Finite Group Databasebeta

← #7470786

Subgroups — #7470786

Order 512.

1693 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45996C211yesmaximal, characteristic
2#39172C211yesmaximal, characteristic, complemented
3#32628C212yesmaximal, complemented
4#31769C214yesmaximal, complemented
5#31270C212yesmaximal, complemented
6#31004C214yesmaximal, complemented
7#30213C212yesmaximal, complemented
8#29561C212yesmaximal, complemented
9#28998C214yesmaximal, complemented
10#27273C214yesmaximal, complemented
11#25932C212yesmaximal, complemented
12#23979C212yesmaximal, complemented
13#23271C211yesmaximal, characteristic, complemented
14C2×H(3,𝔽4)—21no—
15C2×Q82C2211yescharacteristic, complemented
16C2×#556C2211yescharacteristic
17C2×#486C2212yes—
18C2×#460C2212yes—
19C2×#450C2212yescomplemented
20C2×#418C2212yescomplemented
21#2862C2211yescharacteristic, complemented
22#2858C2212yes—
23#2819—21no—
24#2733—21no—
25#2716—24no—
26#2679—21no—
27#2670—24no—
28#2640—21no—
29#2568—21no—
30#2563C2211yescharacteristic
31#2548C2212yes—
32#2546C2211yescharacteristic, complemented
33#2545—21no—
34#2491C2214yes—
35#2439C2214yescomplemented
36#2433C2214yes—
37#2421C2214yescomplemented
38#2408C2212yes—
39#2327—21no—
40C4×#124C2211yescharacteristic
41#2295C2211yescharacteristic, complemented
42#2076C2214yescomplemented
43#2073C2214yescomplemented
44#2073C2214yescomplemented
45#2067C2214yescomplemented
46#2057C2212yescomplemented
47#2041C2214yes—
48#2041C2212yes—
49#2033C2214yes—
50#2024C2212yescomplemented
51#2023C2214yescomplemented
52#2023C2214yescomplemented
53#2023C2214yescomplemented
54#2014C2212yes—
55#2000C2214yes—
56#1993C2214yescomplemented
57#1987C2212yescomplemented
58#1984C2212yescomplemented
59#1973C2214yescomplemented
60#1961C2214yes—
61#1936C2214yescomplemented
62#1934C2214yescomplemented
63#1925C2214yescomplemented
64#1924C2214yescomplemented
65#1918C2214yescomplemented
66#1895C2214yes—
67#1894C2212yescomplemented
68#1880C2214yescomplemented
69#1880C2214yescomplemented
70#1875C2214yes—
71#1870C2212yes—
72#1825C2212yescomplemented
73#1786C2214yescomplemented
74C4×#103C2212yescomplemented
75#1753C2214yescomplemented
76#1745C2212yescomplemented
77C23×D4C2312yes—
78C22×SD16C2314yes—
79C22×M16C2312yes—
80H(3,𝔽4)—48no—
81Q82—42no—
82Q82—42no—
83#556—42no—
84#556—42no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2312yes—
93C2×#127C2312yes—
94C2×#126—22no—
95C2×#124—22no—
96C2×#124—21no—
97C2×#124—24no—
98C2×#124—22no—
99C2×#124—22no—
100C2×#124D411yescharacteristic
101C2×C22⋊Q8C2312yes—
102C2×#121C2314yes—
103C2×#120—22no—
104C2×#120—24no—
105C2×#120—24no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C2×#117C2312yes—
113#486—44no—
114#460—44no—
115#450—44no—
116C2×#107C2314yes—
117C2×C4⋊C8C2312yes—
118#418—44no—
119C2×#103C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2312yes—
129C2×#97C2314yes—
130C2×C4×C8C2312yes—
131#400—24no—
132#398—22no—
133#397—24no—
134#396—24no—
135#394—22no—
136#394—22no—
137#394—22no—
138#394—24no—
139#393—22no—
140#393—24no—
141#393—24no—
142#392—22no—
143#392—24no—
144#392—24no—
145#391—21no—
146#391—21no—
147#391—21no—
148#391—22no—
149#391—24no—
150#391D411yescharacteristic
151#391C2312yes—
152#390—21no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390D411yescharacteristic
157#390C2314yes—
158#390C2314yes—
159#390C2314yes—
160#389C2312yes—
161#389C2314yes—
162#388—24no—
163#388—24no—
164#388D414yes—
165#387—22no—
166#387—24no—
167#387—24no—
168#382—21no—
169#382D411yescharacteristic
170#382C2314yes—
171#380C2314yes—
172#380C2312yes—
173#379—22no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4C2311yescharacteristic
176C4×C4⋊C4C2312yes—
177C4×C22⋊C4C2314yes—
178#376—21no—
179C43C2311yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2312yes—
187#336C2312yes—
188#336C2314yes—
189#336C2314yes—
190C25—22no—
191C22×Q8—21no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—21no—
195C22×Q8C2×D411yescharacteristic
196C22×Q8C2411yescharacteristic
197C22×D4—24no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—44no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—42no—
207C22×D4C2×D411yescharacteristic
208C22×D4C2414yes—
209C23×C4C2412yes—
210C2×SD16—28no—
211C2×SD16—44no—
212C2×SD16—44no—
213C2×M16—44no—
214C2×M16—42no—
215C22×C8C2414yes—
216C22×C8C2414yes—
217C4⋊Q8—24no—
218C4⋊Q8—41no—
219C4⋊Q8—41no—
220C4⋊Q8—42no—
221C4⋊Q8—41no—
222C4⋊Q8—41no—
223C4⋊Q8—44no—
224C4⋊Q8—44no—
225C4⋊Q8—48no—
226C4⋊Q8—44no—
227#127—44no—
228#127—42no—
229#127—42no—
230#126—48no—
231#124—24no—
232#124—44no—
233#124—48no—
234#124—416no—
235#124—48no—
236#124—84no—
237C22⋊Q8—24no—
238C22⋊Q8—42no—
239#121—44no—
240#121—44no—
241#120—416no—
242#120—48no—
