82no—
336C2×Q8—88no—
337C2×Q8H(5,𝔽2)14yes—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—22no—
342C2×D4—416no—
343C2×D4—48no—
344C2×D4—42no—
345C2×D4—48no—
346C2×D4—432no—
347C2×D4—48no—
348C2×D4—42no—
349C2×D4H(5,𝔽2)14yes—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—416no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—48no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4—432no—
379C22×C4—42no—
380C22×C4—48no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4#12414yes—
394C22×C4#12418yes—
395C22×C4AGL(1,ℤ/8)14yes—
396C22×C4#13714yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C2×#4114yes—
404C22×C4C2×#4114yes—
405C22×C4C2×#4114yes—
406C22×C4C2511yescharacteristic
407M16—88no—
408C2×C8—28no—
409C2×C8—28no—
410C2×C8—28no—
411C4⋊C4—24no—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C22⋊C4—28no—
429C22⋊C4—28no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C42—24no—
438C42—24no—
439C42—42no—
440C42—42no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—28no—
452C23—24no—
453C23—44no—
454C23—44no—
455C23—48no—
456C23—48no—
457C23—48no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23C2×#12414yes—
473C23C2×#12412yes—
474C23C2×AGL(1,ℤ/8)12yes—
475C23C2×#13712yes—
476C23C23×D411yescharacteristic
477Q8—28no—
478Q8—48no—
479Q8—44no—
480Q8—48no—
481Q8—84no—
482Q8—164no—
483Q8—164no—
484D4—28no—
485D4—48no—
486D4—432no—
487D4—44no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—432no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—164no—
531C2×C4—164no—
532C2×C4C2×#12414yes—
533C2×C4C2×#12412yes—
534C2×C4C2×AGL(1,ℤ/8)14yes—
535C2×C4C2×#13714yes—
536C2×C4C22×#4112yes—
537C2×C4C2×H(5,𝔽2)12yes—
538C2×C4C2×H(5,𝔽2)12yes—
539C8—48no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—88no—
557C22—816no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22#314114yes—
563C22C22×#12411yescharacteristic
564C22#348411yescharacteristic
565C22#348411yescharacteristic
566C22C22×AGL(1,ℤ/8)11yescharacteristic
567C22C22×#13711yescharacteristic
568C22#358611yescharacteristic
569C22#358611yescharacteristic
570C4—24no—
571C4—22no—
572C4—24no—
573C4—24no—
574C4—44no—
575C4—42no—
576C4—42no—
577C4—82no—
578C4—84no—
579C4—84no—
580C4—82no—
581C4#349014yes—
582C2—24no—
583C2—44no—
584C2—84no—
585C2#3536711yescharacteristic
586C2C2×#314111yescharacteristic
587C2#6062011yescharacteristic
588C2#6157211yescharacteristic
589C2#6157311yescharacteristic
590C2#6158511yescharacteristic
591C2#6158611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35002C2semidirect
3. #29111C2semidirect
4. #29103C2semidirect
5. #27278C2semidirect
6. #27274C2semidirect
7. #25864C2semidirect
8. #25856C2semidirect
9. #21765C2semidirect
11. #20524C2semidirect
17. C2×#489C22semidirect
18. C2×#487C22semidirect
27. #2440C22semidirect
28. #2440C22semidirect
40. #2072C22semidirect
41. #2071C22semidirect
42. #2043C22semidirect
43. #2042C22semidirect
44. #2024C22semidirect
45. #2023C22semidirect
46. #1959C22semidirect
47. #1956C22semidirect
48. #1935C22semidirect
49. #1934C22semidirect
50. #1894C22semidirect
51. #1872C22semidirect
52. #1872C22semidirect
58. #1753C22semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#24322
#291112
#291032
#272782
#272742
#258642
#258562
#217652
#206012
#205242