Finite Group Databasebeta

← #6461916

Subgroups — #6461916

Order 512.

1950 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35341C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic, complemented
3#29568C214yesmaximal, complemented
4#29564C214yesmaximal, complemented
5#29562C214yesmaximal, complemented
6#29556C214yesmaximal, complemented
7#23553C211yesmaximal, characteristic, complemented
8#23504C211yesmaximal, characteristic, complemented
9C2×#2025C211yesmaximal, characteristic, complemented
10#23428C211yesmaximal, characteristic
11#21235C212yesmaximal, complemented
12#21087C212yesmaximal, complemented
13#21062C212yesmaximal, complemented
14#20929C212yesmaximal, complemented
15#20368C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C2212yes—
17C2×#532C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C2×#518C2211yescharacteristic
20C2×#428C2212yes—
21C2×#417C2212yes—
22C2×#416C2212yes—
23C2×#416C2212yes—
24C22×#102C2212yes—
25C2×#405C2212yes—
26#2861C2214yescomplemented
27#2642C2212yescomplemented
28#2641C2212yescomplemented
29#2591—24no—
30#2562—22no—
31#2562—22no—
32#2357—22no—
33#2354—22no—
34#2342—24no—
35#2319C2214yes—
36#2317C2211yescharacteristic
37#2311C2212yes—
38#2311C2212yes—
39#2309C2211yescharacteristic
40#2296C2211yescharacteristic
41#2293C2214yes—
42C2×#390C2211yescharacteristic
43#2274C2211yescharacteristic
44C4×#117C2211yescharacteristic
45#2059C2214yescomplemented
46#2059C2214yescomplemented
47#2043C2212yescomplemented
48#2042C2212yescomplemented
49#2042C2212yescomplemented
50#2041C2212yescomplemented
51#2025C2212yes—
52#2025C2212yes—
53#2025C2214yes—
54#2013C2212yes—
55#2012C2214yes—
56#2011C2212yes—
57#1973C2214yes—
58#1972C2214yes—
59#1970C2214yes—
60#1968C2214yes—
61#1936C2214yescomplemented
62#1935C2214yescomplemented
63#1934C2214yescomplemented
64#1933C2214yescomplemented
65#1928C2214yescomplemented
66#1927C2214yescomplemented
67#1926C2214yescomplemented
68#1925C2214yescomplemented
69#1879C2214yes—
70#1879C2214yes—
71#1877C2214yes—
72#1877C2214yes—
73#1746C2214yescomplemented
74#1745C2212yescomplemented
75#1744C2212yescomplemented
76#1736C2212yes—
77C2×#340C2212yes—
78#1729C2212yes—
79#1728C2211yescharacteristic
80C2×#336C2211yescharacteristic
81C22×#41C2312yes—
82C23×D4C2311yescharacteristic
83C2×AGL(1,ℤ/8)—28no—
84C2×AGL(1,ℤ/8)—28no—
85C2×AGL(1,ℤ/8)—44no—
86C22×SD16C2314yes—
87C22×D8C2314yes—
88C22×M16C2312yes—
89C23×C8C2312yes—
90#532—44no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#125—22no—
94C2×#125—44no—
95C2×#125C2312yes—
96C2×#125C2312yes—
97C2×#124C2312yes—
98C2×#121—22no—
99#518—24no—
100#518—24no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4C2314yes—
104C2×#117—24no—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C4⋊C4—22no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—22no—
113C22×C22⋊C4—22no—
114C22×C42C2311yescharacteristic
115#428—44no—
116C2×#107C2314yes—
117C2×#106C2314yes—
118#417—28no—
119#416—28no—
120#416—44no—
121C2×#103C2312yes—
122C2×#103C2312yes—
123C2×#103C2314yes—
124C2×#103C2312yes—
125C2×#103C2312yes—
126C2×#102—44no—
127C2×#102C2312yes—
128C2×#102C2312yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2312yes—
133C2×#102C2312yes—
134#405—28no—
135C2×#97C2314yes—
136C2×C4×C8C2314yes—
137#400—24no—
138#400—24no—
139#394—24no—
140#394—24no—
141#391C2314yes—
142#391C2314yes—
143#390—21no—
144#390—21no—
145#390—22no—
146#390C2314yes—
147#390C2314yes—
148#390C2314yes—
149#390C2312yes—
150#390C2312yes—
151#389C2312yes—
152#389C2312yes—
153#389C2314yes—
154#388—24no—
155#388—24no—
156#388—24no—
157#388—24no—
158#388—24no—
159#388—24no—
160#388—24no—
161#388—24no—
162#386—24no—
163#386—24no—
164#385—22no—
165#385—22no—
166#384—21no—
167#382—21no—
168#381—24no—
169#380—22no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—24no—
174#379—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4D412yes—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—24no—
182C4×C22⋊C4D414yes—
183#376C2312yes—
184C43C2312yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2312yes—
190#336C2312yes—
191#336C2314yes—
192#336C2314yes—
193#336C2311yescharacteristic
194#336C2311yescharacteristic
195#336C2312yes—
196C25—21no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—24no—
201C2×#41—24no—
202C2×#41—24no—
203C22×Q8C2412yes—
204C22×D4—22no—
205C22×D4—21no—
206C22×D4—22no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4#4114yes—
211C22×D4C2412yes—
212C22×D4C2412yes—
213C23×C4—24no—
214C23×C4—21no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4C2×D412yes—
218C23×C4C2411yescharacteristic
219AGL(1,ℤ/8)—432no—
220C2×SD16—28no—
221C2×SD16—28no—
222C2×SD16—44no—
223C2×D8—28no—
224C2×D8—28no—
225C2×D8—44no—
226C2×M16—24no—
