Finite Group Databasebeta

← #6461579

Subgroups — #6461579

Order 512.

1958 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic, complemented
2#34208C211yesmaximal, characteristic, complemented
3#29508C214yesmaximal, complemented
4#29505C214yesmaximal, complemented
5#29503C214yesmaximal, complemented
6#29497C214yesmaximal, complemented
7#23659C211yesmaximal, characteristic, complemented
8C2×#2073C211yesmaximal, characteristic, complemented
9#23504C211yesmaximal, characteristic, complemented
10#23428C211yesmaximal, characteristic
11#21235C212yesmaximal, complemented
12#21087C212yesmaximal, complemented
13#20932C212yesmaximal, complemented
14#20878C212yesmaximal, complemented
15#20368C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C2212yes—
17C2×#532C2211yescharacteristic
18C2×#520C2212yes—
19C2×#428C2212yes—
20C2×#417C2212yes—
21C2×#416C2212yes—
22C2×#416C2212yes—
23C22×#102C2212yes—
24C2×#405C2212yes—
25#2860C2214yes—
26#2644C2214yes—
27#2513—22no—
28#2512—22no—
29#2480—24no—
30#2364—24no—
31#2352—22no—
32#2352—22no—
33#2328C2212yescomplemented
34#2327C2212yescomplemented
35#2317C2211yescharacteristic
36#2316C2211yescharacteristic
37#2311C2212yes—
38#2309C2211yescharacteristic
39#2309C2211yescharacteristic
40C4×#124C2214yescomplemented
41#2296C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C4×#117C2211yescharacteristic
45#2085C2214yescomplemented
46#2085C2214yescomplemented
47#2073C2212yes—
48#2073C2212yes—
49#2073C2214yes—
50#2043C2212yescomplemented
51#2042C2212yescomplemented
52#2042C2212yescomplemented
53#2041C2212yescomplemented
54#2013C2212yes—
55#2012C2214yes—
56#2011C2212yes—
57#1973C2214yes—
58#1972C2214yes—
59#1970C2214yes—
60#1968C2214yes—
61#1936C2214yescomplemented
62#1935C2214yescomplemented
63#1934C2214yescomplemented
64#1933C2214yescomplemented
65#1880C2214yes—
66#1880C2214yes—
67#1879C2214yes—
68#1879C2214yes—
69#1865C2214yescomplemented
70#1864C2214yescomplemented
71#1863C2214yescomplemented
72#1862C2214yescomplemented
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—28no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#126—24no—
94C2×#124C2312yes—
95C2×#121—22no—
96#520—44no—
97#520—44no—
98C2×C4×Q8C2314yes—
99C2×C4×D4—24no—
100C2×C4×D4—22no—
101C2×C4×D4C2314yes—
102C2×#117—22no—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—22no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—22no—
111C22×C42C2311yescharacteristic
112#428—44no—
113C2×#107C2314yes—
114C2×#106C2314yes—
115#417—28no—
116#416—28no—
117#416—44no—
118C2×#103C2314yes—
119C2×#103C2312yes—
120C2×#103C2312yes—
121C2×#103C2312yes—
122C2×#103C2312yes—
123C2×#102—44no—
124C2×#102C2314yes—
125C2×#102C2312yes—
126C2×#102C2312yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2312yes—
130C2×#102C2312yes—
131#405—28no—
132C2×#97C2314yes—
133C2×C4×C8C2314yes—
134#397—24no—
135#397—24no—
136#397—24no—
137#396—24no—
138#391C2314yes—
139#390—21no—
140#390—22no—
141#390—21no—
142#390—22no—
143#390C2314yes—
144#390C2314yes—
145#390C2314yes—
146#390C2312yes—
147#390C2312yes—
148#389—22no—
149#389C2314yes—
150#389C2314yes—
151#389C2312yes—
152#389C2312yes—
153#388—24no—
154#388—24no—
155#388—24no—
156#388—24no—
157#388—24no—
158#387—24no—
159#387—24no—
160#386—24no—
161#386—24no—
162#386—24no—
163#385—22no—
164#385—22no—
165#384—21no—
166#382—21no—
167#382—22no—
168#382C2312yes—
169#382C2312yes—
170#380—22no—
171#380—22no—
172#380—22no—
173#379—22no—
174#379—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4D412yes—
178C4×C22⋊C4—24no—
179C4×C22⋊C4—24no—
180C4×C22⋊C4D414yes—
181#376C2312yes—
182C2×#95—22no—
183C43C2312yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2311yescharacteristic
189#336C2311yescharacteristic
190#336C2312yes—
191#336C2312yes—
192#336C2312yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C2×#41—24no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—24no—
201C2×#41#4118yes—
202C22×Q8C2412yes—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—21no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C2412yes—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—21no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4C2×D412yes—
217C23×C4C2411yescharacteristic
218AGL(1,ℤ/8)—432no—
219C2×SD16—28no—
220C2×SD16—28no—
221C2×SD16—44no—
222C2×D8—28no—
223C2×D8—28no—
224C2×D8—44no—
225C2×M16—24no—
226C2×M16—24no—
227C2×M16#4118yes—
