Finite Group Databasebeta

← #7598756

Subgroups — #7598756

Order 512.

2151 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45375C211yesmaximal, characteristic, complemented
2#45355C211yesmaximal, characteristic
3#31681C212yesmaximal, complemented
4#31680C212yesmaximal, complemented
5#28292C212yesmaximal, complemented
6#28097C212yesmaximal, complemented
7#28094C212yesmaximal, complemented
8#25891C214yesmaximal, complemented
9#25342C214yesmaximal, complemented
10#24977C214yesmaximal, complemented
11D4×#102C214yesmaximal, complemented
12#21683C211yesmaximal, characteristic, complemented
13#21652C212yesmaximal, complemented
14C2×D42C2214yes—
15C2×#490C2212yes—
16C2×#478C2214yescomplemented
17C2×C8×D4C2214yescomplemented
18C2×C8×D4C2211yescharacteristic, complemented
19C2×C8×D4C2211yescharacteristic, complemented
20C22×#102C2212yescomplemented
21C22×#102C2212yescomplemented
22#2807—22no—
23#2771—22no—
24#2668—22no—
25#2668—22no—
26#2600—22no—
27#2562C2214yescomplemented
28#2535C2214yes—
29#2487C2212yescomplemented
30#2486C2212yescomplemented
31#2455—22no—
32#2408C2212yes—
33#2408C2212yes—
34#2403—22no—
35#2403—22no—
36#2397C2211yescharacteristic
37D4×C4⋊C4C2214yescomplemented
38#2330C2212yescomplemented
39#2328C2212yes—
40#2077C2214yescomplemented
41#2076C2214yescomplemented
42#2067C2214yescomplemented
43#2065C2214yescomplemented
44#2034C2214yescomplemented
45#2034C2214yescomplemented
46#2033C2214yescomplemented
47#2033C2214yescomplemented
48#1991C2212yescomplemented
49#1982C2214yes—
50#1957C2212yes—
51#1955C2214yescomplemented
52#1955C2211yescharacteristic, complemented
53#1955C2211yescharacteristic, complemented
54#1949C2214yescomplemented
55#1936C2214yescomplemented
56#1934C2214yescomplemented
57#1893C2214yescomplemented
58#1893C2214yescomplemented
59#1874C2214yescomplemented
60#1874C2214yescomplemented
61#1843C2211yescharacteristic, complemented
62#1836C2212yescomplemented
63#1824C2212yescomplemented
64#1823C2212yescomplemented
65#1808C2212yescomplemented
66#1807C2212yescomplemented
67#1802C2214yescomplemented
68#1786C2212yescomplemented
69#1786C2212yescomplemented
70#1786C2214yescomplemented
71#1786C2214yescomplemented
72#1762C2214yescomplemented
73C4×#102C2214yescomplemented
74#1726C2211yescharacteristic, complemented
75C23×D4—24no—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C23×C8C2312yes—
79C23×C8C2312yes—
80D42—216no—
81D42—48no—
82C2×#127C2312yes—
83C2×#125C2312yes—
84C2×#123—22no—
85C2×#121—24no—
86C2×#121C2314yes—
87C2×#121C2314yes—
88C2×#120—22no—
89C2×#120—22no—
90C2×#120—24no—
91C2×#120—24no—
92C2×C4×D4—24no—
93C2×C4×D4C2314yes—
94C2×C4×D4C2314yes—
95C2×C4×D4C2311yescharacteristic
96C2×C4×D4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99#490—28no—
100#478—48no—
101C8×D4—48no—
102C8×D4—42no—
103C8×D4—42no—
104C2×#107C2314yes—
105C2×#106C2314yes—
106C2×C4⋊C8C2314yes—
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×#102—44no—
110C2×#102—44no—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×C22⋊C8C2314yes—
120C2×C22⋊C8C2314yes—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2312yes—
123C2×#97C2312yes—
124C2×#97C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2312yes—
127#400—24no—
128#400—22no—
129#399—22no—
130#399D412yes—
131#399D412yes—
132#398—22no—
133#397—24no—
134#397—22no—
135#394—24no—
136#394—22no—
137#394—22no—
138#392—24no—
139#392—22no—
140#392—22no—
141#392D412yes—
142#392D412yes—
143#390C2314yes—
144#389—24no—
145#389—24no—
146#389C2314yes—
147#388—24no—
148#388—24no—
149#387—24no—
150#386—22no—
151#386—24no—
152#386—24no—
153#386—24no—
154#386—24no—
155#385—24no—
156#385—24no—
157#385—22no—
158#384C2312yes—
159#382C2312yes—
160#381—22no—
161#380C2312yes—
162#380C2312yes—
163#380C2314yes—
164#380C2314yes—
165#380C2312yes—
166#380C2312yes—
167#379—22no—
168#379—22no—
169C4×C4⋊C4C2314yes—
170#376C2311yescharacteristic
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177C25—22no—
178C25—22no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—416no—
193C22×D4—88no—
194C22×D4—88no—
195C22×D4C2×D414yes—
196C22×D4C2×D411yescharacteristic
197C22×D4C2×D411yescharacteristic
198C22×D4C2414yes—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C22×C8—24no—
205C22×C8—24no—
206C22×C8—44no—
207C22×C8—44no—
208C22×C8C2412yes—
209C22×C8C2412yes—
210C22×C8C2411yescharacteristic
211C22×C8C2412yes—
212C22×C8C2411yescharacteristic
213#127—24no—
214#127#4118yes—
215#125—24no—
216#125—24no—
217#123—84no—
218#121—28no—
219#121—28no—
220#121—44no—
221#121—44no—
222#121—416no—
223#120—416no—
224#120—416no—
225#120—84no—
226#120—84no—
227C4×D4—28no—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—88no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2414yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4C2×D414yes—
263C2×C22⋊C4C2×D414yes—