227C2×M16—24no—
228C2×M16#4118yes—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8—24no—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2414yes—
235C4⋊Q8—42no—
236#127—42no—
237#125—44no—
238#125—44no—
239#125—48no—
240#124—42no—
241#124—42no—
242#121—48no—
243C4×D4—216no—
244C4×D4—28no—
245C4×D4—28no—
246C4×D4—48no—
247#117—22no—
248#117—22no—
249#117—22no—
250#117—24no—
251#117—24no—
252#117—88no—
253#117#4114yes—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D411yescharacteristic
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—84no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C42—21no—
291C2×C42—24no—
292C2×C42—21no—
293C2×C42—24no—
294C2×C42C2×D412yes—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42C2412yes—
298C2×C42C2412yes—
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301#107—44no—
302#106—44no—
303#103—28no—
304#103—24no—
305#103—24no—
306#103—42no—
307#103—42no—
308#102—28no—
309#102—24no—
310#102—24no—
311#102—44no—
312#102—44no—
313#102—42no—
314#102—42no—
315#97—28no—
316C4×C8—28no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95C2×D414yes—
326#95C2×D414yes—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—42no—
331C24—42no—
332C24—82no—
333C24—82no—
334C24C22×D411yescharacteristic
335#41—48no—
336#41—48no—
337#41—48no—
338#41—48no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×D4—24no—
346C2×D4—216no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—48no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4C2×#4114yes—
366C22×C4—22no—
367C22×C4—216no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4C22×D412yes—
386C22×C4C22×D414yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4C2×#4114yes—
394C22×C4H(5,𝔽2)14yes—
395C22×C4C2511yescharacteristic
396SD16—416no—
397D8—416no—
398M16—216no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—48no—
408C2×C8C2×#4118yes—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—82no—
425C4⋊C4—84no—
426C4⋊C4C2×#4114yes—
427C22⋊C4—216no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—48no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—42no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—44no—
450C42—44no—
451C42—42no—
452C42—84no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—48no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—88no—
469C23—162no—
470C23—162no—
471C23—164no—
472C23#52512yes—
473C23#53212yes—
474C23#53512yes—
475C23C23×D411yescharacteristic
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480D4—28no—
481D4—48no—
482D4—42no—
483D4—42no—
484D4—44no—
485D4—42no—
486D4—42no—
487D4—88no—
488D4—88no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—84no—
517C2×C4—816no—
518C2×C4—88no—
519C2×C4—88no—
520C2×C4—162no—
521C2×C4#52512yes—
522C2×C4#53212yes—
523C2×C4#53512yes—
524C2×C4#55118yes—
525C2×C4#55518yes—
526C2×C4C22×#4111yescharacteristic
527C2×C4C22×#4112yes—
528C2×C4C22×#4111yescharacteristic
529C2×C4C2×H(5,𝔽2)12yes—
530C8—28no—
531C8—42no—
532C8—42no—
533C22—22no—
534C22—22no—
535C22—24no—
536C22—24no—
537C22—44no—
538C22—48no—
539C22—82no—
540C22—82no—
541C22—84no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—168no—
550C22#261014yes—
551C22C2×#52511yescharacteristic
552C22C2×#53211yescharacteristic
553C22C2×#53511yescharacteristic
554C22#358412yes—
555C22#358512yes—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—44no—
561C4—44no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—84no—
566C4#261014yes—
567C4C2×#55112yes—
568C4C2×#55512yes—
569C2—24no—
570C2—84no—
571C2—162no—
572C2C2×#261011yescharacteristic
573C2#6060112yes—
574C2#6072411yescharacteristic
575C2#6072511yescharacteristic
576C2#6081512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23428C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35341C2semidirect
2. #34255C2semidirect
3. #29568C2semidirect
4. #29564C2semidirect
5. #29562C2semidirect
6. #29556C2semidirect
7. #23553C2semidirect
8. #23504C2semidirect
9. C2×#2025C2semidirect
11. #21235C2semidirect
12. #21087C2semidirect
13. #21062C2semidirect
14. #20929C2semidirect
15. #20368C2semidirect
26. #2861C22semidirect
27. #2642C22semidirect
28. #2641C22semidirect
45. #2059C22semidirect
46. #2059C22semidirect
47. #2043C22semidirect
48. #2042C22semidirect
49. #2042C22semidirect
50. #2041C22semidirect
61. #1936C22semidirect
62. #1935C22semidirect
63. #1934C22semidirect
64. #1933C22semidirect
65. #1928C22semidirect
66. #1927C22semidirect
67. #1926C22semidirect
68. #1925C22semidirect
73. #1746C22semidirect
74. #1745C22semidirect
75. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#353412
#342552
#295682
#295642
#295622
#295562
#235532
#235042
C2×#20252
#234282
#212352
#210872
#210622
#209292
#203682