243#120—88no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—24no—
247C4×Q8—41no—
248C4×Q8—41no—
249C4×Q8—42no—
250C4×Q8—44no—
251C4×Q8—44no—
252C4×D4—41no—
253C4×D4—41no—
254#117—24no—
255#117—42no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2411yescharacteristic
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
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⋊C4C2×D414yes—
285C2×C22⋊C4C2×D412yes—
286C2×C22⋊C4C2×D414yes—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D412yes—
294C2×C42C2×D411yescharacteristic
295C2×C42C2411yescharacteristic
296C2×C42C2412yes—
297C2×C42C2412yes—
298#107—44no—
299C4⋊C8—42no—
300#103—44no—
301#102—28no—
302#102—44no—
303#102—44no—
304#102—44no—
305#102—44no—
306#102—44no—
307#102—44no—
308C22⋊C8—44no—
309C22⋊C8—42no—
310#97—44no—
311C4×C8—42no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—21no—
316#95—22no—
317#95—24no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95C2×D414yes—
323#95C2×D414yes—
324C24—24no—
325C24—44no—
326C24—82no—
327C24—84no—
328C24—84no—
329C24—82no—
330C24C22×D412yes—
331C2×Q8—21no—
332C2×Q8—24no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—44no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—44no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—82no—
342C2×Q8—82no—
343C2×Q8—82no—
344C2×Q8—84no—
345C2×Q8AGL(1,ℤ/8)14yes—
346C2×Q8H(5,𝔽2)12yes—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—44no—
361C2×D4—82no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—84no—
372C2×D4—88no—
373C2×D4—84no—
374C2×D4—84no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C2511yescharacteristic
393SD16—88no—
394M16—84no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C4⋊C4—24no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—48no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4H(5,𝔽2)12yes—
422C22⋊C4—28no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—44no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C22⋊C4—84no—
435C22⋊C4—84no—
436C42—24no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—21no—
442C42—24no—
443C42—41no—
444C42—42no—
445C42—44no—
446C42—44no—
447C42—44no—
448C42—42no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—22no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—88no—
462C23—168no—
463C23—164no—
464C23—168no—
465C23—164no—
466C23C23×D411yescharacteristic
467Q8—24no—
468Q8—44no—
469Q8—42no—
470Q8—84no—
471Q8—164no—
472Q8—164no—
473Q8—162no—
474D4—44no—
475D4—44no—
476D4—44no—
477D4—44no—
478D4—82no—
479D4—88no—
480D4—164no—
481D4—164no—
482D4—164no—
483D4—164no—
484D4—168no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4C2×AGL(1,ℤ/8)12yes—
520C2×C4C2×AGL(1,ℤ/8)14yes—
521C2×C4C2×H(5,𝔽2)11yescharacteristic
522C2×C4C2×H(5,𝔽2)11yescharacteristic
523C2×C4#58514yes—
524C8—44no—
525C8—44no—
526C22—42no—
527C22—44no—
528C22—44no—
529C22—42no—
530C22—42no—
531C22—84no—
532C22—84no—
533C22—84no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—1616no—
539C22#348411yescharacteristic
540C22#350012yes—
541C22C22×AGL(1,ℤ/8)11yescharacteristic
542C22#358512yes—
543C22#358611yescharacteristic
544C4—22no—
545C4—24no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—41no—
552C4—42no—
553C4—82no—
554C4—84no—
555C4—82no—
556C4—82no—
557C4#354012yes—
558C4#354512yes—
559C2—82no—
560C2—164no—
561C2#5174911yescharacteristic
562C2#6157211yescharacteristic
563C2#6167311yescharacteristic
564C2#6256812yes—
565C2#6258812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#45996C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #39172C2semidirect
3. #32628C2semidirect
4. #31769C2semidirect
5. #31270C2semidirect
6. #31004C2semidirect
7. #30213C2semidirect
8. #29561C2semidirect
9. #28998C2semidirect
10. #27273C2semidirect
11. #25932C2semidirect
12. #23979C2semidirect
13. #23271C2semidirect
15. C2×Q82C22semidirect
19. C2×#450C22semidirect
20. C2×#418C22semidirect
21. #2862C22semidirect
32. #2546C22semidirect
35. #2439C22semidirect
37. #2421C22semidirect
41. #2295C22semidirect
42. #2076C22semidirect
43. #2073C22semidirect
44. #2073C22semidirect
45. #2067C22semidirect
46. #2057C22semidirect
50. #2024C22semidirect
51. #2023C22semidirect
52. #2023C22semidirect
53. #2023C22semidirect
56. #1993C22semidirect
57. #1987C22semidirect
58. #1984C22semidirect
59. #1973C22semidirect
61. #1936C22semidirect
62. #1934C22semidirect
63. #1925C22semidirect
64. #1924C22semidirect
65. #1918C22semidirect
67. #1894C22semidirect
68. #1880C22semidirect
69. #1880C22semidirect
72. #1825C22semidirect
73. #1786C22semidirect
74. C4×#103C22semidirect
75. #1753C22semidirect
76. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#459962
#391722
#326282
#317692
#312702
#310042
#302132
#295612
#289982
#272732
#259322
#239792
#232712