228C22×C8—24no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8C2412yes—
232C22×C8C2412yes—
233C22×C8C2414yes—
234C4⋊Q8—24no—
235#127—24no—
236#126—88no—
237#124—24no—
238#124—24no—
239#121—216no—
240C4×Q8—44no—
241C4×Q8—44no—
242C4×D4—216no—
243C4×D4—44no—
244C4×D4—44no—
245C4×D4—416no—
246#117—24no—
247#117—22no—
248#117—22no—
249#117—24no—
250#117—22no—
251#117—22no—
252#117—48no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—84no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42C2×D412yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42C2412yes—
296C2×C42C2412yes—
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299#107—44no—
300#106—44no—
301#103—24no—
302#103—24no—
303#103—28no—
304#103—42no—
305#103—42no—
306#102—24no—
307#102—24no—
308#102—28no—
309#102—44no—
310#102—44no—
311#102—42no—
312#102—42no—
313#97—28no—
314C4×C8—28no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95C2×D414yes—
324#95C2×D414yes—
325C24—22no—
326C24—22no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—82no—
331C24—82no—
332C24C22×D411yescharacteristic
333#41—216no—
334#41—216no—
335#41—48no—
336#41—48no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8C2×#4114yes—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—82no—
362C2×D4—82no—
363C2×D4C2×#4114yes—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—28no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—48no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D414yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C2×#4114yes—
390C22×C4C2×#4114yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C4C2511yescharacteristic
394SD16—416no—
395D8—416no—
396M16—216no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—48no—
406C2×C8C2×#4118yes—
407C4⋊C4—28no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—82no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C42—24no—
441C42—24no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—44no—
447C42—44no—
448C42—48no—
449C42—42no—
450C42—42no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—88no—
466C23—84no—
467C23—162no—
468C23—162no—
469C23—164no—
470C23#52512yes—
471C23#53212yes—
472C23#53512yes—
473C23C23×D411yescharacteristic
474Q8—24no—
475Q8—24no—
476Q8—42no—
477Q8—42no—
478D4—24no—
479D4—24no—
480D4—416no—
481D4—42no—
482D4—44no—
483D4—42no—
484D4—44no—
485D4—88no—
486D4—84no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—416no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—84no—
516C2×C4—168no—
517C2×C4—162no—
518C2×C4—164no—
519C2×C4#52512yes—
520C2×C4#53212yes—
521C2×C4#53512yes—
522C2×C4#55118yes—
523C2×C4#55318yes—
524C2×C4C22×#4112yes—
525C2×C4C22×#4112yes—
526C2×C4C2×H(5,𝔽2)11yescharacteristic
527C2×C4C2×H(5,𝔽2)11yescharacteristic
528C8—28no—
529C8—42no—
530C8—42no—
531C22—22no—
532C22—22no—
533C22—24no—
534C22—24no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—84no—
546C22—84no—
547C22—168no—
548C22#260314yes—
549C22C2×#52511yescharacteristic
550C22C2×#53211yescharacteristic
551C22C2×#53511yescharacteristic
552C22#358412yes—
553C22#358512yes—
554C4—44no—
555C4—42no—
556C4—42no—
557C4—44no—
558C4—42no—
559C4—42no—
560C4—82no—
561C4—82no—
562C4—84no—
563C4—82no—
564C4#260314yes—
565C4C2×#55112yes—
566C4C2×#55312yes—
567C2—24no—
568C2—84no—
569C2—162no—
570C2C2×#260311yescharacteristic
571C2#6058712yes—
572C2#6072411yescharacteristic
573C2#6072511yescharacteristic
574C2#6083412yes—
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. #35266C2semidirect
2. #34208C2semidirect
3. #29508C2semidirect
4. #29505C2semidirect
5. #29503C2semidirect
6. #29497C2semidirect
7. #23659C2semidirect
8. C2×#2073C2semidirect
9. #23504C2semidirect
11. #21235C2semidirect
12. #21087C2semidirect
13. #20932C2semidirect
14. #20878C2semidirect
15. #20368C2semidirect
33. #2328C22semidirect
34. #2327C22semidirect
40. C4×#124C22semidirect
45. #2085C22semidirect
46. #2085C22semidirect
50. #2043C22semidirect
51. #2042C22semidirect
52. #2042C22semidirect
53. #2041C22semidirect
61. #1936C22semidirect
62. #1935C22semidirect
63. #1934C22semidirect
64. #1933C22semidirect
69. #1865C22semidirect
70. #1864C22semidirect
71. #1863C22semidirect
72. #1862C22semidirect
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
#352662
#342082
#295082
#295052
#295032
#294972
#236592
C2×#20732
#235042
#234282
#212352
#210872
#209322
#208782
#203682