264C2×C22⋊C4C2×D412yes—
265C2×C22⋊C4C2×D412yes—
266C2×C42—24no—
267C2×C42C2411yescharacteristic
268C2×C42C2412yes—
269#107—44no—
270#106—44no—
271C4⋊C8—28no—
272C4⋊C8—41no—
273C4⋊C8—41no—
274#102—28no—
275#102—28no—
276#102—28no—
277#102—44no—
278#102—44no—
279#102—44no—
280#102—44no—
281#102—44no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284C22⋊C8—44no—
285C22⋊C8—44no—
286#97—24no—
287#97—41no—
288C4×C8—22no—
289C4×C8—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—24no—
298#95—24no—
299#95C2×D412yes—
300#95C2×D412yes—
301C24—24no—
302C24—48no—
303C24—48no—
304C24—44no—
305C24—44no—
306C24—82no—
307C24—82no—
308C24—84no—
309C24—84no—
310C24—82no—
311C24—82no—
312C24C22×D412yes—
313C24C22×D412yes—
314C2×D4—28no—
315C2×D4—28no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—28no—
319C2×D4—216no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—82no—
336C2×D4—82no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4—88no—
340C2×D4—88no—
341C2×D4—84no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—864no—
346C2×D4C2×#4118yes—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—44no—
359C22×C4—48no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4—84no—
365C22×C4—82no—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D414yes—
371C22×C4C2511yescharacteristic
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—84no—
383C2×C8—84no—
384C2×C8H(5,𝔽2)12yes—
385C4⋊C4—28no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—48no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—48no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C22⋊C4—84no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C42—21no—
415C42—21no—
416C42—84no—
417C42C2×#4114yes—
418C42H(5,𝔽2)14yes—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—22no—
426C23—24no—
427C23—28no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—816no—
435C23—816no—
436C23—88no—
437C23—88no—
438C23—164no—
439C23—164no—
440C23—168no—
441C23—168no—
442C23—164no—
443C23—164no—
444C23#57614yes—
445C23#57714yes—
446C23C23×D411yescharacteristic
447D4—232no—
448D4—28no—
449D4—28no—
450D4—44no—
451D4—44no—
452D4—44no—
453D4—44no—
454D4—816no—
455D4—88no—
456D4—88no—
457D4—816no—
458D4—816no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—816no—
485C2×C4—164no—
486C2×C4—164no—
487C2×C4#54618yes—
488C2×C4C22×#4112yes—
489C2×C4C2×H(5,𝔽2)11yescharacteristic
490C2×C4C2×H(5,𝔽2)12yes—
491C2×C4#58511yescharacteristic
492C8—22no—
493C8—41no—
494C8—42no—
495C8—42no—
496C8—42no—
497C22—24no—
498C22—24no—
499C22—28no—
500C22—28no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22—48no—
508C22—48no—
509C22—48no—
510C22—84no—
511C22—84no—
512C22—84no—
513C22—84no—
514C22—1616no—
515C22—1616no—
516C22C2×#54612yes—
517C22#349611yescharacteristic
518C22C2×#57612yes—
519C22C2×#57712yes—
520C4—28no—
521C4—42no—
522C4—42no—
523C4—41no—
524C4—41no—
525C4—84no—
526C4—84no—
527C4—84no—
528C4#348111yescharacteristic
529C4#351412yes—
530C4#354311yescharacteristic
531C2—44no—
532C2—44no—
533C2—88no—
534C2#5942011yescharacteristic
535C2#6116714yes—
536C2#6226811yescharacteristic
537C2#6227011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#45355C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2397 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #45375C2semidirect
3. #31681C2semidirect
4. #31680C2semidirect
5. #28292C2semidirect
6. #28097C2semidirect
7. #28094C2semidirect
8. #25891C2semidirect
9. #25342C2semidirect
10. #24977C2semidirect
11. D4×#102C2semidirect
12. #21683C2semidirect
13. #21652C2semidirect
16. C2×#478C22semidirect
17. C2×C8×D4C22semidirect
18. C2×C8×D4C22semidirect
19. C2×C8×D4C22semidirect
20. C22×#102C22semidirect
21. C22×#102C22semidirect
27. #2562C22semidirect
29. #2487C22semidirect
30. #2486C22semidirect
37. D4×C4⋊C4C22semidirect
38. #2330C22semidirect
40. #2077C22semidirect
41. #2076C22semidirect
42. #2067C22semidirect
43. #2065C22semidirect
44. #2034C22semidirect
45. #2034C22semidirect
46. #2033C22semidirect
47. #2033C22semidirect
48. #1991C22semidirect
51. #1955C22semidirect
52. #1955C22semidirect
53. #1955C22semidirect
54. #1949C22semidirect
55. #1936C22semidirect
56. #1934C22semidirect
57. #1893C22semidirect
58. #1893C22semidirect
59. #1874C22semidirect
60. #1874C22semidirect
61. #1843C22semidirect
62. #1836C22semidirect
63. #1824C22semidirect
64. #1823C22semidirect
65. #1808C22semidirect
66. #1807C22semidirect
67. #1802C22semidirect
68. #1786C22semidirect
69. #1786C22semidirect
70. #1786C22semidirect
71. #1786C22semidirect
72. #1762C22semidirect
73. C4×#102C22semidirect
74. #1726C22semidirect

Maximal subgroups

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

subgroupindex
#453752
#453552
#316812
#316802
#282922
#280972
#280942
#258912
#253422
#249772
D4×#1022
#216832
#216522